/
Author: Rautian S.G. Shalagin A.M.
Tags: physics lasers spectroscopy kinetics
ISBN: 0-444-88357-8
Year: 1991
Text
Non-linear Spectroscopy
S. G. Rautian and A. M. Shalagin
Kinetic Problems of
Non-linear Spectroscopy
S.G. Rautian and A.M. Shalagin
Institute of Automation and Electrometry
Siberian Branch of the USSR Academy of Sciences
Novosibirsk, U.S.S.R
1991
NORTH-HOLLAND
AMSTERDAM NEW YORK OXFORD TOKYO
© Elsevier Science Publishers B.V., 1991
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Library of Congress Cataloging-in-Publication Data
Rautian, Sergei Glebovich.
Kinetic problems of non-linear spectroscopy / S.G. Rautlan and
A.M. Shalagin.
p. ca.
Includes bibliographical references and index.
ISBN 0-444-88357-8
1. Laser spectroscopy. 2. Nonlinear optics. I. Shalagin,
Anatolil Mikhailovich. II. Title.
QC454.L3R38 1990
535.8'4—dc20 90-42874
CIP
This volume is printed on acid-free paper.
Printed in The Netherlands
Preface
The advent of the non-linear laser spectroscopy of gases dates back to the
early 1960s. Over about the three decades of its existence a host of new
and quite peculiar phenomena have been discovered and studied, and
their simple and transparent physical interpretation has been worked out.
There is no doubt that a new, independent field of gas spectroscopy has
been formed with its specific problems, physical images, laws and
investigative methods. Therefore we hope that our attempt to summarize
in the present book the general results of non-linear gas spectroscopy is
fully justified and that not many essential topics have been left out.
Two aspects can be considered to be critical. On the one hand, these
are dynamic processes due to the interaction of atoms and molecules with
a coherent electromagnetic field. On the other hand, there are stochastic
processes caused by the interaction of radiating particles with the medium
and leading to the relaxation of numerous varying non-equilibrium states
produced by dynamic processes. The close, often bizarre, tangle of
dynamic and stochastic kinetics gives the specific colouring which is
inherent in modem non-linear spectroscopy and kinetic problems turned
out to be most important for it.
In the course of their development, non-linear spectroscopy and
physical kinetics in general and, particularly, the kinetic theory of gases
were brought closer and closer together. The most important result of this
trend was the discovery of light-induced drift and a new extensive field of
gas kinetics in the field of laser radiation. Spectroscopy and physical
kinetics, not long ago having been independent branches of physics, now
can be said to benefit mutually from their interaction.
Our book is devoted to the theoretical non-linear spectroscopy of
atomic and molecular gases with special emphasis being placed on the
analysis of dynamic and stochastic kinetic processes. The entire presenta-
tion is based on the quantum kinetic equation for the one-particle density
matrix with a generalized collision integral of the Boltzmann type.
Consistent sequential analysis of spectroscopic, optical and gas kinetics
phenomena is universally believed to be possible only on the basis of this
equation. The apparatus of the quantum kinetic equation has been
vi
Preface
significantly developed over the last two decades in connection with
problems of non-linear spectroscopy and gas kinetics in the field of laser
radiation. The aforegoing justifies the comparatively large size of chapter
2, which is devoted to the substantiation and general analysis of the
kinetic equation and its relative complexity.
Chapter 3 deals mostly with dynamic processes, and relaxation is
treated under the most simple assumptions. By contrast, chapters 4 and 5
discuss phenomena due to different relaxation types of the translational,
rotational, vibrational and electronic degrees of freedom, and the dynamic
interaction with the coherent field is only slightly varied. Chapters 6 and 7
deal with light-induced drift and related phenomena.
The choice of material was determined by our wish to dwell mostly on
general laws of universal importance. Specific results and numeric values
of the parameters characterizing different objects were necessarily rele-
gated to the background and served mostly as illustrations. The research
interests of the authors also influenced this choice.
The references are by no means exhaustive. We limited ourselves to
monographs and surveys as well as to the original articles. We also bore in
mind that the Western reading public is little acquainted with Soviet
studies.
The present monograph is related to the book Nelinejnye Resonansy v
Spektrakh Atomov i Molecul (Nauka, Siberian Branch, Novosibirsk,
1979) (Nonlinear Resonances in Spectra of Atoms and Molecules) written
together with G.I. Smirnov. The structure of chapters 2-5 has been
retained, but they were substantially rewritten. The most radical revision
has been carried out in chapters 1 and 2, chapter 2 having been extended
at the suggestion of North-Holland. New chapters 6 and 7 have been added.
The book is addressed to specialists familiar with university courses on
quantum mechanics and statistical physics. The brilliant books on theoret-
ical physics by L.D. Landau and E.M. Lifshitz were considered standard.
We hope this book will prove useful for a wide range of readers:
engineers and research workers, students and postgraduates, and all those
interested in optics, spectroscopy, quantum electronics and physical
kinetics.
We are grateful to Professor L.J.F. Hermans and North-Holland
who suggested the publication of this book. We should like to express our
deep gratitude to I.V. Pertsovskaya for the translation of the book, and to
the editor D.M. Norris whose clear understanding and thoroughness helped
us greatly to improve the text.
Contents
Preface v
Contents vii
List of Symbols ix
1. Introduction 1
References 9
2. Quantum Kinetic Equation for the Density Matrix 11
2.1. Quantum system interaction with an external electromagnetic field: description by
means of probability amphtudes 11
2.2. Quantum kinetic equation 17
2.3. Radiation relaxation 27
2.4. Collision integral 31
2.5. Frequencies and kernels of the collision integral 61
v 2.6. Transport frequency of collisions 90
2.7. Macroscopic equations of gas kinetics involving interaction with laser radiation 95
References 106
3. Resonance Radiation Processes 109
3.1. Spectral line broadening in the absence of non-linear phenomena 109
3.2. Interaction of atoms with a strong resonance field 136
3.3. The method of a probe field 154
3.4. Spontaneous emission of atoms interacting with the external electromagnetic field 186
References 193
4. Bennett Structure for Systems with Large Doppler Broadening 195
4.1. Velocity distribution of atoms under the interaction with a plane monochromatic wave 195
4.2. Non-linear resonances due to Bennett holes 207
4.3. The effect of collisions on Bennett holes and non-linear resonances 218
References 260
5. Probe Field Spectroscopy under Large Doppler Broadening 261
5.1. Non-linear resonances in three-level systems 261
5.2. The probe field method in two-level systems 272
5.3. Spectrum of spontaneous emission 287
5.4. Polarization phenomena 291
5.5. Recoil effect 311
References 317
6. Light-induced Drift of Gases 319
6.1. Qualitative description of the light-induced drift 319
vii
6.2. General laws of light-induced drift 321
6.3. Light-induced drift as the realization of Maxwell’s demon 329
6.4. Drift velocity dependence of medium and radiation characteristics 333
6.5. Light-induced drift in the field of non-monochromatic radiation: highest attainable drift
velocity 339
6.6. Spatial distribution of the concentration 344
6.7. Optical pistons and “solitons” 349
6.8. Light-induced drift of molecules and multilevel atoms 355
6.9. Light-induced drift in physics, astrophysics and technology 373
References 377
7. Light-induced Gas Kinetics 379
7.1. Modifications of the light-induced drift effect 379
7.2. Light pressure and light-induced drift 384
7.3. Light-induced pulling and pushing 389
7.4. Anisotropy of the pressure tensor: “cooling” and “heating” of gas components 393
7.5. Kinetic effects in a one-component gas 396
References 407
Appendix I. Some Properties of Clebsch-Gordan Coefficients and of 6/ Symbols 409
Appendix II. Wigner D matrices 411
Problems 413
Subject Index 437
List of symbols
a: set of quantum numbers of internal degrees of freedom with the
exception of JM
a(t): column of amplitudes
ay(t): amplitude of a stationary state
ajK, ввк, арк‘ polarization coefficients
A(y | vj, Лтл(г | vO, A(aa'v | Ar(v | v,), Ayir(v | v,): kernel of
the collision integral
AK_s(v “ УЦ): Keilson-Storer kernel
Amn: Einstein coefficient for the spontaneous transition m—
kernel of the in-term R(2)
Лт„(к): coefficient of R(r2)
Fourier transform of the correlation function
BjK, BK: spectral polarization functions of rank к
c: light velocity in vacuum
CaP(y): component of the tensor of the second moment of a kernel
C = (-1)", Cj = (-1)"1: phase factors
C(v | vj: statistical multiplier of a kernel
Caft: component of the tensor of the second moment of the collision
integral
Й: operator of the dipole moment
dif: matrix element of Й
dnn -. reduced matrix element of d
D, Df. diffusion coefficients
D“ft: component of the tensor of the fourth moment of the distribution
function
DMMl(apY): Wigner matrix
e, ea: unit vector along the given direction
E: strength of the electric field
Ef. energy of the stationary state j
Ex, Ey> Ег: Cartesian components of E
E„: circular component of E
’S: amplitude of E
f(afiu | /(и | «J: scattering amplitudes
ix
X
List of Symbols
fmi(y | v'): Green’s function
fm„: oscillator strength for the transition m—
F: external force acting on an atom
F, Fi, Fiji density of the internal friction force
F(yrt | v'), F(xkQ | v'), F(ykQ | v'), Ffyrcpt | v'): Green’s function
F^v | v'): diagonal Green’s function
F^fy I v'): regular part of Ря(у | v')
F(y | v'): non-diagonal Green’s function
F(y | v'): regular part of F(y | v')
F(a, P; y; z): hypergeometric function
g„ g(Jj): statistical weight of the state i
G, G^, G„, Gpa, Gnn’> G(nM, n'M'), G(M), G^, G4: constant of
interaction of an atom and a field
G'tjfKq | r^i), G'ij(Kq | Kiqt): matrix element of the interaction of an atom
and a field in the icq representation
ft: Planck constant 4
Й: Hamiltonian
Й„: Hamiltonian of atomic internal degrees of freedom
Йь: Hamiltonian of particle b
Й': Hamiltonian of interaction of an atom with an electromagnetic field
Й'ц: Hamiltonian of interaction of an atom with the (i mode of the field
Hn(z): Struve’s function of order n
i: index of the gas components
i: state index
i: imaginary unit
I (кд): polarization tensor of the field
IfAxq)'. polarization tensor of the probe field
Z(co), Z(Z2): normalized profile of a spectral line
I(Z2): spectral density of radiation
/: index of the gas components
j: state index
J: density of flow of particles (flow of particles)
Jd: diffusion flow
Jt: density of flow of particles in the state i
juo: light-induced flow of particles
J: angular momentum operator
J, J', Jm, Jn' quantum number of angular momentum
J(z): Bessel function
J (icq): cross-polarization tensor
k: de Broglie wavenumber
List of Symbols
xi
k: wavenumber
к: wave vector
к/. wavevector of the probe field
I: mean free path
I: state index
m: state index
m, ma, mb: masses of particles a and b
M: quantum number of angular momentum projection
M: atomic (molecular) weight
Mit M2: memory factor
n: state index
n: set of quantum numbers with the exception of M
nf. number of collisions
N: concentration of absorbing particles
Nb: concentration of buffer particles
Nf. population of the state j
p: momentum operator
p,pa, pb: momentum
p(v): number of absorption acts per unit time and unit velocity interval
{p): number of absorption acts per unit time
P(&>), P(£2): work done by a field per unit time
РДЦ,), Р„ : work done by a probe field per unit time
P“; absorption power
Pe: emission power
/?: pressure tensor of particles i
q: momentum variable in Wigner representation
qt flow of translational energy
g,: flow of translational energy of component i
qi(y): excitation rate of state j, v
qaft, q“p- tensor of the third moment of the distribution function
б,а2,ез: first, second and third moments of the function p(y)
Q2: trace of Q2
Qf. excitation rate of state j
r: spatial variable in the Wigner representation
R: matrix of spontaneous transitions
R(1), R(2): out- and in-terms of R
Rtf. third momentum of the collision integral
work done by a field per excitation act of state j
s, sf. halfwidth of a one-dimensional kernel
S: collision integral
xii
List of Symbols
S(I), S(2): out- and in-terms of the collision integral
Se: elastic part of the collision integral
^(v): collision integral for component i
S/jiy): integral of collisions of component i with component j
t: time
T: temperature
T: matrix of transitions under scattering
T(a0p | : element of the T matrix
Tb: temperature of the buffer gas
и, «p relative velocity of colliding particles
и: velocity of light-induced drift
u(z): real part of the probability integral
U'mn(Kq | element of the dynamic interaction matrix Uzm„ in Kq
representation
v(z): imaginary part of the probability integral 4
v, Vp velocity of particle a
v, vb: thermal velocities of particles a and b
va, t%: velocities of particles a and b
Vii> va: projection of velocity v
V: volume
ftC, Hamiltonian of the interaction of a particle with external and
probe fields
Vm„(Ar): matrix element of the operator V in the Kq representation
w(z): probability integral
wtj = i Tr Cv: rate of collisional exchange of translational energy between
gas components i and j
wmj'. rate of transitions j^>m
W(v): Maxwellian distribution
WB(J): Boltzmann distribution
Wp density of translational energy of gas component i
x,y,z,xa: Cartesian coordinates
a, a', a(: set of quantum numbers of internal motions of atom a
a: Euler angle
a(co): absorption coefficient
»i, a2: roots of the characteristic equation
: absorption coefficient for the probe field
a„: integral absorption coefficient
Д, P', Pi, Pi- set of quantum numbers of the internal motions of atom b
P: Euler angle
List of Symbols
xiii
fljK: spectral polarization function of non-linear interference effects of
rank к for a two-level system
spectral polarization function of non-linear interference effects of
rank к for a three-level system
y: radiation halfwidth of a spectral line
y: Euler angle
y: matrix of spontaneous damping
у/ constant of spontaneous damping of a,(t)
Г: halfwidth of a spectral line
F(z): Г function
rjK: constant of damping of the polarization moment of rank к of state j
Гт- constant of damping of the state m
Г,: halfwidth of the Bennett structure
<5: shift due to recoil effect
<5(z): d function
d,/ Kronecker symbol
A, Ay: collisional shift of spectral line
e = Ц, - Q
e' = е-(к„- k)'V
ёа, ёь: mean translational energies of particles a and b
% = v - Vp difference between velocities after and before collision
tj: set of internal coordinates of an atom b
f): momentum operator canonically conjugated with §
в: scattering angle
в, &: polar angles
к: rank of an irreducible tensor
к: saturation parameter
k(M): saturation parameter for the transition M-M'
Л: wavelength
Л: mode index of the strong external field
Л: de Broglie wavelength
ц: reduced mass of colliding particles
ц: mode index of the probe field
v>, v,y, v(aa' | v): out-frequency of the collision integral
v,, Vy, v(aa' | v): in-frequency of the collision integral
vtr(v), vi(v) (j = m,n; к = 1,2,3,4); transport frequency of collisions
v**/ (k = 1,2,3,4): frequencies of the generalized model of strong
collisions
set of internal coordinates of an atom a
xiv
List of Symbols
£ = hk/m.'. recoil velocity
p: impact parameter
p: density matrix
p(r§, /•'£')> р(лл, r'n), p(aa'vrt), Pytyrt), Pijixq, v): element of the
density matrix
p(g, r): density matrix in the Wigner representation
p0: collision radius
Pt>: density matrix of buffer particles
pw: Weisskopf radius x
a: index of the circular component of a vector
a(u | «1), cr(u, |u - mJ), <т(и, в): differential cross-section
<т(и): cross-section
<7tr(u), сг^Хи) (к = 1, 2): transport cross-section
r, Tt, r2, Tj, Tij, T2j, tin, r2n: characteristic relaxation times
r0: mean free time
rc: time of collision
<p: azimuthal angle
<p, Ф1> Ф2, фз- spectral factors of light-induced drift
ф,(О„): function describing the contour of a non-linear resonance
0(z): probability integral
ф(т), ф,у(т): correlation function
ф(а, y;z): confluent hypergeometric function
row of functions
Wf. eigenfunction of the operator Йо
V(t): wavefunction
co: frequency of the electromagnetic field
o)m„: Bohr frequency for a transition m-n
frequency of the probe field
Q = a>-o)mn
= a)ml
Q’ = co - &)m„ - k v
£2’^ = 0^- a)mt -k^-v
1
Introduction
The interaction between radiation and matter has two main aspects. One
of these includes the problems of the propagation of waves in a medium,
and the other is connected with elementary acts of absorption, emission
and scattering by microscopic systems constituting the medium. When the
problem is more or less completely reduced to the analysis of wave
propagation, the medium is characterized by macroscopic quantities such
as dielectric constant, polarization and refractive index. The wave
propagation is described by a pair of universal Maxwell equations for the
strengths of the electric (E) and magnetic (H) fields, inductions D and B,
where
IdD IdB
curlH = -—-, curlE =------—, (1.1)
cdt c dt
and material equations reflecting specific properties of the medium. For
example, in the case of a stationary isotropic non-magnetic medium, with
no spatial dispersion under sufficiently small field intensities, the material
equations have the form [1]
В = H, D(t) = E(t) + f f(v)E(t - r) dr, (1.2)
Jo
where the function/(r) serves as an optical characteristic of the medium.
For monochromatic plane waves which are time- and coordinate-
dependent by the law
exp(-i(ot + ik • r) (1.3)
it follows from eqs (1.1) and (1.2) that
к X H =——e(oj)E, kXE = —H, D = e((o)E,
c c
, <o2 Г
k = — £(co), e(io) = 1 + I /(r)e‘“TdT. (1.4)
c Jo
In this case the medium properties are concentrated in the dielectric
2
Introduction
[Ch. 1
constant e(w). In particular, the specific radiation power P absorbed by
the medium is given by the relation
P = -^-e"(co)E -E*, e((o) = e'((o) + ie"((o), (1.5)
8л
i.e. P is determined by the imaginary part of the dielectric constant. With
e"((o) < 0 the medium under the action of an external field emits energy
but does not absorb it. If the imaginary and real parts of the wavevector
k = k' + ik" are parallel and, as is often the case, |e"|<e' the phase
velocity v = aj/k' and absorption coefficients a(co) = 2 |Jt"| are expressed
in terms of e'((o) and e"(oj) as follows:
с (O £"(a))
v = / 4 , a((o) =-----, . (1.6)
Ve (<o) c Ve (<o)
Equations (1.1)-(1.6) describe phenomena which are the subject of
linear phenomenological optics. These are space effects arising at macro-
scopic distances. By contrast, in molecular optics and spectroscopy special
attention is paid to the processes of radiation emission, absorption and
scattering due to electromagnetic field interaction with the atoms and
molecules of the medium.
One of the basic problems of molecular optics consists in finding the
relation between the macroscopic characteristics of the medium (dielectric
constant etc.) and the properties of its microscopic parts. Spectroscopy
deals with the studies of spontaneous and induced emission, scattering and
absorption of electromagnetic waves by the medium. The above-
mentioned phenomena are due to numerous processes’ resulting in
conversion of other energies (internal energy of atoms and molecules,
chemical, thermal, mechanic energy etc.) to the field energy, where as in
spectroscopy attention is focused just on the study of elementary acts of
various types of interactions.
In the simplest case of spontaneous emission by either atomic or
molecular gases the energy distribution over the spectrum has the form of
sharp spectral lines and the specific radiation intensity is given by the
relation
-00
Pe = ft(omnAmnNmI{(o), Z((o)d(o = l. (1.7)
J —00
Here to™ and Лт„ are the Bohr frequency and the first Einstein coefficient
for the transition m-n between the energy levels Em and En corresponding
to the spectral line in question, and Nm is the number of atoms in the
Introduction
3
upper state tn related to unit volume of the gas. The function
differing from zero in the narrow range of frequencies close to
describes the spectral distribution of the emitted energy (the spectral line
contour) and is called the spectral density of the first Einstein coefficient.
The specific form of the function I (co) depends on the relaxation processes
disturbing the regularity of electromagnetic wave emission by the atom.
The absorption and induced emission line contours are known to be
described by the same function /(co); in other words, the imaginary part
of the dielectric constant e"(co) is proportional to I (co) in the frequency
range close to wm„ (see eqs (1.5) and (1.6)).
Needless to say, it is not always possible to distinguish between the
problems of propagation of waves and those of their interaction with the
atoms of the medium. By way of example consider cooperative phenom-
ena under spontaneous emission or self-induced transparency. Formally
speaking, it is impossible to separate optical and spectroscopic problems
when the Maxwell and Schrodinger equations appear coupled. If the
Schrodinger equation can be considered for a given electromagnetic field,
the spectroscopic aspect of the radiation and substance interaction
problem can be distinguished from the optical aspect.
From general physical considerations it is quite clear that to obtain data
on acts of elementary interaction the conditions are to be non-
equilibrium. In the opposite case the system state is described thermo-
dynamically by means of such quantities as pressure, temperature and
spectral density of equilibrium radiation. In particular, use of the relation
(1.7) implies that the spectral density of gas radiation Pe considerably
exceeds that of the surrounding objects (container walls, detector).
Spectroscopy has accumulated a wide range of means and methods of
creating non-equilibrium conditions, for example the difference between
medium and radiation temperatures, optically thin media, quasi-
monochromaticity, polarization and directivity of the radiation acting on
matter, external quasi-static fields, pulse and beam methods, and excita-
tion in gaseous discharge and by means of chemical reactions. A wealth of
knowledge has been amassed on the spectroscopic properties of different
objects, both terrestrial and astronomical. Therefore, spectroscopy in its
essence is one of the most “kinetic” fields of physics*.
* The terms “kinetic” and “kinetics”, generally speaking, can have different meanings. In mechanics,
kinetics is understood as a description of the motion regardless of its cause. Often the terms “kinetics”
or “physical kinetics” denote the area of knowledge of non-equilibrium systems, of the processes
leading to near equilibrium or, by contrast, of the creation processes of non-equilibrium states. In onr
book the second meaning of the term “kinetics” is used.
4
Introduction
[Ch. 1
Paradoxical as it may seem, physical kinetics and spectroscopy until
recently have been practically independent fields of physics. Anyway,
classical textbooks on physical kinetics pay little attention to spectroscopy
whereas the authors of thick volumes on spectroscopy fail even to
mention physical kinetics. This situation seems to be accounted for by a
clear distinction drawn between spheres of interest. Traditional physical
kinetics was concerned mostly with macroscopic kinetic phenomena which
are closely associated with translational degrees of freedom. Spectros-
copy, in contrast, concentrates almost exclusively on the kinetics of
internal degrees of freedom.
Within the scope of linear spectroscopy there are several large classes of
kinetic problems. The simplest and most common kinetic model is
described by the system of equations
(T, + Г" = S Wmjty’ Гт = Z ™km, (1.8)
XOt / j к
where »vz„ are the rates of transitions n-+l. Under equilibrium conditions
it is not necessary to know w/n since the ratio of populations Nj/ty can be
Obtained from Boltzmann’s formula
W = (gt/gj) exp[(£, - Е,.)/Г], (1.9)
where g,, g, are statistical weights of the levels i, j and T is the
temperature. However, under non-equilibrium conditions the formula
(1.9) is not valid and the system of equations (1.8) must be solved in order
to calculate populations which according to the relation (1.7) specify
the integral intensities of spectral lines.
The second class of kinetic problems which are fundamental for
spectroscopy is connected with the relaxation of the dipole moment or of
the polarization. The kinetics of the dipole relaxation specifies the shape
of the spectral line profile (i.e. the function I(oj) in the relation (1.7)) and
is given by both inelastic and elastic processes. In the impact approxima-
tion dipole moment relaxation is exponential and the function I(w) is
я л 4 nv(v) dv
л J r^ + ta) — (omn — A — к - v)
(110)
where Г and A are the halfwidth and line shift due to collisions and W(v)
is the velocity (v) distribution of the atomic oscillators.
Relations (1.8)-(1.10) constitute the simplest kinetic model of spectros-
copy. Fluxes and gradients of macroscopic quantities treated mainly by
physical kinetics are out of place here.
Introduction
5
The above discussion referred to linear optics and linear spectroscopy.
At sufficiently high values of the electromagnetic field intensity the
influence of non-linear phenomena becomes pronounced and in order to
analyse them it is also advisable to differentiate between the processes of
wave propagation (phenomenological non-linear optics) and elementary
acts of field interaction with atoms and molecules of the medium
(non-linear molecular optics and non-linear spectroscopy).
Generally speaking, the electromagnetic field may affect any degrees of
freedom of the particles—translational, rotational, electronic and nuclear.
Consequently, a wide variety of non-linear phenomena is observed which
can hardly be treated from the unified point of view.
This book deals with comparatively simple non-linear phenomena
occurring under resonance conditions at relatively low values of the
interaction energy of atoms and the field, which are much less than the
distances between the energy levels. In addition, gas systems under low
pressures will be studied when the mean free time is much greater than
the duration of collisions.
Under the aforementioned conditions the general picture of all these
phenomena can be naturally based on the notions of stationary states of
an isolated atom (molecule) and transitions between them induced by the
external field. One of the basic effects associated with such transitions
consists in the equalization of the mean populations of combining levels
which, in its turn, leads to the changed absorption coefficient, specific
absorbed power etc. Such phenomena, called the saturation effect, have
been known for quite a long time. In particular, they implicitly appear in
Einstein’s derivation of Planck’s formula for the energy distribution in the
spectrum of equilibrium radiation: transitions induced by the equilibrium
radiation result in a Boltzmann’s level distribution of atoms. More
particularly, the saturation effect was revealed experimentally in 1926 [2].
Later it was studied in detail in radiospectroscopy [3, 4] and in the optical
spectrum of phosphors [2].
The high degree of the time and space coherence of laser radiation
results in the saturation effect’s becoming much more peculiar and
acquiring essentially new properties. First of all it must be emphasized
that the action of coherent radiation on atoms is selective. For example,
the interaction is most effective with the atoms for which the projection of
the velocity v on the direction of the wavevector к satisfies the condition
(resonance shift due to the Doppler effect.)
k’V = a)-ajmn, (111)
6
Introduction
[Ch. 1
where co and a)mn are the field frequency and the Bohr frequency. Such
selectivity with respect to velocities results in the fact that population
equalization refers not to the entire v distribution of atoms but to some
narrow interval close to the velocity given by the relation (1.11). The
characteristic width of this interval is Au = Г Ik. When the pressure is not
too high the inequality Г «kv (v is the mean thermal velocity) holds and
therefore Au/v = Г/kv «1. Thus the structure of the velocity distribu-
tion becomes sharp and non-equilibrium and depends on the field
intensity, its frequency and polarization, and the types of relaxation
processes. This structure, called the Bennett structure, is fundamental for
all non-linear gas spectroscopy under low pressures and in processes
taking place in gas lasers.
The second fundamental physical factor closely connected with the fact
that the external field is monochromatic consists in the specific kinetics of
induced transitions. Let at time t = 0 an atom be excited to some state m,
interacting in resonance with the monochromatic field. In the absence of
relaxation processes the ensuing evolution could be of a completely
dynamic character. Relaxation processes disturb the strictly dynamic
characteristics of the evolution but it retains the properties of the dynamic
processes and essentially depends on the field’s frequency, polarization,
geometry and other circumstances. In particular, when the field intensity
is sufficiently high the probability amplitudes of atoms’ being in combining
states prove to be oscillating functions of time, the oscillation frequency
being proportional to the field amplitude. Note that such oscillations may
be interpreted as level splitting of the atom interacting with the external
field.
The above-described properties of evolution evidently reflect the
kinetics of induced transitions caused by the coherent field and the
saturation effect can be treated as a consequence of this kinetics, allowing
for the averaging over the lifetime of the atom’s excited states.
More immediately, the kinetics of induced transitions is manifested in
the spectrum of spontaneous emission. The spectral line shape is, roughly
speaking, specified by the Fourier transform of the decay curve of the
excited states. Since this curve changes as a result of the interaction with
the external field the line shape of spontaneous emission will be changed
as well. In particular, the oscillatory dependence of an atom’s being in an
excited state must imply that the line is split. We can arrive at this
conclusion in a different way, regarding the level splitting as due to the
interaction of the atom with the coherent field. If this level splitting is
large enough, we must observe the splitting of spectral lines corresponding
Introduction
7
to the transitions for which initial or final levels or both are perturbed by
the external field.
Similar phenomena can also be observed in the absorption spectra as
well as in those of the induced emission of the weak (“probe”) radiation
which is resonant to the transition perturbed by the strong external field
or to the adjacent transitions.
Field-induced transitions not only bring about population changes of the
corresponding atomic states but also lead to the appearance of the
induced dipole moment or the correlation of combining states or their
coherent “mixing”. This fact drastically influences the absorption
spectrum of the probe field and spontaneous emission, which can be
proved as follows. On deriving the fundamental relation (1.7) of linear
spectroscopy it was supposed that optically combining states evolve
independently of each other and of all the other atomic states. This
supposition determines the general structure of the relation (1.7), i.e. Pe
being proportional to the number of atoms Nm in the initial state m for the
given radiation process, the line contour being described by a single
function /(to), and the proportionality of the spectral densities of the
Einstein coefficients for absorption and emission. In contrast, the exist-
ence of the induced transitions evidently implies that the evolution of
states mixed by the external field is not independent, i.e. the above-
presented supposition is not fulfilled and, as may be seen, the most
general laws of linear spectroscopy no longer hold: Pe is not proportional
to Nm, the spectral line contour is described by several functions of the
type I (co), and Einstein coefficient spectral densities for absorption and
emission are not proportional to each other.
Non-linear phenomena due to mutual dependence of the evolution of
the states mixed by the external field were called non-linear interference
effects.
There is a wide variety of particular non-linear phenomena under
different physical conditions. However, all of them could be reduced to
three fundamental phenomena already discussed, i.e. to the saturation
effect, level splitting and non-linear interference effects.
For better understanding of the peculiarities of relaxation processes in
non-linear spectroscopy note that the Bennett structure can be treated as
a circular, disk-shaped atomic beam. This beam has an equilibrium
distribution in projections of the velocity v perpendicular to the wave-
vector к and is collimated with respect to the v projection on k. The
collimiation degree is Au/v = r/kv.
The Bennett structure or an atomic beam as any non-equilibrium
8
Introduction
[Ch. 1
structure is subject to velocity-changing collisions and so such collisions
are especially important in non-linear spectroscopy. Unlike typical condi-
tions of traditional gas kinetics the Bennett structure is as a rule very
sharp (in the scale of the Maxwellian distribution width, Au = Г Ik « v).
Therefore in non-linear spectroscopy even very small collisional velocity
changes can be pronounced, e.g. those due to diffraction effects which
usually do not contribute to diffusion, thermal conduction and other
transport phenomena. This distinction is quite understandable if we draw
on the analogy between atomic beams and Bennett structures.
Certainly, in addition to the elastic scattering there are other important
relaxation processes which are also significant in non-linear spectroscopy,
i.e. inelastic transitions between states, disorientation, phase shifts of the
atomic oscillator, spontaneous damping. However, it is the velocity
change under collisions that is characteristic of all the relaxation processes
in non-linear resonance problems.
The analogy with a beam points to the generation of a macroscopic
flow of excited atoms along' (or opposite to) the wavevector k. In the
absence of collisions, flows of atoms populating different levels cancel
each other out (if the light pressure is neglected) and the gas as a whole
remains at rest. Collisions with the buffer gas accompanied by velocity
changes scatter and decelerate Bennett atomic beams. If this deceleration
varies for different levels the above-mentioned cancellation fails to take
place and the absorbing gas begins to drift (the so-called light-induced
drift).
The above considerations show that the problems of non-linear spec-
troscopy and gas kinetics tend to converge and sometimes overlap. The
physical basis of this trend is Bennett beams. The common language (or
methodological basis) for non-linear spectroscopy and gas kinetics was
found within the scope of the formalism of the quantum kinetic equation
with a collision integral of the Boltzmann type.
The famous integrodifferential Boltzmann equation was first derived in
1872 [5]. Extensive studies were devoted to this equation and it is
certainly impossible even to attempt to cover the references. We shall
note only some studies which proved to be important for the kinetic
problems of spectroscopy.
The influence of collisions on the distribution function of gas particles is
described in the Boltzmann equation by means of the collision integral, by
which for many decades particles have been considered to be devoid of
internal degrees of freedom. Only within the scope of the quantum theory
were the internal motions and collision-induced transitions between
stationary states of atoms and molecules taken into account (see, for
References
9
example, ref. [6]). However, the Boltzmann description of collisions even
in the quantum theory has been explicitly and implicitly considered to be
applicable only to states diagonal in the energy indices, i.e. only to the
populations of energy levels. Such a viewpoint was, for example,
expressed by Snider [7] where the operator expression (2.98) for the
collision integral was obtained, this expression as will later be shown [8]
being actually applicable also to the polarized states non-diagonal in
energy indices.
The psychological barrier which was an obstacle to the extension of
Boltzmann’s ideas to the sphere of coherent polarized states and,
consequently, to the problems of spectroscopy was successfully overcome
in refs [9, 10]. In the first of these studies the kinetic equation for the
distribution function of classical oscillators was introduced. The collision
integral of this equation allowed for collision-induced changes of velocity
and phases of atomic oscillators. This kinetic equation made it possible to
consider from a unified viewpoint all the basic problems of the phenom-
enological theory of spectral line broadening. In ref. [10] a system of
coupled kinetic equations was first suggested for the population distribu-
tions of optically combining levels and the corresponding polarizations
(see eqs (3.10) and (3.11)). The system of kinetic equations suggested in
ref. [10] was later derived and confirmed with varying degrees of accuracy
(see, in particular, refs [11-13]) and is nowadays commonly accepted as
the basis of the theoretical description, on the one hand, of the
phenomena of non-linear gas spectroscopy and, on the other hand, of the
light-induced phenomena of gas kinetics and gas dynamics.
References
[1] L.D. Landau and E.M. Liftshitz, Elektrodinamika Sploshnykh Sred, 2nd edn (Nauka, Moscow,
1982) [Electrodynamics of Continuous Media (Pergamon, Oxford, 1985)].
[2] S.I. Vavilov, Mikrostruktura Sveta (Akadyemya Nauk SSSR, Moscow, 1950).
[3] C.H. Townes and A.L. Schawlow, Microwave Spectroscopy (Pergamon, New York, 1955).
[4] R. Karplus and J.A. Schwinger, Phys. Rev. 73 (1948) 1020.
[5] L. Boltzmann, Wien. Ber. 66 (1872) 275.
[6] S. Chapman and T.G. Cowling, The Mathematical Theory of Non-uniform Gases, 3rd edn
(Cambridge University Press, Cambridge, 1970).
[7] R.F. Snider, J. Chem. Phys. 32 (1960) 1051.
[8] E.G. Pestov and S.G. Rautian, Zh. Eksp. Teor. Fiz. 64 (1973) 2032 [Sov. Phys. JETP 37 (1973)
1025].
10
Introduction
[Ch. 1
[9] S.G. Rautian and I.I. Sobel’man, Usp. Fiz. Nauk 90 (1966) 209 [Sov. Phys. Usp. 9 (1967) 701].
[10] S.G. Rautian, Zh. Eksp. Teor. Fiz. 51 (1967) 1176 [Sov. Phys. JETP 24 (1967) 788].
[11] V.A. Aleksejev, T.L. Andrejeva and I.I. Sobel’man, Zh. Eksp. Teor. Fiz. 62 (1972) 64 [Sov.
Phys. JETP 35 (1972) 325].
[12] E. Smith, J. Cooper, W.R. Chappell and T. Dillon, J. Stat. Phys. 3 (1971) 401.
[13] P.R. Berman, Phys. Rev. A 5 (1972) 927; 6 (1972) 2157.
Quantum kinetic equation for the density
matrix
2.1. Quantum system interaction with an external
electromagnetic field: description by means of
probability amplitudes
The behaviour of an atom* in an external field, including an electromag-
netic field, is given by the Schrodinger equation [1]
ih^-V = AV, Й = Й>+К$, (2.1)
at
where Ф is the wavefunction, Йо is the Hamiltonian of an isolated atom
and the term hV corresponds to the interaction with the external field. If
the latter’s intensity is comparable with the intensities of the interatomic
fields or exceeds them it is better to analyse the system states in the total
field which is the sum of internal and external fields. This is, for instance,
the situation with a superstrong magnetic field (of the order of 10’G)
when the atom retains its integrity but its properties differ greatly from
those in the absence of the field. In the event that external fields are small
compared with the interatomic fields, it will be natural first to solve the
problem of obtaining the states of an isolated atom and to use these states
as the basis for analysing atomic behaviour in an external field.
For most phenomena to be considered there is no necessity to take into
account the quantum properties of an electromagnetic field. Therefore the
external field will be described classically throughout the book. Spon-
taneous emission will be somewhat of an exception. Nevertheless, for this
case too a quasi-classical “prescription” will be given (see section 3.4).
Let Щ denote the eigenfunctions of the operator Йо satisfying the
* For the sake of simplicity the term “atom” will be used, although almost everything in the following
refers to arbitrary systems with quantized internal degrees of freedom. The more exact terms will be
used for the cases when the properties of specific systems (of atoms, ions, molecules) appear essential.
11
12
Quantum kinetic equation for the density matrix
[Ch. 2
equations
Э
ih —
dt ’ ’ ’
Wj = ipj exp(—iE//ft) (2.2)
= Е№.
To obtain the wavefunctions of the stationary states and the values of
the energies E, corresponding to them is a difficult problem which
throughout the book will be assumed solved and will not be studied.
Let us represent the wavefunction of an atom in an external field as a
linear combination of wavefunctions of stationary states:
«40 = S “,(')«( (2.3)
/
For state amplitudes a;(t), from the Schrodinger equation (2.1) there
follows the system of equations
i^XO = 2 Vik(t)ak(t)-, Vik(t) = Wk). (2.4)
Thus the external field “mixes” stationary states and determines the
transitions between them.
In the representation (2.3) we do not take into account the states with a
continuous spectrum (ionized atoms and molecules, dissociated mole-
cules). Also, in the present monograph the transitions to the continuum
which under certain conditions may prove essential and be of special
interest are not treated. Such phenomena are considered in various books
[2,3] and reviews [4,5].
The interaction of an atom with an external field described by the
right-hand sides of eqs (2.4) is of a dynamical character. The system’s
evolution depends also on relaxation effects which may be due to
spontaneous transitions and various processes taking place when an atom
collides with other gas particles or the walls of a container. If relaxation is
caused by spontaneous transitions the system of eqs (2.4) must be
substituted by (see section 2.3)
^«XO = ~YiaM ~ i S Vik(t)ak(t). (2.5)
The quantity 2y;- is the total probability of spontaneous decay of the state
§2.1]
Quantum system interaction
13
j, i.e. the sums of the first Einstein coefficients
2r, = SA>, (2.6)
I
the sum being taken over all the states with energy Et less than Er
The above method of taking the spontaneous transition into account has
a limited field of application. Physically, this method’s being approximate
can be easily explained by the following example. Let the external field be
absent, = 0. Then, according to eqs (2.5), we have
ay(t) = ay(t0) exp[-y;(Z - r0)],
i.e. each state exponentially decays independently of the others. Hence in
eqs (2.5) we do not take into account the possible population of some
level as a result of spontaneous transitions from the higher levels. Thus
the system of eqs (2.5) may be applied to these cases only when we are
interested in the evolution of a group of states that are mixed by an
external field and have spontaneously decayed mostly to states which do
not belong to this group. If the Ajk denote the probabilities of spon-
taneous transitions inside the group of states of interest the following
inequalities may serve as an applicability criterion of the system of eqs
(2.5):
A*«2y;, (2.7)
It follows from the above discussion in particular that eqs (2.5) cannot
be applied to the ground state, as y, = 0 for the ground state. Physically
this is quite clear, as the ground state is known to be populated as a result
of direct or successive spontaneous transitions.
In section 2.2 we present a method of taking into account spontaneous
relaxation which is free from the above-mentioned drawback and is based
on describing atom evolution by means of the density matrix.
In discussing various problems it may prove useful to write quantities
and their interrelations in the form of matrices. Let us introduce the row
of wavefunctions
М» = (^1^2-..), V = (‘P1‘P2...) (2.8)
and the column of amplitudes
/«i(0\
•(0 = | a2(0 |. (2.9)
14
Quantum kinetic equation for the density matrix
[Ch. 2
These quantities enable us to write the relations (2.3) and (2.5) as
V(r) = Va(r) (2.10)
and
T«(0 = -[? + iV(r)]a(0; V(0 = <4*| V IV), (2.11)
dr
where у is the diagonal matrix of spontaneous decay:
/ У1 o • • \
y = l 0 y2 ••• I. (2.12)
Let us introduce the matrix (evolution operator) S(r, t0) of eq. (2.11)
according to relation
a(r) = S(r,r0)a(r0), (2.13)
where a(r0) is the column of initial values of the state amplitudes ay(r) at
t = t0. The matrix S(r, r0) evidently obeys the equation
^S(Z, r0) = -[? + iV(r)]S(r, Го) (2.14)
dr
and initial conditions
S(r0, r0) = 1 (2.15)
where 1 is the unit matrix. The matrix S(r, r0) contains the solutions of eq.
(2.11) or (2.5), corresponding to all possible sets of initial values of the
amplitudes a,(r). From eq. (2.13) we can conclude that the elements of the
/th column of the matrix S(r, r0) (i.e. Ski(t, t0)) give the solution of eq.
(2.11) under the initial conditions ay(r0) = 1, ak(t0) = 0, кФ].
According to the general principles of quantum theory the mean
quantum mechanical value is
L(r) = <V*|£|V>, (2.16)
where L is an operator corresponding to the physical quantity L.
Substituting expression (2.3) into eq. (2.16) we obtain the relation
L{t) = ^a*(t)Likak(t) (2.17)
i,k
or in matrix form
L(r) = Tr{L<r}; u = a(r)a+(r), L=<4*|£|V>. (2.18)
§2.1]
Quantum system interaction
15
Subsequently we will be interested mainly in the components of the dipole
moment d(t) induced in an atom by the external field and the work done
by the external field P(t) per time unit. For these quantities, according to
eqs (2.17) and (2.18) we have
d(0 = S a*(t)dikak(t) = Tr{dn} (2.19)
j,k
P(t) = S a*(t)djk • Eak(t) = Tr{d • Eq}, (2.20)
i.k
where djk is a matrix element of the dipole moment operator and E is the
strength of the external field.
The choice of initial conditions for the probability amplitudes is
determined by the physical statement of a problem. For instance, the
initial conditions
am(to) = l, a;(to) = 0, j±m (2.21)
correspond to the excitation of an atom to the level m at a time t0.
-The subsequent evolution due to the interaction with an external field
and relaxation processes results in the fact that after a long enough time
the system will be in its ground state, i.e. the singled-out group of states
completely decays. For some quantities it is not their instantaneous values
L(f) that are of interest, but the integrated values over the total evolution
time, i.e.
<£ = f L(t)dt. (2.22)
Ло
For example, the integral
&=f P(t)dt (2.23)
•'to
determines the field work performed over the total evolution time. If
$%>0 the field energy is absorbed by an atom. In the opposite case,
<%<0, the value —signifies the energy emitted by an atom. For initial
conditions (2.21) this work equals the energy absorbed or emitted by an
atom for each excitation of state m.
All the above formulae are referred to a coordinate system centred on
an atom, i.e. to a moving coordinate system. When translational degrees
of freedom are described classically, which is sufficient for an understand-
ing of the majority of the phenomena of non-linear spectroscopy, the
atomic motion may be taken into account in the following way. If an atom
16
Quantum kinetic equation for the density matrix
[Ch. 2
has a constant velocity v, the radius vectors r and r' in the laboratory and
atomic systems are connected by the relation
r = r' + vt. (2.24)
The interaction with an external field V(t) is usually given in a laboratory
coordinate system, i.e. it is a function of r, t. At the same time, the
amplitudes of states a(t) are defined in the atomic system. Consequently,
the matrix V(r' + vt, t) appears in eq. (2.11).
If an atom undergoes velocity-changing collisions the equations should
be written
r = r+ v(t)dt,
(2.25)
ТХГ’’О =
dt
у + iV( r' + I v dt, t
a(r, t).
(2.26)
Therefore, owing to random velocity changes due to collisions, the
interaction matrix V turns out to be a random time function.
Let us substitute the variables in the eq. (2.26) according to formula
(2.25) which corresponds to the transition to the laboratory coordinate
system and introduce the designation
(2.27)
a( r — v dt, 11 = b(r, t).
The matrix b(r, t) obeys the following equation that results from eqs
(2.26) and (2.27):
3
3t
+ v • V )b(r, f) = — [y + iV(r, t)]b(r, t)-
(2.28)
We thus eliminate the random time dependence of the right-hand side of
the eq. (2.28), the dependence being transferred to the term v • V.
When the processes taking place as results of collisions (velocity
changes, dumping, phase shift) are considered within the framework of
the state amplitude apparatus it is necessary to solve the equation of
motion with random coefficients and to average the observed quantities
over various parameters describing collisions. Consequently, the relation
(2.18) takes the form
L(t) = Tr{L(n)}=Tr{Lp}, (2.29)
p=(n), (2.30)
§2.2]
Quantum kinetic equation
17
where the angular brackets denote the averaging of the quantity in
brackets over the interaction with the medium. The matrix p defined in
formula (2.30) is called the density matrix.
Quite a different method of analysing the role of collisions is based on
setting up an equation directly for the density matrix, which is called a
quantum kinetic equation. Such an approach is hence different from that
mentioned above in the order of the solution procedures of the equations
describing the system evolution and the averaging over the interaction
with the medium.
2.2. Quantum kinetic equation
Relaxation processes can be treated as a result of the interaction of an
emitting atom with a large ensemble of other perturbing particles (atoms,
molecules, walls of a container). Such an interaction leads to the mixing of
states. The wavefunction is not sufficient for their description, so one has
to employ the density matrix apparatus.
Unlike the approach in section 2.1, a quantum treatment will now be
applied not only to interatomic but also to the translational degrees of
freedom. This treatment of the translational degrees of freedom is
necessary for an understanding of some specific phenomena such as the
role of the recoil effect and proves to be useful for analysing various
general problems.
Let r and § denote the centre-of-inertia coordinate and set of internal
coordinates of an atom. The Hamiltonian in the absence of perturbing
particles will be represented as
fi = fi0+ti$(p,r, fio=p2/2m + O(fj,£), (2.31)
p = -ihV, =
where p,f) are operators of momenta corresponding to r and §, m is a
mass, and l7(?),£) denotes the Hamiltonian of the internal degrees of
freedom. As distinct from eq. (2.1), the expression for the Hamiltonian of
a single atom includes the momentum of translational motion.
The density matrix in the coordinate representation obeys the equation
18
Quantum kinetic equation for the density matrix
[Ch. 2
(see, for instance, ref. [1])
at
= R + S-\-t~ \Й(р, r, r), t) - Й*(р', r', ft', 0]p(r§, r'£).
In
(2.32)
The term S in the right-hand side of eq. (2.32) represents the interaction
of this particular atom with other gas particles and is called a collision
integral. The term R describes transitions due to spontaneous emission. In
the absence of the terms S and R, which can be called statistical, the
evolution of an atom is given by the Hamiltonian Й and is of a dynamic
character. Therefore the terms of eq. (2.32) containing Й and Й* are
called dynamic. Statistical terms will be discussed in sections 2.3 and 2.4.
When the coordinate representation is being employed, the mean quan-
tum mechanical values of physical quantities are obtained from the formula
L(r) = J[£p(r§,r'§')]^:,drd§.
The condition of constant number of particles
| p(r^,r^)drd^ = N
leads to an obvious equality
[ p(r?, r?) dr d? = 0.
at J
(2.33)
(2-34)
(2.35)
By virtue of the fact that the processes described by the terins of the
right-hand side of eq. (2.32) are different and independent, the trace of
each of them must become zero. With reference to the dynamic terms,
this requirement as can be seen from eq. (2.32) is fulfilled. The
corresponding property of statistical terms is discussed in sections 2.3 and
2.4.
Consider eq. (2.32) in various representations which will be further
employed. As stated in section 2.1, it will be advantageous to expand the
density matrix in terms of eigenfunctions 4^ of the Hamiltonian of internal
motions of an isolated atom:
= ^р(гт,Г'п) (2.36)
§2.2]
Quantum kinetic equation
19
The matrix p(r, r') of coefficients p(rm,r'n) satisfies the following
equation obtained from eq. (2.32) by general rules:
= R + S - i{V(p, r)p(r, r') - [V(p, r')p(r, r')]*},
(2.37)
V(p,r) = (4*4 $ |V>; R= (ФЧЯ |V>; 8=<ФЧ5|Ч»).
Elements of the matrix V may depend on r, iftV and t.
The mean quantum mechanical values of physical quantities in the
representation (2.36) are obtained according to the following relation
which is a consequence of eqs (2.33) and (2.36):
L(t) =Tr|J [Lp(r, r')]r.=rdrj . (2.38)
For translational degrees of freedom, in addition to the coordinate
representation the Wigner and momentum representations may also prove
useful. Density matrices in coordinate and momentum representations are
connected by the relations
p(p,pr) = (2jtft)-6J p(r,r’)exp[—i(p • r—p’• r’)/ft]drdr’, (2.39)
p(r, r') = | p(p,p) exp[i(p • r-p' • г')/й] dp dp'. (2.40)
The kinetic eq. (2.32), when the momentum representation is used for
translational degrees of freedom, takes the form
/3 p2 — p'2\ f
IT - > э A )p(p>p') = R + S — i I [V(p -P1)P(P1,P') -
\at £tnn / J
p(P>Pi)V(pi-p')]dPi, (2.41)
where the notation
V(p) = V(—iftV, r) exp(—ip • г/й) dr (2-42)
is introduced for the matrix of the interaction with an external field in the
momentum representation. Mean quantum mechanical values are ob-
tained in the following way:
L(t) =Tr{ J [Lp(p,pU,=,dp) . (2.43)
20
Quantum kinetic equation for the density matrix
[Ch. 2
According to eq. (2.41) a dynamic term is an integral of the convolution
type of the density matrix and the interaction matrix. This evidently
represents the exchange of momenta between a field and an atom. Let,
for example, V(p, r) have the form of a plane wave:
V(p, r) = Vexp(ifc • r); V(p) = V<5(p - Лк). (2.44)
In this case instead of eq. (2.41) we have
Э n2 — p'2\
T~)P(P,P') =~R+S-i[Vp(p-Лк,р)-p(p,p+ hk)V],
dt Zmn /
(2-45)
i.e. the kinetic equation couples density matrix elements whose argu-
ments’ difference is equal to the value of the photon momentum Лк.
When seeking the solutions of some problems using the coordinate
representation p(r, r'), the quantities R = (r + r')/2 and q = r — r' proved
more advantageous. Similarly, in the momentum representation the
variables q = (p + p')!2, t—p —p' are sometimes introduced instead of p
and p'. The quantities q and r evidently prove useful for describing the
correlation properties of a system.
For a wide range of questions the Wigner representation for transla-
tional degrees of freedom seems advantageous (see, for example, refe
[6-8]). The Wigner matrix p(q, r) is connected with p(r, r') and p(p,p')
by the relations
p(q, r) = (2лй) 3 p(r + if /2, r - q/2) exp(-ig • ij/ft) dij
= J P(9 + *72, q — *72) exp(ir • т/й) dr, (2.46)
p(p,p) = (2лй)-3| r) exp[-i(p -p') • г/й] dr, (2.47)
Р(Г, r') = | p(«,'^-y") exp[i(r - r') • q/Л] dq, (2.48)
i.e. the Wigner representation is a combination of the coordinate and
momentum representations. After transformation (2.46) of the kinetic eq.
§2.2]
Quantum kinetic equation
21
(2.37) we can obtain the following equation for p(g, r):
/ Э Q \
(ft+m
П/ 0 n\
v(gi--,ri+y)p(gi,ri)-
\ £ f
(0 n\ '
91 + 2’Г1-2/ X
expH
[(9 - 91) • 4 + (r - Fl) • 0П dij d0 dtf! dFp
(2-49)
The Wigner variable q is in many respects similar to the classical
momentum. In particular, the operator (qlm) • V in eq. (2.49) is similar to
the term v • V in eq. (2.28) for state amplitudes, where the classical
description of atomic translational motion was used. Mean quantum
mechanical values are calculated in the Wigner representation from the
formula
L = Tri L(q, r)p(q, r) dq dr
(2.50)
i.e. with respect to translational degrees of freedom in a way similar to
averaging by means of the classical distribution function over momenta
and coordinates.
We now consider a special case which is, nevertheless, of practical
importance when V(p, f) is independent of p. This is, for example, the
case when an atom interacts with an external electromagnetic field. Then
integration over 0 in eq. (2.49) yields (2лй)3 d(r — rt) and the kinetic
equation for the Wigner function appears local with respect to the
coordinate:
d Q \ f
—r + — • V)p(g, f) = R + S — 1(2лй)-31 dr/ d^ x
dt m / J
[V(r + ?)p(«1»F)-p(9bF)v(F-y) X
L \ £ I \ и/.
exp[i(9i-9)n/ft]- (2.51)
The integral character of the dynamic term in eq. (2.51) (with respect to
the momentum variable) is due to the recoil effect. Its interpretation is
most illustrative when the field is expanded in terms of plane waves, i.e.
V(r) = f V(k) exp(ik • r) dk, (2.52)
22
Quantum kinetic equation for the density matrix
[Ch. 2
after which eq. (2.51) becomes
(T,+ ~ ’ V)P^’= R + S “ 1 “ V’ r
\dt m / J L \ 2
/ tik
p|5 + V’
X X
r ) V(Jt) exp(i£ • r) dit.
(2.53)
It can be seen from eq. (2.53) that the electromagnetic field couples the
states for which the difference of momenta is equal to the momentum of
the photon tik.
The Wigner representation is advantageous for going over to the
classical description of the translational motion of atoms. Under such a
transition, formally corresponding to й—>0, the magnitudes of q and 0
may be assumed small and V may be expanded in terms of powers of q
and ft Expansion up to linear terms, i.e.
V(g T 9/2, r ± q/2) = V(g, r) T f • V,V(g, r) ± %- VrV(q, r), (2-54)
corresponds to the classical approximation. Substituting expression (2.54)
into eq. (2.49) and performing all the integrations we can obtain
(S^-v> = R + s-.(Vp-pV)-
(V, V V,p + V,p V,V - V,V V,p - V,p ,V). (2.55)
The dynamic terms of the zeroth approximation (the third term) in the
right-hand side of eq. (2.55), describe the action of the external field only
on the internal variables of an atom. In an electromagnetic field the
matrix V is non-diagonal, which corresponds to the mixing of stationary
states of an isolated atom. It is these phenomena that will subsequently be
of interest to us. If V does not depend on internal variables, the external
field cannot influence the internal motions and the term under considera-
tion becomes zero.
The other terms in the right-hand side of eq. (2.55) describe the action
of the external field on the translational motion of an atom as a whole. In
the simplest case when V is independent of internal coordinates we have
V = VI, where V is a value independent of the quantum numbers of
stationary states and 1 is the unit matrix. As it has already been noted, the
terms of the zero approximation become zero and eq. (2.55) takes the
§2.2]
Quantum kinetic equation
23
form
(2.56)
F = -й VrV.
Here F evidently represents the force acting on the inertia centre of an
atom, and the quantity q + mh V,V has the meaning of a generalized
momentum.
The more complicated form of the general eq. (2.55) is due to the fact
that forces and generalized momenta can vary in different stationary
states. In conformity with the above statement, the forces are represented
in eq. (2.55) by the matrix й VrV, and generalized momenta by the matrix
mh V, V.
In the spfectroscopy of non-linear resonance the recoil effect has been
first considered as an example of the manifestation of external forces [9]
(see section 5.5). Emission or absorption spectra of particles moving with
acceleration under the action of non-quantum forces are analysed in refs
[10-13].
In most spectroscopic phenomena the influence of an external field on
translational degrees of freedom is insignificant and we may retain only
the first line of eq. (2.55). We consider in more detail the structure of the
dynamic terms of this equation and write its matrix elements as
+ v v)p(w'> vrt)= R(aa') + S(atx') —
i 2 [V(aa1? rt)p(<x1<x', vrt) —
at
p(aa1, vrt)V(a1a', it)]; v = q/m.
(2-57)
We shall confine ourselves to the analysis of systems whose states can be
characterized by the total momentum J and its projection M (atoms,
molecules, spherical top molecules, linear molecules), i.e.
a-aJM=nM, n = aJ, (2.58)
where a and n are sets of quantum numbers, with the exception of JM and
M respectively. The elements of the density matrix and interaction
Hamiltonian have the following structures:
p(a, a';vrt) = p(aJM,a'J'M’; vrt) = p(nM, n'M'; vrt),
V(a, a';rt) = V(aJM, a'J'M';rt) = V{nM, n'M'; rt).
(2-59)
24 Quantum kinetic equation for the density matrix [Ch. 2
Depending on the phenomenon under consideration, different quantum
numbers and continuous variables are essential. For simplicity we shall
subsequently write down only the variables which are of importance for an
understanding of the problem being discussed; others will be omitted.
In the theoretical analysis of some effects it appears possible not to take
the degeneration of states into consideration (model of non-degenerate
states). In such cases the system of eqs. (2.57) will be written as
+ V • v)pmn = Rmn + Smn - i 2 (Vmjpjn - PmiVjn)- (2.60)
vat / j
The model of non-degenerate states is most widely used for the solution of
concrete problems because it is relatively simple and easy to interpret.
In order to describe phenomena where state degeneration is essential
(e.g. polarization effects), in addition to the JM representation (2.59) the
so-called representation of polarization moments (or representation of
irreducible tensor operators or Kq representation) is widely used. This is
connected with the JM representation by the following relations [14,15]:
Lnn (Kq)= X -M' | кд)ЦаШ',а7'М'), (2.61)
MM'
L(aJM, a'J'M') = £ (—1)J ~M {JMJ' - M' | Kq)Lnn{Kq), (2.62)
where L{aJM, a'J'M') is a matrix element of the arbitrary operator L in
the basis of angular momentum J eigenfunctions; {JMJ' — M' \Kq) is the
Clebsch-Gordan coefficient*. The mean quantum mechanical value L(t)
is expressed in terms of polarization moments by the formula
L(0 = S [ Lnn {Kq)p*n (Kq) dv dr. (2.63)
пп’кд J
In order to explain the transformation (2.61) we explicitly write the
density matrix expansion (2.36) in terms of eigenfunctions of the internal
motions of an isolated atom, i.e.
p(£,?) = S p{aJM,a'J'M')4/{aJM\ %)4f*{a'J'M' | £), (2.64)
nMn'M’
where the W{aJM | £) are simultaneously eigenfunctions of the momen-
tum operator. After substituting the expansion (2.62) into eq. (2.64) we
* See Appendix I for properties of the Clebsch-Gordan coefficients {JMJ'M' | Kq} and their relation
to Wigner 3/ symbols and other similar quantities.
52.2]
Quantum kinetic equation
25
have
P<£, S') = S pnAxqyV'inn’Kq | ££'),
nn’xq
(2.65)
W(nn'Kq | &') = 2 (-I/'-"' {ШГ -M’\Kq)
MM’
The function Wtnn'xq | &') is an eigenfunction of the momentum к,
which is obtained by addition of the momenta J and J'. Thus the
transition from eq. (2.64) to eq. (2.65) represents the transformation of
the product of the J and J' momenta eigenfunctions to the basis of
wavefunctions of the total momentum. In accordance with general rules к
takes values |J — J'\ =s к =s J + J', and the quantity q, which is interpreted
as the projection of к, can change within the range -к =s q =s к.
According to Appendix I:
(J MJ' - M* | 00) = (-!/-" djr dMM./y/2J + l.
Therefore from formula (2.61) it follows that
p„„.(00) = djr 2 p(nM, n (2.66)
M
i.e. the polarization moment of the zeroth order is meaningful only for
J = J', and when n=n' it describes a sum (over M) of the population of
the nth level. The quantity pnn(lq) is called the vector of state
orientation; p„„(2q) is called the alignment tensor [14].
Equations (2.57) in the Kq representation can be written as (see
problem (2))
1 + v . V U , = Rnn, + S„„. + i 2 (^1ЛрЛЛ1 - С^Л1рЛ1Л), (2.67)
/ Л1
where p„„ is a column vector with elements р„„(кц), and U„,„. is a matrix
with elements U"t„ (Kq | k^i). The matrix СЛЛ1 is connected with by
the relation
| = (-l)Ji-J+4~4'Unn"(Kq I Mi)- (2.68)
The explicit expression for the elements of the matrix is
I w) = S (-ly-'-' W+T V2k7T1 x
ka
f Al
Ir г т к-1)г,_’,(*фГ1-Я1|Аа)Уяп.(Аа),
J J)
(2-69)
26
Quantum kinetic equation for the density matrix
[Ch. 2
where j
Ut
к A]
' | is a 6/ symbol and Ип„(Асг) are the coefficients of the
expanded interaction Hamiltonian matrix element in terms of irreducible
tensor operators:
V„„.(Ao) = X - M' | Ao) V(JM, J'M'), (2.70)
MM’
a = —A, —A + 1,. . . , A — 1, A.
Since the interaction V is Hermitian the relations
V„V(Ao) = (-iy-"'-X’»(A - a), (2.71)
UZ'^q | Klqt) = - qx | к - q) (2.72)
are valid.
The physical meaning of the quantities Vnn.(Ao) which determine the
dynamic term of the kinetic eq. (2.67) can be clarified by using as an
example the dipole interaction for which the Hamiltonian has the form:
= = (2.73)
where Eo,clo are the spherical components of the vectors of the electric
field E and dipole moment a = 0, ±1. For example,
Ей = Ег-, Ex = -^{Ex + iEyy, E_x=^{Ex-iEyy (2.74)
According to the Wigner-Eckart theorem, the matrix elements da are
given by the expressions
da(aJM, = dnn(-l)J'~M'{JMJx - Mx | lo) exp(iaw), (2.75)
dnnt = (n||d||n1)/V3,
where (n|| d Unj) is a reduced matrix element of the operator Thus
V(aJM, S - Mx | la)Eo exp(iwn„1t).
Л a
(2-76)
Substituting eq. (2.76) into eq. (2.70), we obtain for the dipole interaction
Кт,(Ао) = -8kx(EadnJty exp(ito„„,0, (2.77)
i.e. only elements with A = 1 differ from zero and the Vnni(Aa) are
proportional to the spherical components of the electric field. In the
§2.3]
Radiation relaxation
27
general case the quantities A and a characterize a spherical interaction
tensor of order A.
Since the principal subsequent results will refer to the dipole interaction
with an electromagnetic field we give the expression for the matrix U^n, in
this particular case. From the expansion (2.69) with eq. (2.77) taken into
consideration it follows that
Unn'n (Kq | = (-l)1-y,-J'-KV2ic + 1 V2ki + 1 exp(ito„„7) x
Г
* p1 J f S (-1)K,_”<K?K1 - 9i1 la)dnn.Ea/h.
{J J JiJ a
(2.78)
If the electromagnetic field is polarized linearly or circularly, the
expression (2.78) may be further simplified. For linear polarization when
the quantization axis is chosen along the electric vector E, only the
component Eo = E is different from zero:
Unn'n {Kq | Kxqx) = + 1 \J2kx + 1 exp(ito„„7) x
r* r |(-l)K,-’1<^9*’i “ 9i 110)dn„.E/fi.
VJ J J\)
(2.79)
For circular polarization, direction of the quantization axis along the
wavevector of the field is convenient. Then only one circular component,
e.g. Ex = E, is different from zero:
Un„'n (Kq | Kxqx) = (-iy-J'-J ~Ky/2K + 1 \J2kx + 1 exp(i«on„ t) x
(r Kj> f l(_1)K1_’^K^Ki_9il M}dnn Elti. (2.80)
V.J J J\)
2.3. Radiation relaxation
The matrix R in the right-hand side of the kinetic equation (2.37)
describes the transitions due to spontaneous emission and is called the
matrix of spontaneous transitions. R is of the simplest form in the Wigner
representation: \
R(g, r) = -R(1)(9, r) + R(2)(9, r); \2.8l)
R(1)(9, r) = yp(9, r) + p(9, r)y; R<J> = (ym + yjp™; (2.82)
^^(9, r) = дтп X f Aml(q, qx)pu(qx, r) dqx, (2.83)
28 Quantum kinetic equation for the density matrix [Ch. 2
Amfa, 9i) = £ <5[(9i - Я)2 ~ b2<rfmlc2\
ZJtn G)lm
= d(k2 - ^m/c2)d(q -qx- ЙЛ) dk. (2.84)
ZJtCOfrn J
Here Alm and to,„ are the Einstein coefficient for spontaneous emission and
the Bohr frequency for the l-^m transition. The summation in the
expression (2.83) is carried out over the states whose energy Et exceeds
the energy Em; the diagonal matrix у containing the rates of spontaneous
decay y„ of states n is determined by the formula (2.12).
The term R(1) evidently describes the density matrix damping owing to
the spontaneous emission, i.e. R(1) corresponds to the same processes that
have been considered in section 2.1 (see formulae (2.5) and (2.6) and
subsequent discussion). This term will be referred to as an “out-term” of
the R matrix. If the equation for the density matrix is derived from the
eqs (2.11) for the probability amplitudes, R will contain only the term
R(1).
The matrix R(2), which will be referred to as an “in-term”, is diagonal;
its elements R^(q,r) give the excitation rate of the state m, q, r as a
result of spontaneous transitions from the states /, qx, r with greater
energies. The number of such transitions per unit time must evidently be
proportional to Alm and to the population pu(qx, r) of the state I, qx, r,
which can be seen in the relation (2.83). The integral form of the
quantity R(2)m(q, r) is due to the recoil effect, taking place under
spontaneous emission. The expression (2.84) reflects the fact that the
momentum q of the atom which emitted a photon of wavevector к and
which underwent a transition to the state m differs from its momentum
9i = 9 + hk before the emission process when the atom was inHhe state /.
As well as the law of conservation of momentum, it is assumed in the
expression for Am/(991) that the photon frequency is approximately equal
to the transition frequency a>lm. Such an approximation is quite adequate
because of the small widths of the spectral line radiation.
It can easily be verified that the equality
Tr( f R(9, r) dq dr] = 0 (2.85)
follows from the relations (2.81)-(2.84), which means that spontaneous
transitions do not change the total number of particles. Therefore the
theory of spontaneous relaxation based on the kinetic equation is free
§2.3]
Radiation relaxation
29
from the restrictions typical of the simplified approach to this problem by
the method of probability amplitudes (see section 2.1).
It must be emphasized that the relation (2.85) implies summation over
all the states of the atom. In the analysis of concrete situations, especially
under resonance interaction of the electromagnetic field with an atom, not
all the states are taken into consideration; only a group of them are
considered, those most strongly interacting with the field. In such a
formulation the trace of R over the selected states, naturally, is not zero.
In most problems of non-linear spectroscopy it is not necessary to take
the recoil effect into account and we can confine ourselves to a classical
description of the translational degrees of freedom. In such an ap-
proximation it should be assumed that fik = 0 (see eq. (2.84)), as a result
of which Ami(q,q1)=AlmS(q-ql) and the expression for R(2) in the
Wigner representation is simplified:
R™(q, r) = dmn S AlmPll(q, r). (2.86)
i
The matrix R in the coordinate representation, as calculations based on
eqs (2.81)-(2.84) and the transformation formulae (2.46) and (2.48) show,
is given by the following:
R= —R(1) + R(2); (2.87)
R(1) = Yp + pY; R<^,(r,r') = (Ym + Y„)pmn(r,r'); (2.88)
Rfitfr, r') = дтп 2 Aml{r - r')pu(r, r'); (2.89)
/
Аиг-Н = л/"(|Г~<!а>7/С). (2.90)
|r-r'| (Olm/c
The out-term R(1) the retains the structure which it had in the Wigner
representation (compare eqs (2.88) and (2.82)). The role of the recoil
effect is given in the coordinate representation by the factor Am,(r — r') in
the expression (2.89) for the in-term R^n(r, r'). The width of this factor is
equal to cla>im, which according to the uncertainty principle yields
\q — qx\~ h(Dlmlc = tik,
i.e. such a width of the factor Aml(r — r') corresponds to a change in
momentum by a value equal to the photon momentum.
The classical description of the translational motion corresponds to the
equality Aml(r — r') = Alm and the in-term R(2) takes the form
R^(r, r') = 8mn X A/mpu(r, r'),
(2.91)
30
Quantum kinetic equation for the density matrix
[Ch. 2
similar to that in the Wigner representation (compare eqs (2.91) and
(2.86)).
The foregoing referred to non-degenerate states. In the degenerate case
spontaneous cascade transitions carry out the “transfer” not only of the
population but also of the density matrix elements non-diagonal in
degeneracy indices (coherence between sublevels). Let a = aJM = nM,
where M is the projection of the momentum J; then (see problem (14))
R('\nM, n'M') = (y„ + y„.)p(nM, n'M')-,
(2.92)
R<2\nM, n'M') = <5„„. X A(nMnM'\n1M1n1M'1)p(nxM1,n1M'x).
The out-term R(1) has the same structure since the spontaneous decay
constants are the same for all the magnetic sublevels. The in-term R(2) is
diagonal in scalar quantum numbers n and n' and contains elements
p(nxMx | nxM\) diagonal in nx. However, R(2) is non-diagonal in degeneracy
indices.
Calculation of the coefficients A(nMnM' | n^M^M'x), carried out in
problem (14), yields the expressions
A(nMnM' \nlMlnfM'1) = ЛЛ1П^ JXM'X),
(2.93)
where ЛЛ1Л is the first Einstein coefficient for the transition nx-+n. Thus in
the JM representation the in-term R(2) of spontaneous transitions is
essentially non-diagonal. The properties of the Clebsch-Gordan
coefficients suggest only the condition M — M' = Mx — M{.
In the Kq representation (see section 2.2) the in-term turns out to be
diagonal in n and degeneracy indices [16]. Simple manipulation yields
the expressions
R^ (Kq) = дт. 2 Anin(K)p„int(icq), (2.94)
Л1
Л.,.(к) = Л.„(2/ + 1)(-1У«'*"‘{1 2 2}, (2.95)
(K J J)
where i [ is a 6/ symbol. Thus, as a result of spontaneous
11 Jx Jx)
cascade transitions the polarization moments of different orders are
transferred to the other levels independently (diagonalization in Kq). The
rates ЛЛ1Л(к) are not the same for different к. For к = 0 from the
§2.4]
Collision integral
31
expression (2.95) it follows that the summed (over M) population of the
level characterized by the element рЛ1Л1/(00), excites the element
p„„(00), as would be expected, with a rate equal to A„,„. For k=#0 the
coefficients АЛ1Л(к) have different values and may be both positive and
negative (in the case of transitions where J = Л). When АЛ1Л(к) is negative
it means that the cascade transition induces on the level n a polarization
moment of reversed sign compared with that of the пг level.
2.4. Collision integral
The derivation of an expression for the collision integral S is one of the
fundamental problems of quantum kinetics not yet completely solved.
However, for the problems of non-linear spectroscopy of sufficiently dilute
gases, the impact approximation is applicable within the range of which
the collision integral can be obtained with satisfactory rigour.
Consider a gas consisting of two components one of which interacts with
an external field and is described by a density matrix p. The second
component acts as a thermostat; it will be called a perturbing or buffer
component and will be described by a density matrix pb. Let the
interaction energy W of colliding particles decrease rapidly enough as the
distance between them increases so that we can speak of an effective
radius of interaction pc and an effective time of collision tc ~ pju (u is the
relative velocity). Assume the concentration of perturbing particles Nb to
be sufficiently small and the volume pl of the range of interaction is then
significantly smaller than the specific volume 1/Nb, i.e. the condition
pX«l (2.96)
is satisfied. The inequality (2.96) also means that the time of collision rc is
considerably shorter than the mean free time r0:
t0 ~ 1/Nbupl-, tJ t0 ~ p3cNb« 1 (2.97)
When the above-mentioned conditions, which are the essence of the
impact approximation, are fulfilled the collision integral S in the operator
form may be written as follows* (see problem (3)):
S = 1 Trb{T(p X рь)Я+ - Я(р X p^T}. (2.98)
In
* The collision integral in such a general form was to our knowledge first obtained by Snider [17].
32
Quantum kinetic equation for the density matrix
[Ch. 2
The symbol Trb indicates the trace is calculated only over variables of the
perturbing particle; p X рь is the direct product of p and pb; $2 denotes the
Moller operator describing a binary collision and obeying the equation
(# + #b)£2-£2(# + #b) +W£2 = 0, (2.99)
where Йь is the Hamiltonian of an isolated perturbating particle and W is
the interaction Hamiltonian of colliding particles. The scattering matrix T
is given by the relation
T = W£2 (2.100)
If a gas contains several perturbing components (atoms, molecules,
electrons, ions) S will be a sum of terms similar to those in eq. (2.98) and
characterizing collisions with perturbing particles of each type.
The external field, generally speaking, influences the processes occur-
ring under collisions if its intensity is sufficiently high. According to
problem (3), allowing for this influence does not alter the general form of
the collision integral (2.98) but the Moller operator must obey the
equation similar to eq. (2.99) where Й + Йь is substituted for Й + Йь +
ftv(t); here ftV (t) is the interaction Hamiltonian with an external field
[18]. In this book there will be no detailed treatment of collisional
processes taking place in an external field, i.e. it is implied that the field
intensity is not high enough (for details see refs [18—20]).
The collision integral (2.98) is expressed in terms of standard operators
T and S2 of theory of collisions. It must be noted that it is the Moller
operator $2 and not a scattering matrix (S matrix) that enters into eq.
(2.98). This can be explained by the equal probability of collisions over
the time interval until t (in the impact approximation over the time
interval — °°, t). The Moller operator also describes the evolution just over
this interval. The scattering matrix on the contrary contains the evolution
result over the interval — it being implied that collision took place at
a finite t.
To obtain T and S2 both theoretically and experimentally is quite a
difficult task that has been solved for comparatively few particular
problems. It is especially complicated to solve this problem for excited
states which are of major interest from the viewpoint of non-linear
spectroscopy. Therefore, data on T and S2 for any particular situation
cannot be relied on as exhaustive. Consequently the data on the analytic
structure of the collision integral assume special importance, as they
enable one to use various models and to find solutions of particular kinetic
problems. Similarly important are the general results of collision theory
§2.4]
Collision integral
33
which help one to predict the qualitative character of the matrix elements
of the T and $2 operators from the form of the potential. These
considerations stimulated us to analyse the collision integral in detail,
which will be done in this section and in section 2.5.
The suffix a will denote that a quantity pertains to a particle for the
density matrix of which a kinetic equation is set up. Subsequently this
suffix will be dropped but in this section its introduction proves useful.
The trace of the collision integral (2.98) over particle variables is shown
to be identically equal to zero:
Tra S = Trab{T(p X pb)ftt _ ft(p x pb)T}
in
= ^ТгаЬ{(ЯТ'-ТЯ)(рХрь)} =0, (2.101)
in
since
ТЯ = Я1УЯ = Я1УЯ = ЯТ
as the interaction Hamiltonian W is Hermitian. Thus, collision integral
(2.98) ensures the fulfilment of the condition (2.35) which means the
conservation of the total number of type a particles.
The Moller operator contains a unity operator,
Я = 1 + К, (2.102)
as a component (see problem (3)); owing to this the collision integral may
be written as
S=-S(1) + S(2), (2.103)
S(1) = ITrb{T(p X Ph) - (p x pb)T}, (2.104)
s(2) = ^Trb{T(p X ph)K* - K(p X ph)T}. (2.105)
The components S(1) and S(2) will be respectively called the out-term and
the in-term.
For subsequent analysis it will be more convenient to use the internal
degrees of freedom of an atom in the energy representation and the
variables of its centre of inertia in the momentum representation. Let a
and (3 designate the set of quantum numbers of particles a and b and pa
and pb denote the momenta of their centres of inertia. On introducing
these designations, the matrix element of the out-term Sm(aa'pap'a) may
34
Quantum kinetic equation for the density matrix
[Ch. 2
be written as
Sw(aa'pap'a) =2 I dPai dPai v(aa'pap'a | а^Ра^Лр^а^ХО
aiori J
(2.106)
where v(aa'pap'a | а}а\ра1р'а1) is defined by
v(aoc'pap'a | ajalPaiPai)
= T 2 I dPb Ф>ы x
И 00, J
[T(afipapb I ai01PalPbl)pb(0j3PblPb) ^a’a; 8(j>a ~ Pal) ~
daa,8(pa-pal)T*(a’fip'apb| al^p^PbOpb^iPbPw)].
(2.107)
Unlike the radiation decay the out-term of the collision integral, generally
speaking, “mixes” (see eq. (2.106)) the atomic states but only “over half
the variables”. However, under certain conditions satisfied in the great
majority of problems the quantities v(... |...) appear diagonal in both
internal and external variables, which will be discussed later. In such
situations Sm(aa'pap'a) determines the decay of the density matrix
element p(aa'pap'a) or removal of an atom from the point aa'pap'a to
other points of state space. This accounts for the chosen name of the
component S(1), i.e. the out-term.
The matrix elements of the T and К operators are connected by the
relation (see problem (3))
K(a0papb I (XlPlPalPbl)
= -2тТ(сфрарь | a,fi,pa,pM) x
- Ea + EPi - Ep + —(2.108)
\ dLfTt /
where d+(x) = 2 d(x) + i/2nx; Ea, Ep are the energies of the states a,
m, mb denote the masses of particles a and b. Therefore the matrix
elements of the component S(2) may be represented as
S<2\aa'pap'a) = 2 I dPai dPai A(aa'pap'a | a^p^p'^) x
aiori J
р{осха[ра1р'а1), (2.109)
§2.4]
Collision integral
35
where the function А(аа'рар'а | a^a^p^p'^), called the kernel of the
collision integral, is given by the relation
A(aa'pap'a | a^p^p^)
= T E [ dPbdPbi dPbipbGMIPbiPbi) x
И pptp\ J
Т\сфрарь I a$iPaiPbi)T\a'fip'apb | а^р'^р'ы] X
/ и'2 _ и/2 и/2 _ r»2\
fi/rr rr t tt tt । P а1 P& i Pbl Pb\ (
- E.. + £„ - E, + +
(n2 — n2 n2 — n2\~
Ea-Ea + EPt-Ep+^—^+^-^) . (2.110)
2m 2mb /
From the very structure of the expression (2.109) it is clear that
5(2)(aa'pap') gives the rate of transitions to the state aa'pap'a from all
the other points of the state space. Obviously, the kernel
A(jxa'pap'a | aialpaiPai) gives the density of the number of transitions
<xioc'ipaxp'ai-^ aa'pap'a per unit time.
Within the scope of the impact approximation, the expressions (2.106)
and (2.107) are the most general, including any processes that occur
during collisions, i.e. inelastic as well as elastic collisions, such as
excitation, disorientation, velocity change, phase shift of atomic oscillator,
de-excitation. Perturbing particles may be in arbitrary states and, in
particular, in coherent states (non-diagonal matrix elements of pb are
non-zero). When, for example, perturbing particles interact with an
external field, they may prove to be oriented and possess an induced
dipole moment which will be exhibited as a result of collisions. The
abundance of the above-mentioned circumstances makes the formulae
(2.106), (2.107), (2.109) and (2.110) comparatively complicated. If under
certain conditions the main role is played by one of the above-discussed
processes the expression for the collision integral is substantially sim-
plified. In addition, some simplification is attained as a result of the laws
of conservation.
Consider first the general consequences of spatial homogeneity of the
system for the collision integral. If the interaction Hamiltonian of colliding
particles W is a function of the coordinate difference of their centres of
inertia, i.e.
W(ra, rb, g, t|) = W(ra - rb, g, t|) (2.111)
36
Quantum kinetic equation for the density matrix
[Ch. 2
(g and r? are the internal coordinates), the translational motion is
separated into the motion of the common centre of inertia of particles a
and b and into their relative motion. It follows from relation (2.111) that
in the coordinate representation (see problem (4))
T(aprarb | = d(R -R^Tiapr | (2.112)
r = Га - Гъ, R = ЦаГа + рьГь,
ра = p/nv, pb = p/mb, р = тть/(т + ть), (2.113)
where R, г are the coordinates of the centre of inertia and the relative
position of the colliding particles. In the momentum representation the
analogue of the formula (2.112) has the following form:
T(afipapb | a^iPaiPbi) = ^(P - Pi)T(afy> | а&рд, (2.114)
P=Pa+Pb, P = MbPa- MaPb, (2.115)
where P and p are the momentum of the centre of inertia motion and the
momentum of relative motion.
As under the spatial homogeneity condition (2.111) (or its equivalents
(2.112) and (2.114)) the collision integral can be considerably simplified if
the distribution of perturbing particles is assumed spatially homogeneous,
i.e.
Pb(»'b,»,b) = Pb(»,b-»,0, (2.116)
or in the momentum representation
Pb(Pb,Pb) = й(рь ~Pb)Pb(Pb). (2.117)
If the assumptions (2.114) and (2.117) are used, the relations (2.106),
(2.107), (2.109) and (2.110) are somewhat simplified:
Sm(aa'pap'a) = 2 v(aa' | Л1Л;,раРа)р(а1а1РаРа); (2.118)
aia\
v(aa'| a1a[,pap;) = -^S dP x
"Ma fifii J
ГT(app I -
даа1рь(ррЛьР'а'-Р)т\а'Рр | a^1P)l;
' Ma ' -I
(2.119)
§2.4]
Collision integral
37
S(2\aa'pap'a} = S I dPai А(аа'рар'а | а^р^р^ + Apa) x
aiori J
p{axa[palpal + Apa); Apa =p'a-pa, (2.120)
A(aa’pap'a | aia{palpai + Apa)
= 2 f dpt pbifafil 1ЛЬР‘1 P1) <XPa ~Pal ~p +Pi) x
«Ma PPiPi J ' Ma '
Tfafip | aipip1)T*(a'ftp + Apa | a^ipi + Apa) x
<5_
pj -p2 + 2Mb Apa • (pt -p)
Eal — Ea, + EP\ — Ep. +
2м
(»2 — d2\ 1
Eai — Ea + EPl - Ep +E\JL) • (2.121)
£Ll f J
It is clear from the expression (2.118) that the conservation of total
momentum and spatial homogeneity of the distribution of perturbing
particles lead to the locality of the out-term with respect to the
momentum variable. The quantity v(aa' | papa) (so called out-
frequency of collision integral) is by relation (2.119) specified by the T
matrix element diagonal in p, i.e. is given by the amplitude of forward
scattering. Summation and integration in relation (2.119) imply averaging
over all the states of perturbing particles with the weight function pb(Pifi,
(МьРа _p)/Ma)) (it can be seen from eq. (2.115) that (pbpa — p)/pa is the
momentum of the perturbing particle expressed in terms of pa and p).
If we abandon the condition of spatial homogeneity of perturbing
particles then the momentum arguments of the T matrix and pb in relation
(2.119) will differ from each other by the same quantity:
Pl-P=Pbl-Pb
At the same time d(pbl —pb) in eq. (2.117) will be replaced by a function
with the effective width Apb == h/Lb, where Lb is the characteristic scale of
the non-homogeneity. The width of the sharpest structure of the T matrix
as a function of p — p^ is also determined by the uncertainty principle and
is equal in order of magnitude to h/pc (pc is the radius of collision).
Consequently, when pc«Lb, p and px in T(a(3p | a^pi) may be
38 Quantum kinetic equation for the density matrix [Ch. 2
assumed practically equal. On the contrary, Lb cannot be less than the
mean free path /. Therefore* formulae (2.118)-(2.121) hold true when
pc«l. However, and the inequality pc«l is identical to the
condition (2.96) for the applicability of the impact approximation.
Therefore, within the scope of the impact theory the conditions of
practical spatial homogeneity are fulfilled and formulae (2.118)-(2.121)
are applicable.
The in-term (2.120) retains its integral form with respect to pal even
under the conditions of spatial homogeneity, which must be a reflection of
momentum changes under collisions and collision-induced transitions
PaiPai~*PaPa- Spatial homogeneity leads only to the equality p'aX —paX =
p'a—pa = &pa, which means that spatial homogeneity is not affected by
collisions.
Pay attention to the momentum 8 function in the expression (2.121) of
the collision integral, reflecting the law of conservation of the total
momentum of colliding particles. The sum of and <5 functions is
reduced, as will be seen below, to the law of conservation of energy in
processes taking place under collisions. Summation over p,fix,P'x and
integration over p, px have the meaning of averaging the production
T(... |.. .)T*(... |...) over all the states of the perturbing particles, 8
functions and the distribution pb acting as weight factors.
The difference p'a — pa = Apa in the arguments of T*(... |...) reflecting
the spatial inhomogeneity of particles a in order of magnitude equals h/La
where La is the scale of spatial inhomogeneity; in most of the problems
discussed below spatial inhomogeneity is caused by an external electrom-
agnetic field and its scale equals the wavelength Л; in other words the
difference Apa is due to the recoil effect. In the optical spectrum and for
its not too excited states the conditions
Л»рс, Л»й/|р|, (2.122)
are fulfilled and the differences Apa can be dropped in the 8 function and
in the arguments of T(... |...). The inequalities (2.122) refer to the case
when Л«/ which is equivalent to the condition when the Doppler
linewidth kv considerably exceeds the impact linewidth (к = 2л/Л, i>2 =
2T/m, T denotes an absolute temperature in energy units). Otherwise the
minimum scale of inhomogeneity may coincide with I. In such a case the
* We abstract from the case of perturbing particles interaction with a powerful electromagnetic field
which can create spatial inhomogeneity with a scale equal to the wavelength.
§2.4]
Collision integral
39
difference Apa in the expression (2.121) for the kernel may be dropped by
virtue of the conditions
pX«l; Pc«*/\P\. (2.123)
Thus, within the scope of the impact theory the in-term of the collision
integral has the form
Sm(aa'pap'a) = 2 dPai А(аа'ра | a^p^p^a^p^p^ + Apa);
atari J
A(aafpa | ajaipai)
2л f
=7-5 2 J dpi x
"Pa fl fl, fl] J
PbWl, (MbPal -Pl)Ma) X
<5(Pa-Pal~P+Pl)X
T(afip | aipipl)T*(a'Pp | x
{<5_[£ai - Ea. + E„ — Ep + (p2 - p2)/2p] +
<5+[£ai - Ea + EPi — Ep + (p2 — p2)/2p]}. (2.124)
As has already been noted, when treating the evolution of an atom, the
quasi-classical description of translational degrees of freedom is
sufficient*. Under such conditions the Wigner representation defined by
relations (2.46)-(2.48) can prove useful. Using these transformations the
following expression for the collision integral in the Wigner repre-
sentation can be obtained:
S(g, r) = —S(1)(g, r) + S(2)(g, r); (2.125)
Sm(aa’qr) = 2 v(aa' | q}p{axa\qt)\ (2.126)
aiaj
. , i , . i /2лй\3 v, f ,
v(aa' | axax,q) =-I-----1 >, dp x
" ' Ma > flfl, J
[T(aPp | a1j51p)pb(j31/3, (pbq-p)lpa)da^ -
даа,Ръ(001, (РъЯ-p)/pa)T*(a'0p | a'iPiP)];
(2.127)
S(2)(aa'qr) = 2 | A(aa'q | a1a[q1)p(aia[q1r); (2.128)
aiai J
* It refers to the quasi-classical description of the variables ra, ra or pa, p' of the density matrix p, but
not of course of the arguments of the T matrix. Quantum effects under scattering can be essential and
are discussed later.
40
Quantum kinetic equation for the density matrix
[Ch. 2
(4л^й)^ г
A(aa'q | axa\qx) =------— 2 dpdpj Рь(0&, (pbqi - Pi)/P*) X
Ma fiflip'i J
T(a0p | al0lpl)T*(a'0p | aM x
<5(«-9i-P+Pi)x
{<5_[Eal - Ea. + EPK - Ep. + (pl — p2)/2p] +
<5+[£ai — Ea + Ep, — Ep + (p2 — p2)/2p]}. (2.129)
In the calculations resulting in eq. (2.129) we made one more
assumption, i.e.
Л » mbhlmp, (2.130)
which is a little different from inequality (2.122) but also valid for
situations of practical interest.
On comparing eqs (2.126)-(2.129) with eqs (2.118), (2.119) and (2.124)
it can be seen that passing to the Wigner representation was formally
reduced to substitution of the momenta pa and pal by Wigner variables q
and q^ analogous to classical momentum. The conditions of spatial
homogeneity discussed and employed above appeared as collision integral
locality in the coordinate r. The physical meaning of such locality is
obvious: collisional processes take place in a region with linear dimensions
pc much less than mean free paths. Therefore they may be treated as
occurring at a point if only comparatively slow variations (in space and
time) in the particles’ distribution functions are of interest. With regard to
radiation processes which are described by non-diagonal elements of the
density matrix, locality in coordinates means that radiation (absorption)
only over the mean free path is allowed for in the kinetic equation, and
radiation processes proceeding in the interaction region of the colliding
particles itself are not taken into consideration.
Apart from the foregoing conditions of spatial homogeneity of the
colliding particles’ interaction, the collision integral may be simplified for
other general reasons. As a rule, perturbing particles are not polarized,
i.e.
Рь(0Р' > Ръ) = &рр’Ръ(0>Ръ), (2.131)
which leads to a decreasing multiplicity of the sums in the expressions
(2.127) and (2.129) for v and A(aa'q | a^q^). In particular, elements of
the T matrix in eq. (2.127) are diagonal in variables of internal motion of
particle b.
Violation of the condition (2.131) may be due to interaction of the
§2.4]
Collision integral
41
perturbing particles with a powerful external field (see problem (3)). Such
an interaction results in a wide range of non-linear phenomena which are
beyond the scope of the present book and will not be discussed.
Now let us turn to an analysis of the phase factors entering into the
collision integral and to the associated time dependence. It will be recalled
that elements of the T matrix were defined on the basis of wavefunctions
4za(?) = exp(-iEar//i); 9^(rj) = ехр(-1Е^Г/й); (2.132)
consequently, the T(afip | a^pO contain the factors
exp[i(Ea - Eai + Ep — Ep,)t/h], (2.133)
which determine certain oscillations of components of the in-terms and
out-terms. If the frequencies of these oscillations are large enough the
contribution of the corresponding components will be small and they may
be dropped. The latter conclusion obviously refers to oscillation fre-
quencies exceeding l/re because, by the main assumption of collision
theory, relaxation of the elements p(aa'qr) must be considerably slower,
with a characteristic scale of the order of the mean free time.
According to eq. (2.132) the quantity v contains the factor (see eq.
(2.127))
eia; йе = Ea + Ea.x - Eai - Ea.. (2.134)
If we restrict ourselves to the cases when the distance between levels
exceeds й/тс, i.e.
|Ea -EJ» й/тс; \Ea. - EJ » й/тс, (2.135)
then of all the components of the out-term only those must be retained
where the quantum numbers a and differ only as a result of
degeneracy. Consequently, the out-term is determined by the amplitudes
of forward elastic scattering.
The product T(... |.. .)T*(... |...) in the expression (2.129) for the
kernel of the collision integral also contains in accordance with eqs (2.133)
and (2.131) the factor (2.134). Note that the difference of arguments of
the <5_ and functions in expression (2.129) is equal to he. Therefore
under the condition (2.135) the components with e = 0 are retained in S(2)
and the sum <5_ + in expression (2.129) is reduced to the 8 function.
The equality e = 0 is satisfied for the most interesting components of the
in-term, i.e.
Ea = Ea., Ea=Ea(, (2.136)
Ea = Eai; Ea. = Eai. (2.137)
42
Quantum kinetic equation for the density matrix
[Ch. 2
The case (2.136) corresponds to the excitation of the element p(aa'qr)
diagonal in energy (Ea = Ea.) by elements p(a\a\qr) also diagonal
(Ea, = Eai, not necessarily Ea = Ea. = Eat = Eai). In the case (2.137) the
non-diagonal element is excited by the elements also non-diagonal at the
same Bohr frequencies of the corresponding transitions: toaa. = coatai. The
latter equality will probably be violated but the difference will be much
less than l/rc. These are suitable conditions when the contours of
overlapping spectral lines are discussed and the polarization exchange
between the corresponding transitions is allowed for (this problem is
studied in section 3.1). Under these conditions the sum <5_ + is usually
also substituted for the 8 function although possible ensuing calculation
errors have, to our knowledge, not yet been analysed, so that the validity
of such a procedure is open to question.
Summing up everything said about and d+ functions, we may
suggest a mnemonic rule according to which “like is excited by like”, i.e.
populations are excited by populations and polarizations by polarizations.
It must be emphasized once again that this rule is essentially based on the
supposition of perturbing particles’ not being polarized and otherwise
does not hold.
Now some substitutions of variables will be performed, owing to which
the collision integral can be written in a reduced form (simplification being
purely technical). The quantities connected with q, qb and p by the
relations
q = mv, 9b = "ibt%, p = pu, (2.138)
turn out to be more convenient variables than momenta. When transla-
tional motion is described classically the quantities v and obviously
refer to the colliding particles’ velocities and и is their relative velocity.
Instead of T matrix elements, scattering amplitudes with a dimension of
length and given by the formula
T(a$p | I x
exp[i(£a - Eat + EP — EP1)t/h] (2.139)
are often used. The appearance of the exponential factor in eq. (2.139) is
due to the fact that elements of the T matrix are defined on the basis of
the functions Фа and whereas for scattering amplitudes the basis
and is more commonly used.
On passing from momentum variables to velocities the density matrix
§2.4]
Collision integral
43
normalization must be changed
p(v,r) = m3p(q,r); Pb(i%, rb) = mJpb(gb, rb).
Then
TrJ I dv drp(v,r)\ = N
(2.140)
and the formula for calculating mean quantum mechanical values retains
its usual form:
L(r) = Tri I Lp(v, r) dv dr к
(2.141)
Taking the foregoing into consideration, we write the resultant form of
the kinetic equation as it will be used further throughout this book:
/ э \
I — + и • V )p = -i(Vp - pV) + R + S;
\at /
R = —R(1) + R(2); S = -S(1) + S(2);
S(1)(aa'w) = 2 eiav(aa' | v)p(a1a[vr)-,
a\a\
2ith f
v(aa' | atia'i, v) = -— >. dw рь(Д, v - и) x
ip p J
(2.142)
(2.143)
(2.144)
[ffafiu | - 8aatf*(a’fiu | а]Ди)];
(2.145)
S(2)(aa'vr)= 2 eietj dv, A(aa'v | a1a'1vl)p(a1a'1v1r); (2.146)
oriori J
A(aa'v | (Xia'iVi) = 2^ i du di/, pb(fii, Vi - «0 x
pp< J
<5[u — Vj — p(u — Ui)/m]8(u2 — uj + 2 AE/p) x
o(aa'$3u | apx'ifitfitU!);
| aia'&fiiUi) =f(afiu | а^и^^а'ри | (2.147)
He = Ea - Em + - Ea.\ AE = Ea- Eai + Ep- EPl. (2.148)
The dynamic term in eq. (2.142) is written on the assumption that the
classical description of the translational degrees of freedom is valid and
without allowing for external forces acting on the centre of inertia of an
44
Quantum kinetic equation for the density matrix
[Ch. 2
atom. More general expressions free from the above-mentioned restric-
tions were discussed in section 2.2 (see eqs (2.49) and (2.55)). Furthermore,
in expressions (2.144) and (2.146) phase factors are retained which are
essential when overlapping line profiles are being analysed. The frequency
v(aa' | v) in eq. (2.144) and the kernel A(aa'v | aiajvj differ
from the quantities introduced earlier by the factor exp(ier) and we hope
that this substitution will cause no misunderstanding.
The product of scattering amplitudes in relation (2.147) is called a
differential scattering cross-section for the process aitx'iPiPiUi—> aa'fiflu.
It may describe both scattering of particles (ax = a'x, <x = a') and various
kinds of coherence (aj #= a')-
Below are given expressions for the collision integral in the most
common particular cases. By way of a first example, consider the so-called
model of relaxation constants corresponding to conditions when the
change in the velocity of particle a during collisions may be neglected. It
follows from the law of conservation of total momentum that the velocity
changes v — v, and и — и, are connected by the relation
v - Vi = (и - ux)plm.
Let Ди, Ди and Диь designate the effective widths of the scattering
amplitudes and velocity distributions of particles a and b. If we assume
that
Ди » Ди р/т; Дvb » Ди р/т (2.149)
p(a1alvir) may be factored outside the integral sign in relation (2.146)
and, in the distribution pb(Pi,Vi~ »i), entering into eq. (2.147), may
be substituted for v. As a result, the collision integral takes the form
S(aa’vr) = — S(1)(aa'w) + S(2)(aa'w)
= 2 eu'[v(aa' | v) — v(aa' | v)] x
aia\
p(ala'lvry, (2.150)
v(aa'| axa'x, v) = J A(aa'vx | axa'xv) dv!
= 2 2 f du dui рь(Д1, v — Ui)<5(u2 — u? + 2 &E/p) x
ppi J
f(afiu | а1Д1и1)/*(а'Ди | a^juJ; (2.151)
AE = Ea + Ep- Eai - EPl.
§2.4]
Collision integral
45
The quantity v(aa' | v) has dimension c1 and is called the
in-frequency. As it can be seen from the inequalities (2.149) the model of
relaxation constants can be realized as a result of both the small width of
the scattering amplitude as well as the relatively small mass of the
perturbing particles (mb«m) or for both of those reasons. It was
supposed in eq. (2.150) that the model of relaxation constants is
applicable to all the components of the in-term. As a rule, this is not so,
i.e. the conditions (2.149) are fulfilled only for some components. Then
the in-term contains both integral components and those of the type of eq.
(2.150).
The in-frequency v defined by the formula (2.151) may be treated
regardless of the model of relaxation constants. It is a zero moment of the
kernel as a function of a final velocity (velocity after collision), i.e. it
characterizes, the scattering result integral over final velocities. Therefore
the frequency v is an important characteristic of the collision integral S
and its kernel.
Now consider another particular case corresponding to collisions of
structureless particles a and b. Such a model is a basic model in the kinetic
theory of gases when transport processes are being treated, i.e. it
corresponds to the initial formulation of the Boltzmann collision integral.
In the model of structureless particles no internal degrees of freedom
appear and the out- and in-terms are
S(1)(v, r) = v(v)p(v, r), (2.152)
v(v) = f du [/(и | и) — f*(u | w)]pb(v ~ «), (2.153)
№ J
S(2)(v ,r) = J A(y | Vi)p(yu r) dvr; (2.154)
A(v | vj = 2 J du duj рь(ц — »!)d(u2 — u2) x
<5[v - Vj - (u - «г)д/т]/(« | «1)У*(и | «j). (2.155)
In these expressions, unlike in the ordinary Boltzmann collision integral,
collisions of particles a with the particles of another type b are considered.
For a description of collisions of particles a with each other it is sufficient
to substitute pb(vj - иг) for р(ух - »i).
Subsequently, a model of non-degenerate states will be widely used; in
this model the set of quantum numbers a is reduced to one number
46
Quantum kinetic equation for the density matrix
[Ch. 2
enumerating states with different energies:
SMy, r) = Vmn(v)pm„(v, r); (2.156)
vm„(y) = — S f d« Рь(Д, v - и) x
Щ p J
| л/Ju) | лДи)]; (2.157)
S™(y, r) = f Am(v | r) dVi +
2 e,e,l A(mnv Im^jvOp^XVbOdVo (2.158)
mini^mn J
Amn{v | ц) = 2 S I d« du, рь(Дъ Vi - uj x
flfly J
<5[u — — (u — ux)p/m]8(u2 — uj + 2 &E/y) x
f(mpu\mpux)f*(npu\nfiux); (2.159)
A(mnv | mxnxv^ = 2 I d» d»i Рь(Дь Vi— »i) x
flfly J
<5[u — Vi — (u — ux)y/m]8(u2 — ux + 2 AE/y) x
f(mflu | mxpxux)f*(npu | nxf}xux). (2.160)
In the relation (2.158) the term with m = mx and n = nx is specially written
down in order to emphasize that the kernel Amn(y 1u,) is due to the elastic
part of the scattering for both diagonal (m=n) and non-diagonal (m #= n)
elements. Inelastic processes described by the kernels A(mnv | mxnxvx)
(т*тип*п^) determine excitation and de-excitation of levels (m =
n, mx = Л1) and polarization transfer (m ^=n, mx =£nx) from one transition
to the other.
Collision integrals are more cumbersome when degenerate states are
treated. Let us for a definite example consider the states of colliding
particles a and b with fixed values of angular momenta J and Jb and their
projections M and Mb on the axis Oz in the laboratory coordinate system.
In the general formulae (2.145) and (2.147) the projections M and Mb will
be made explicit and the set of other quantum numbers will be designated
as n and nb:
2лй f
vfnMn'M' | nxMxn'xM'x, v) = -— >, du pb{nb, v — u) x
iP пьМь '
[f(nMnbMbu | nxMxnbMbu)8a.ai -
8aaif*(n'M'nbMbu | n'xM'xnbMbu)]‘,
(2.161)
§2.4]
Collision integral
47
AfnMn'M'v | плМ1п'1М\у1') = 2 2 I du d«! x
пъМъпыМы
Рь(пы, V! - Ui)d
V - V,---(w - Ml)
m
x
<5(u2 — uj + 2 &E/p) x
а(пМп'М'пьМьпьМьи |
MiAfi/ijAf i/ibiAfbiMbiAfbjUi); (2.162)
а^пМп'М'пьМъП^М'ъи | п1М1п']М[пыМып^Мыи]')
=f(nMnbMbu | ПгМгПыМыи^^п'М'п^М^и | n;Af;«biAfbi«i).
(2.163)
When the equalities
a=a'; <x1 = <x'i-, <x = nM\ = Э1 = Эн fi = nbMb
are satisfied, i.e. when the left-hand and right-hand arguments of the
amplitudes f and f* coincide in pairs, the quantity о = |/|2 has the
meaning of an “ordinary” differential cross-section of scattering. In the
general case о must be treated as a generalized differential cross-section
characterizing the scattering of polarized particles and coherences.
On the basis of relation (2.151) we may write for the in-frequency in the
model of relaxation constants
v(nMn'M'\n1M1niM\,v) = 2 IdwduiX
льМьлыЛ/ы
a(nMn'M'nbMbnbMbu |
Рь(лы, v - Ui)8(u2 -uj + 2
(2.164)
In the Kq representation (see section 2.2) the characteristics of the
collision integral take the form
, , i , v 2лй v* Г j z \
vfnnKq | п^к^, v) = -— 2 du pb(nb, v - u) x
пъМь
S -M'\xq)x
MM'MM
(-1Лл«(лад-м;|^)х
[f(MMbu\M1Mbu)da.a,-
daaJ*(M’Mbu | М[Мьи)]-, (2.165)
48
Quantum kinetic equation for the density matrix
[Ch. 2
A(nn'Kqv | n1n'xK1qxvx) = 2 2 I d« d«i pb(nbx, Ц - uj x
льпы
<5[v — Vj — (u — ux)p/m] x
<5(u2 — u2 + 2 &E/p) x
y/2Jb+ 1 V2/bi + 1 a(KqOOu | Ki^OOuj);
(2.166)
v(nn’Kq | nxn'xKxqx, v) = 2 I du du! pb{nbx, v, - uj x
ЛЬЛЫ
+ \ V2Jbl + 1 x
<5(u2 — u2 + 2 ДЕ I p)(j(Kq00u | Ki^iOOuj);
(2.167)
o(KqKbqbul K1q1Kblqblu1)=
MM' M\M\ МьМьМыМь!
- M' I Kq) x
{JXMXJ! — M11 Kxqx) x
(-^-“’-^’(ЛЛ^Л - M'b | Kbqb) x
(•/ы-Л/ы^Ы ~ Mbx | X
o(MM'MbMbu | MxM'xMbxMbxux).
(2.168)
For simplicity, relations (2.165)-(2.168) are written with all the indices
but к, q and M omitted in the amplitudes. It should be recalled that these
expressions are based on the assumption that the distribution of perturb-
ing particles is diagonal in nb and Mb and is independent of Mb.
The expressions for v(MM' | MXM'X, v), v(MM' | MXM’X, v) and
A(MM'v | AfiAfJvi) can be simplified with respect to angular variables, as
was done for translational degrees of freedom by proceeding from the
assumption of the spatial homogeneity of the distribution of perturbing
particles and from the law of total momentum conservation. In the
present case the latter is analogous to the law of conservation of the total
angular momentum of the colliding particles, which is made up of the
angular momenta of internal and relative motions. This law is fulfilled if it
is possible not to allow for external fields when collisions are considered.
Otherwise, as has already been noted, a wide range of non-linear
phenomena arise which are beyond the scope of the present book.
The analogue of the condition (2.116) of a spatially homogeneous
§2.4]
Collision integral
49
distribution of perturbing particles is the requirement of its isotropy with
respect to any of the perturbing particles, i.e.
рь(пь, 1>ь) = pb(nb, |t%|) = pb(nb, |v - h|) = pb(nb, |uI). (2.169)
The relations (2.169) are valid if pb(nb, vb) is isotropic and the relative
velocities и significantly exceed those of particles a. These relations may
be regarded as a formulation of the so-called model of isotropic collisions
[21] which is of major practical importance as within the scope of its
applicability simple solutions of many problems of the spectroscopy
of degenerate states appear possible.
In order to simplify expressions for collision frequencies it is convenient
to employ the physical equivalent of the law of conservation of total
angular momentum, i.e. the space isotropy rather than the law itself. The
derivation of the relations given below, which is based on the law of
conservation of total angular momentum, is discussed in problem (5).
Consider first the expression (2.161) for the out-frequency. The
scattering amplitudes f entering into eq. (2.161) as well as the frequency v
are specified in the laboratory coordinate system. To carry out explicit
integration in eq. (2.161) with respect to the directions of relative velocity
f(MMbu | MiMbUi) may be expressed in terms of scattering amplitudes
f(M'Mbu | M\M'bu) in a coordinate system the axis Oz of which is directed
along и (и system). By the general rule of the transformation of matrix
elements calculated on the basis of the eigenfunctions of the angular
momentum operator (see Appendix II) we have
f(JMJbMbu | hMJbMbU) = £ J(JM'JbM'bu | x
М'М{МьМы
DJMM'(afiy)DJ£M.(<xfiy) x
D^U(^r)^.(^r), (2.170)
where О'тт(сфу) is Wigner matrix, and a/Jy are Eulerian angles
connecting two coordinate systems xyz and x’y'z’. Rotation of axes
determined by the Eulerian angles is performed in three stages (fig. 2.1):
(1) rotation through an angle <x about the axis Oz until the axis Ox
coincides with the projection of the vector и on the plane xOy; (2) rotation
about the axis 0y' through an angle Д until the axis Oz coincides with the
direction u; (3) rotation about и (axis Oz') through an angle y. It must be
clear from the previous discussion that the Eulerian angles a and Д are
identical to the spherical angles of the vector и in the initial coordinate
system xyz\ <x = q>, /3 = в. Consequently, integration with respect to
directions in eq. (2.161) corresponds to integration over the Eulerian
50
Quantum kinetic equation for the density matrix
[Ch. 2
Fig. 2.1. Eulerian angles.
angles a and Д. As a result of the space isotropy, the forward scattering
amplitudes /(.. .u |... u) are not changed under rotation about the
direction и through an arbitrary angle y. The same is true of pb(nb, v —u)
if we assume the model of isotropic collisions (2.169). Therefore in eq.
(2.161) we may add integration with respect to the third Eulerian angle y.
On substituting the expression (2.170) into eq. (2.161) summation over
Mb is possible if the relation
2 DU,-(a/3y)D;mmi(^y) =
(2.171)
is used (see Appendix II).
Integration with respect to Eulerian angles referring only to the
remaining two D matrices is carried out by the formula (see Appendix II)
J sin 0 d£ da dy = — (2.172)
Combining the calculation results we find that
| ЛМЛМ0 = (2.173)
§2.4]
Collision integral
51
V(JJ') = X [ d“ Pb(Jb, l«I) X
[(2J + Vr'f(JMJhMhu | JMJbMbu) -
(2J' + l)~lf\J'MJbMbu I J'MJbMbu)].
(2.174)
Consequently, the out-term of the collision integral in the model of
isotropic collisions has the form
SW(JMJ'M', v) = v(JJ')p(JMJ'M', v). (2.175)
According to relation (2.174) the out-frequency is given by the amplitude
of forward scattering, the values of all its arguments remaining the same.
It should also be emphasized that v is not only diagonal in JJ' and MM'
but also independent of MM'. In other words, v(JJ') is a scalar and may
be calculated from the formula (2.174) using the values of /(. .. |...)
obtained for an arbitrary coordinate system and not only in the и system.
The values of the relative velocities in the model of isotropic collisions
are assumed to exceed significantly those of the velocities of particles a
(see relations (2.169)) or, in other words, the velocities of particles b are
much greater than those of particles a. The latter means that perturbing
particles possess smaller masses or higher temperatures or both. There-
fore it stands to reason that the out-frequency is independent of v as is
clear from expression (2.174).
The fact that the out-frequency is diagonal in M and independent of M
in the Kq representation means, as can be readily shown, that in the Kq
representation it is diagonal in к and q and depends on neither к nor q'.
v(xq I ^i<7i) = 5rridw,v(JJ'). (2.176)
Within the scope of the model of isotropic collisions the in-frequency v
is also diagonal but unlike the out-frequency the diagonality of v exists
only in the Kq representation. This statement can be proved analogously
to the proof of the v case. The generalized differential cross-section о
written in relation (2.167) for the laboratory coordinate system must be
expressed in terms of its d values in the collision coordinate system (or
uui system), i.e. the system whose zOy plane contains vectors и and
a(Kq00u | = 2 ^.(ф^ф) x
d(Kq'00u | K1q[00u1)D;;;i(<p10i(p). (2.177)
The axis Oz in the uut system is directed along un and the axis Oy lies in
52
Quantum kinetic equation for the density matrix
[Ch. 2
Fig. 2.2. Eulerian angles of the plane h, Ht.
the plane uu}. The Eulerian angles вг and <p connect the laboratory
system xyz and the collision system xyz, the angles tp, and 0, being, by a
general rule, the azimuthal and polar angles of the vector in the initial
laboratory system (fig. 2.2). Therefore integration in relation (2.167) with
respect to the direction of the vector is nothing but integration over
and 0i. As for integration over u, it may be performed in the coordinate
system x'y'z (see fig. 2.2) where <p and 0 are azimuthal and polar angles
of the vector u. Therefore
du duj = u2u2 du dU] sin в sin 0t d0 d0i dtp dtpx.
As a result of space isotropy the differential cross-section a depends only
on the angle в and is invariant under the spatial orientation of the system
xyz, i.e. it is independent of the Eulerian angles вг and <p. If collisions
are isotropic (condition (2.169)), the angles <p15 0] and <p in the integrand
of relation (2.167) appear only as the arguments of the D matrices and
integration over them, by eq. (2.172), yields
v(nn'Kq I п^п'гК^г) = dKKIdwv(nn'к |и^к);
Jpoc /.oe fJl/2
I I u2duu2dui| sin в dd x
о Л)____ Jo
V2Jb+l V2Jbl + 1 Ръ(пЬ1, и,) x
d(u2 — u2 + 2 AE/g) x
S a(Kq'OOu | k^'OOui). (2.178)
Zk + 1 a'
§2.4]
Collision integral
53
Thus in-frequencies are diagonal in к and q and independent of q *.
Unlike the out-frequency, in the general case v is к dependent as a result
of which the in-frequency in the JM representation is fundamentally
non-diagonal:
v(nMn'M' | X
(JMJ' -M'\Kq){JMX -M{\Kq)x
vfnn'K I п}п{к). (2.179)
According to the properties of Clebsch-Gordan coefficients (see Appen-
dix I)
= (2.180)
i.e. the isotropy of space and the distribution of perturbing particles are
reflected in JM representation as the above equality.
When radiation relaxation was discussed (section 2.3) a similar result,
i.e. the diagonality of relaxation terms in the Kq representation and its
absence in the JM representation, was mentioned. The theorem proved
demonstrates that the polarization moments representation for the density
matrix is fruitful.
The in-frequency as well as the out-frequency in the model of isotropic
collisions are independent of the velocity v. This was explained in the
discussion of eqs (2.173)-(2.175).
The expression for the in-frequency may be transformed in such a way
that it does not contain the differential cross-section a specified in the
collision system. Because of the summation over q' the expression (2.178)
is a scalar whose value does not depend on the choice of coordinate
system. Therefore instead of eq. (2.178) we can write
у{пп'к | пхпхк) = 2 2 I 5(и2 — и? 4- 2 AE/p) x
«ьлы
\^4+Тл/2ЛГ+Т (
-----Z—“j---------Рь(Лы, wt) x
i* A
a(icq'00u | Kq'00ut), (2.181)
q'
where the coordinate system for о can be selected arbitrarily.
Within the scope of isotropic collisions the in-frequencies possess some
properties which can be proved by means of the above relations (see
* This result was first obtained for elastic scattering [22,23].
54
Quantum kinetic equation for the density matrix
[Ch. 2
problem (6)). Although generalized differential cross-sections (2.163) are
complex, in-frequencies in kinetic equations diagonal in the scalar indices
n = aJ are real:
у(ппк 1 ntntx) = v*(nnK |
(2.182)
The Clebsch-Gordan coefficients are real, and therefore in the M
representation as well
v(nMnM' | nxMxnxM\) = v*(nMnM' | nxMxnxM\). (2.183)
In contrast to eq. (2.182) in-frequencies describing the exchange between
transitions are, generally speaking, complex:
Re vfnn'K | П1П\к) #=0, Im v(nn'K | пАп\к) #=0, (2.184)
пФп', п^п\.
Further, in-frequencies for к = 0 are positive:
v(nn0 1 n j n ,0) > 0. (2.185)
Physically this is an obvious conclusion, since density matrix polarization
moments of the zeroth order are proportional to the populations of levels.
At the same time, in-frequencies у(ппк | пАп}к), к=/=0, describing the
relaxation of anisotropy may be of arbitrary sign. Less trivial are
relationships between frequencies differing in к values:
у(ппк | п^п^к) v(nn0 | n^jO), (2.186)
Re v(nn'K | п}п\к)
~(2J' + 1)(2J, + 1)T/4
. (2J + 1)(2J; + 1).
\/v(nn0 | Л!Л|0)г(л'л'01 nJnJO) . (2.187)
It follows from inequality (2.186) that the rate of population relaxation is
less than that for polarization moments of non-zero order. The inequality
(2.187) is related to the problem of spectral line broadening when к = 1.
Physically it implies the existence of broadening due to phase modulation
and disorienting collisions (see section 3.1).
It is easily verified that the kernel of the collision integral is non-
diagonal in both JM and Kq representations (in M and Kq respectively).
At the same time in the model of isotropic collisions the kernel moments
{|V - vtI'} = [v(k I к)] 1 |v - Vj|' A{Kqv | k^v,) dv
(2.188)
§2.4]
Collision integral
55
of an arbitrary order / are diagonal in к and q in the Kq representation and
are independent of q. The proof of this statement is analogous to that
referring to the in-frequency. This circumstance enables one to hope that
the main role is played by diagonal (in Kq) kernels and that the
non-diagonal kernels can be neglected:
A(Kqv | = dKKXdqqiA(KV | KVj). (2.189)
Relation (2.189) is a kind of postulate and errors due to its violation have
not yet been studied. Nevertheless, the approximation (2.189) is some-
times used since it enables one to solve the problems of the non-linear
spectroscopy of degenerate states comparatively easily.
Together with the frequencies v and v there are such important
characteristics of the collision kernel as the persistence ratio for atoms and
the transport frequency of collisions. After Chapman and Cowling [24]
these quantities may be introduced as follows. The ratio A(y | vt)/v
signifies the probability of atom’s possessing velocity v after a collision
provided that its velocity before the collision was Vj. Consequently, the
mean velocity of an atom after collision is
1
v(vi)
(2.190)
where for simplicity all the indices of the internal states of the atom were
omitted. It must be emphasized that integration in relation (2.190) is
performed with respect to the first argument v of the kernel A(v | Vj), i.e.
averaging is carried out over velocities after the collision and the value of
the velocity v, before the collision is fixed.
The mean velocity {v} as specified by the formula (2.190) has a simple
and clear physical meaning for the case of elastic scattering. As a matter
of fact, Chapman and Cowling [24] introduced {v} for structureless
particles when elastic scattering is the only result of col’isions. In the case
of inelastic scattering the notion may be extended to any scattering
process using the kernel A(nnv | n}n}v}) which describes a transition of
atoms from state n} to state n. The meaning of {v} becomes more
complicated when the kernel describes a velocity change for a coherent
state of an atom, e.g. when the kernel is A(nn'v | nn'v,), n tn'. In this
case the kernel and frequency v(nn' | nn', vj, generally speaking, are
complex and the mean velocity {v} may prove to be complex also.
Consider elastic scattering and the simplest model of non-degenerate
states. Assume that the distribution of buffer particles is isotropic in the
laboratory system, i.e. pb is a function of |t%|. Then by relation (2.159) we
56
Quantum kinetic equation for the density matrix
[Ch. 2
can obtain
{v}=jr(|v1|)v1, (2.191)
Jrdvj) = 1 - v,r(|vi|)/v(vi), (2.192)
where the following designations are introduced:
v,r(vi) = 4 [ v’' “i“ia«X“1)Pb(|vi - Mil) d«i (2.193)
m+mbVi J
<7tr(Uj) = [ (1 - cos в)о(и | «J du, (2.194)
в being the scattering angle (the angle between и and u,). лг(|Г]|) is called
the persistence ratio for an atom of velocity Vj before collision. The
quantities o,r(u) and vtr(vi) are the transport cross-section and the
transport frequency of collisions.
According to relation (2.191) the mean velocity of an atom after
collision is directed along the velocity Vi before the collision. The
coefficient я(|vj) characterizes the persistence of velocities after collision.
It can be seen from the formula (2.192) that я(|Vj|) < 1; this implies that
collisions with the buffer gas are sure to decelerate the atom. The degree
of deceleration is characterized by the quantity 1 - лг(|v,|). If buffer
particles are sufficiently light (m»mb) then 7r([v,|)~ 1, i.e. a single
collision does not significantly change the velocity of the atom and for any
marked deceleration there must be many collisions.
If scattering is mostly through small angles then vtr«v and also
7r(|vi|) «= 1. If the scattering (in the centre-of-mass system) is mostly
isotropic and m«=/nb or m<mb, {v} is decidedly less than v15 i.e. the
atom is decelerated significantly by a single collision.
Under certain conditions the effective velocity change may prove to be
comparatively insignificant. From the expression (2.147) for the kernel it
immediately follows that
v - Vi = ~ (2-195>
m + mb
Therefore the velocity change v - may be small for mb « m or at small
и — ut or when both of these conditions are fulfilled. The first condition is
physically obvious because light buffer particles certainly cannot substan-
tially hinder the motion of a heavy atom. This is reflected in relations
(2.193) and (2.192) according to which v,r(v) decreases as mb/m decreases
and the persistence ratio approximates unity. The second condition (small
§2.4]
Collision integral
57
и — «J implies that scattering in the centre-of-mass system is sharply
directed, as is often the case.
Under the considered conditions the collision integral is readily
transformed to the differential form. On separation of the inelastic part
(v - v)p consider the elastic part Se (for simplicity indices of internal
states are omitted):
Se = -vp(v) + J A(v | v1)p(v1) dv,. (2.196)
Employing formula (2.151) and the relation
A(v | vj exp(-v?/v2) = A(v, | v) exp(-v2/u2), (2.197)
which obviously results in the collision integral’s vanishing under equi-
librium conditions (for details see section 2.5, formula (2.246)), we obtain
5e = -J A(v, | v){p(v) - exp[(v? - v2)/v2]p(vi)} dv,. (2.198)
It is convenient to deal with the function
p(v) = exp(v2/v2)p(v),
because for this function the collision integral is the simplest:
& = 5e exp(v2/v2) = - J A(v, | v)[p(v) - p(v,)] dvt.
Let the kernel A(v, | v) be a sharper function of v, than p(v,). Then
p(v,) can be readily expanded in a power series of v, — v; confining
ourselves to the first three terms of the expansion we obtain
(2199)
ap dv dvp
where the vector b(y) and tensor cap(y) are given by
b(v) = J (v — v,)A(v,1 v) dv, (2.200)
cat>(y) J (v - v,)a(v - ^^(v, I v) dv, (2.201)
In the model of non-degenerate states the vector b(y) can be easily shown
to be proportional to v (compare with eqs (2.190)-(2.192)):
b(y) = v,r(v)v. (2.202)
58
Quantum kinetic equation for the density matrix
[Ch. 2
The tensor cat>(y), which is a set of the second moments of the kernel, is
axially symmetrical (the axis of symmetry is directed along v) and is
expressed in terms of the transport cross-section о,г(и) (2.194) and the
cross-section (see problem (10)):
J (1 - cos2 0)о(и | «J dfi. (2.203)
It must be stressed, however, that the v dependence of v,r(v) and
cafi(y) exceeds the accuracy with which the equation (2.199) is derived.
The corrections to v,r(v) and саР(у) due to their dependence on v are of
the order of magnitude of mb/m and to take them into account it is
necessary to retain in expression (2.199) the omitted expansion terms (at
least of the third and fourth orders). Therefore to the approximation
(2.199) the quantities vtr(v) and cap(y) must be calculated at v—>0. In
this limiting case the tensor cafl is isotropic, i.e. it is equivalent to a scalar
whose value is given by
c = |v2v,r. (2.204)
Now turn from the function p(y) to p(y). The collision integral S for
the latter has the form
S = -(v - v - 3v,r)p(v) + vtrr • V„p(v) + iv(rD2 A„p(v)
= —(v - v)p(v) + v,r div„[vp(v)] + fay2 A„p(v). (2.205)
The term containing A„p evidently describes diffusion in velocity space
and c = vtrv2/2 is the diffusion coefficient. The coefficient v,rv of V„p is the
friction force divided by the mass (compare eqs (2.205) and (2.56)),
describing the action of the buffer gas on the atom and directed opposite
to v.
The kinetic equation with the collision integral (2.205) is called the
Fokker-Planck equation or is referred to as the diffusion approximation.
The diffusion approximation can be applied to collision integrals both
diagonal in internal indices (i.e. for populations) and non-diagonal. The
passage to the diffusion approximation for the elastic part of the collision
integral is at first sight an obvious consequence of the model of isotropic
collisions (2.169)*. This is the usual attitude to analysing traditional
problems of gas kinetics when the scale of the non-equilibrium part of the
* The v independence of the frequencies v, v and vtr, c is of the same origin. It should be borne in
mind that approximation (2.169) is sufficient when calculating the even kernel moments (v, c). In
order to calculate odd moments, e.g. vtr, the first correction of eq. (2.169) due to anisotropy must be
allowed for.
§2.4]
Collision integral
59
velocity distribution is of the order of the thermal velocity v. The situation
in non-linear spectroscopy is quite different. The external electromagnetic
field may lead to a very sharp non-equilibrium structure with a width of
order Г/к where Г is the half-width of the spectral line (spontaneous or
impact) and к is the wavenumber. Under those conditions for the
diffusion approximation to be applicable under an arbitrary scattering law
it is necessary that
\/mblm «Г/кй, (2.206)
which is not very often the case, as Г/кд may, for instance, be of the
order of 10-2. Therefore in order to apply the diffusion approximation to
the problems of non-linear spectroscopy not only must the role of the
mass ratio mb/m be analysed but also the particular properties of the
differential cross-section.
If scattering mostly occurs through small angles with a characteristic
value of 0O (in the centre-of-mass-system) the applicability of the diffusion
approximation is ensured by the inequality
у/ть/(т + тъ) в0« Г/kv (2.207)
where 0O plays the leading role and the value of the mass ratio is not
critical. The condition (2.207) may be fulfilled by small 0O and at mb > m.
As is well known, under collisions of heavy particles the differential
cross-section contains almost isotropic and sharp parts (for more details
see section 2.5). Therefore in certain problems the collision integral is also
broken into almost isotropic and sharp parts and the diffusion approxima-
tion is applied to the latter.
As has already been noted, the model of isotropic perturbation (or
isotropic collisions) discussed above is most frequently used when solving
certain problems and the choice is due to the comparative simplicity of
this model. Now consider the case when the velocities v of atom a cannot
be thought of as much less than the velocities ц, of buffer particles.
To make the matter clear, consider an atom with a certain velocity v
and change over to its coordinate system. In this coordinate system the
buffer gas, isotropically distributed in the laboratory system, will move as
a whole with a flow velocity — v. Thus, atom a acts as if “blown over by
the wind” of buffer particles, the velocity of the “wind” being equal to
-v. It is quite obvious that the perturbation of atom a by the “wind” of
buffer particles is characterized not by spherical but by axial symmetry,
the axis of symmetry being collinear with v. The absence of spherical
symmetry means that there is collision or perturbation anisotropy. The
60
Quantum kinetic equation for the density matrix
[Ch. 2
latter will become more pronounced as the isotropic component of the
velocity “inside the flow” decreases compared with the “wind” velocity,
i.e. the smaller vb is in comparison with v. The friction force v,rv and
anisotropy of the diffusion tensor cap(y) are the simplest manifestations of
the “wind effect”.
The limiting case
v = \/2Tlm » ub = V2Tb/wb (2.208)
is of special interest where collision anisotropy is most pronounced. The
condition (2.208) implies that under closely spaced values of the tempera-
tures T and Tb buffer particles are much heavier than atoms a, i.e.
mb»m, (2.209)
and buffer particles may be assumed to be practically stationary. In other
words, the velocity distribution of buffer particles may be described by <5
functions:
Ръ(Р, IK,) = рь(Д)<5(ц,) = Ръ(Р)д(у - и). (2.210)
Formulae (2.209) and (2.210) enable one to simplify substantially the
expression for the kernel of the collision integral; in the argument of the <5
function in expression (2.147) it will be assumed that fi = m after which
relations (2.145)-(2.147) take the form
z >1 I \ 2лЙ VI ZZ>\
v(aa' | atai, v) = -— 2 Рь(Д) x
W p
[f(a0v | ai£v)<Vai- daa,f*(a'Pv | a[0v)];
(2.211)
A(aa'v | aia[vi) = 2 2 pb(0i) x
PPi
<5(v2 - v? + 2 ДЕ//п) x
ofaa'Pv 1 a^ai/JiVi); (2.212)
v(aa' | »!»;, v) = 2 2 pb(0i) x
pp,
J <5(v? — v2 4- 2 AElm)o{<xa'Pvx [ ax<x'xf}xv) dvt
(2.213)
Therefore, in this model of collisions with maximum anisotropy, integration
with respect to relative velocities is performed in the explicit form,
§ 2.5] Frequencies and kernels of the collision integral 61
scattering amplitudes and differential cross-sections contain v and vt as
arguments, and also the law of conservation of energy is formulated with
the help of v and t^. Physically these conclusions are quite clear as in the
case of almost stationary buffer particles whose relative velocities practi-
cally coincide with the velocities of atoms a(« = v — t%«v).
The model based on approximation (2.209) and (2.210) was introduced
into gas kinetics theory by Lorentz as early as 1905. Therefore the
relations (2.211)-(2.113) will subsequently be referred to as the Lorentz
model. As has been noted, the Lorentz model corresponds to collisions
with maximum anisotropy and the maximum manifestation of the “wind
effect”.
Collision anisotropy may appear, in particular in the v dependence or,
to be more exact, in the |v| dependence of the out- and in-frequencies. In
the Lorentz model this dependence is pronounced to the greatest degree.
From the relations (2.145) and (2.147) it can be seen that the v
dependence is due to the velocity distribution of the buffer particles
pb(J}, v — u) and the argument of this distribution on v — и proves the
above interpretation of collision anisotropy as a consequence of the “wind
effect”.
Within the scope of the model of non-degenerate states the v
dependence of v, v will be the only manifestation of the “wind effect”.
The same is true of the degenerate states but only provided that there is
no collision disorientation. In the opposite case the matrices
v(nn'Kq | прг\к^, v), v(nn'icq | прг'^к^, v)
not only are v dependent but also prove to be non-diagonal in xq. The
last conclusion applies to the collision integral kernel as well.
2.5. Frequencies and kernels of the collision
integral
In the present section some properties of the in- and out-frequencies are
analysed as well as those of the kernels of the collision integral. For
convenience, the necessary relations (2.145), (2.147) and (2.151) will be
62
Quantum kinetic equation for the density matrix
[Ch.‘2
rewritten as
v(aa' | v) = {f(a | аг)да.а. - 8aaif(a' | a\)), (2.214)
v(aa' | »!»;, v) == ((/(a | | »[))), (2.215)
u
8 v - Vi----------(« - «i
. L m
(2.216)
Here angular brackets (...) denote averaging (over fl, u) of bracketed
expressions with a weight function (2nft/ig)pb(/?, v — u) and the double
angular brackets ((...)) designate averaging (over /?,/?!,«,«!) with a
weight function 2рь(/?15 v -th) 8(u2- uj + 2 АЕ/ц) for v and with a
weight function 2pb(/?i, Vi ~ «i) 8(u2 - u2 + 2 AE/g) for A. For sim-
plicity, in relations (2.214)-(2.216) variables fl and и are omitted, e.g. for
(...) and ((...)) respectively
f(a | aO =f(aflu | а^щ), f(ac | a,) =f(acflu | а^и,).
It can be easily seen from eqs (2.214)-(2.216) that simultaneous
transposition of indices a+*a', is equivalent to complex
conjugation:
v(aa' | axa\, v) = v*(a'a | v),
v(aa' | axa\, v) = v*(a'a | oflai, v),
A(aca'v | а^а'м) = A*(a' av | a,[a'1v1).
(2.217)
Relations (2.217) are obviously due to the hermiticity of the density
matrix. It can be also seen from eq. (2.214) that
2 Re v(acac' | aa', v) = v(aa | aa, v) + v{a'a' | a'a', v). (2.218)
Note that Re v may be both positive and negative (see eq. (2.215)).
Consider the conclusions concerning the frequencies of the so-called
optical theorem. Owing to the hermiticity of the interaction Hamiltonian
of colliding particles the equality
$ГТ = ГП (2.219)
is satisfied which ensures that the trace of the collision integral vanishes
(see section 2.4). On introduction of the operator К = 52 — 1 eq. (2.219) is
transformed into the equality
Т-Т = ТК-КТГ, (2.220)
which is the operator form of the optical theorem. In the energy
§2.5]
Frequencies and kernels of the collision integral
63
representation the matrix elements of the operators T and К are
connected by the formula (2.108) which enables one to obtain from eq.
(2.220) a relation containing only the elements of the T matrix. If we
allow for the law of conservation of total momentum (2.114) and take into
consideration scattering amplitudes by using the formula (2.139), it
follows from eq. (2.220) that
Imf(apu | apu) = ^~ 2 f l/(*i0i«i I ^«)l2 x
a]/3, J
d(u2 — и? + 2 AE/g) d«]. (2.221)
Sometimes it is this relation that is called the optical theorem. Each term
of the sum over a} and p} is proportional to the cross-section o(a'1]811 afi)
of the scattering process оф—>аф1 (either elastic or inelastic; see, for
example, ref. [1]), i.e.
u, f 2 AE
о(асфг | аф) = — J \/(аф}и} | a0«)|2 du,; u? = uj + ——,
(2.222)
so that relation (2.221) may be rewritten as
Imf(acpu | apu) = o(ap), a(ap) = 2 o(aip11 ap),
4ЛП atp,
(2.223)
where o(aP) is the total scattering cross-section in the state оф.
The optical theorem is employed to find the relation between the out-
and in-frequencies. From eq. (2.214) we have
v(aa' | ata', v) = {f(a | a) —f*(a' | a'}}. (2.224)
First consider the case a = a'. Substituting in relation (2.224) the
difference f(a | a) —f*(a | a) using eq. (2.221) and comparing the
resulting expression with eq. (2.215) we obtain
v(aa | aa, v) = S v(a’ia'i I aa> »)• (2.225)
«1
Therefore, the out-frequency v(aa | aa, v) is a sum of frequencies
corresponding to the transitions ar—» a, a—>at. Separating the first of
them we obtain
v(aa\ aa,v) — v(aa \ aa,v) = У v(a,a} | aa, v), (2.226)
64
Quantum kinetic equation for the density matrix
[Ch. 2
i.e. the difference between the out- and in-frequencies is caused by the
particle transitions a—> ax, a}^a. Since each term of the sum is positive,
the difference v — v characterizing the relaxation rate of the number of
particles in the state aav is positive too.
In the case a' #= a, the out-frequency v(aa'\aa',v) is complex,
generally speaking. The optical theorem makes it possible to obtain the
relation for its real part:
2 Re v(aa' | aa', v) = У [*(aiai I cca,v) + I oc'a', v)],
(2.227)
i.e. the real part of the out-frequency v(aa' | aa',v) (which, according
to eq. (2.218), equals to the arithmetic average value of the out-
frequencies) is expressed in terms of arithmetic averages of the in-
frequencies v(aiai | aa, v) and v^a, | a'a', v). Unlike relation (2.226)
the difference between the out-frequency v(aa'\aa',v) and the in-
frequency v(aa' | aa', v) is caused not only by a change in the number of
particles as a result of transitions a, a'—Separating from the sum
over a, in eq. (2.227) terms with a} = а, аг = a', we obtain
2 Re[v(aa' | aa', v) — v(aa' | aa', v)]
= У ¥(»!»! | aa, v) + У ₽(»!»! | a'a', v) +
v(aa | aa, v) + v(a'a' | a'a', v) — 2 Re v(aa' | aa', v).
(2.228)
The right-hand sides of the equalities (2.226) and (2.228) differ in terms
outside the signs of summation over a} in eq. (2.228). To find their
structure we employ formula (2.215), which readily yields
v(aa | aa, v) + v(a'a' | a'a', v) — 2 Re v(aa' | aa', v)
= <1Л«I «)-/(«' I (2.229)
Therefore the considered quantity is essentially positive and due to the
difference in the scattering amplitudes of states a and a'. From the
relations (2.226), (2.228) and (2.229) follows the inequality
2 Re[v(a,a'' | aa', v) — v(aa' | aa', v)]
v(aa | aa, v) — v(aa | aa, v) + v(a'a' | a'a', v) —
v(a'a'\ a'a',v), (2.230)
§2.5]
Frequencies and kernels of the collision integral
65
which becomes an equality only when the scattering in states a and a' is
identical.
The relations (2.225)-(2.230) can be directly applied to the model of
non-degenerate states. For example, within the scope of this model the
quantities v(aa | aa, v) — v(aa | aa, v) and Re[v(aa' | aa', v) —
v(aa' | aa', v)] characterize the relaxation of the number of particles in
the state a and the so-called impact width of the spectral line correspond-
ing to the transition a-a' (see section 3.1). Consequently the line width is
not less than the arithmetic average of the decay rates of states a and a'.
In the absence of inelastic processes a, a' —> the impact broadening is
caused entirely by a difference in scattering amplitudes of the states a and
a'. At the same time in this case the scattering amplitudes are determined
only by the phase change of the states a and a' during collision.
Therefote, in terms of the correlation theory of spectral line broadening,
the expression (2.229) may be said to describe a broadening due to phase
modulations or phase shifts of an atomic oscillator taking place under
collisions. If inelastic processes take place the amplitudes of elastic
scattering f(a | a) and f(a' | a') are, naturally, not independent of
inelastic processes and subdivision into amplitude and phase modulation
becomes indefinite. Nevertheless, the sums in the right-hand side of the
formula (2.228) can be treated as the amplitude modulation contribution
to the linewidth and the last three terms as the phase modulation
contribution.
It is to be especially noted that phase modulation is intimately
connected with velocity (or momentum) change under elastic scattering;
that is why both are described by the same scattering amplitudes f(a | a)
and f(a' | a'). By way of illustration, the well-known expressions for the
scattering amplitude in a spherically symmetric field are given (see, for
instance, ref. [1]):
f(a | a) = 2 (2/ + l)[exp(2ir/te) - l]Pz(cos 0), к = p/h.
ZA.K i
Here в is the scattering angle, I denotes an orbital quantum number, and
2rha is a phase shift which is the result of the interaction of an atom in the
a state with the scattering centre. Simple manipulations yield [25]
v(aa' | aa', v) — v(aa' | aa', v)
= p (S (2/ + 1){1 - exp[2i(r//a - .
66
Quantum kinetic equation for the density matrix
[Ch. 2
The differences 2(r/te — T]la.) must be interpreted as phase shifts of the
atomic oscillator corresponding to the transition a — a' under a certain
value of I. In the quasi-classical description of scattering sums over I are
transformed into integrals with respect to the impact parameter (l/k). It
may be asserted that the existence of phase modulation necessarily
involves a velocity change under collision (r}la #= 0, #= 0) and a
difference in scattering of the states a and a' (r}la #= r,/a..).
Thus analysis of the collision integral in a quantum kinetic equation
provides a foundation for a classical picture when collisions lead to a
velocity change of the atomic oscillator as well as amplitude and phase
modulation.
Recall that the collision integral in the Wigner representation is being
considered and by definition v = q/m where q is the momentum Wigner
variable. In the quasi-classical limit v is the velocity of an atom. However,
for a quantum description of translational motion all the above-mentioned
holds too if by v the quantum variable q/m is meant.
There have been numerous attempts to discriminate between “phase
interruption collisions”, “state-dependent collisions” and “velocity-
changing collisions” (see, for example, refs [26, 27]). The above discus-
sion shows that such discrimination between different collision types has
no physical foundation if we are interested in the causes of particular
results of collisions. The term “velocity-changing collisions” can only
specify conditions under which the model of relaxation constants is
inapplicable.
For degenerate states a = aJM, where a denotes other quantum
numbers. From the sum over a} in eqs (2.225)-(2.227) elastic processes
aJM^> aJMA can be separated:
I
v(nMnM |nMnM, v) = X v(nMlnMl |nMnM, v) +
Mi
У v(a}a, | aa, v), (2.231)
ai ,n\^n
2 Re v(nMn'M‘ | nMn'M', v) = У ^(иЛ^пЛ/! | nMnM, v) +
Mi
У vfjt'M^n'M'i | n'M'n'M', v) +
Mi
(inelastic part). (2.232)
Sums over Mx and M'r in eqs (2.231) and (2.232) contain frequencies of
elastic disorienting collisions.
§2.5]
Frequencies and kernels of the collision integral
67
Now assume the applicability conditions of the model of isotropic
collisions to be fulfilled. Within this model the relations
v(aa' | а,а\, v) = 6aaidaaiv(aa' | aa') (2.233)
are valid and all the out- and in-frequencies are independent of the
velocity of atoms v. Recall that in the Kq representation the following
equalities are satisfied:
v(JJ'Kq | = bjjxdJ AdKKidqqiv{JJ'), (2.234)
v(JJ'Kq | JJ\K}qA) = dKK,dqqiv(JJ'K | А-Цк) (2.235)
It can be easily verified that in the Kq representation the relation
(2.231) takes the form
v(JJ) — v(nnO | nnO) = У v(ntnfi | nnO), (2.236)
Л>#Л
according to which the difference between the out-frequency v(JJ) and
the in-frequency v(nnO | nnO) (elastic processes) is caused by inelastic
processes n—This is quite a predictable conclusions as the in-
frequencies v(nnK | ппк) under к = 0 characterize a variation of particle
number in the state n.
The impact width of the spectral line due to the multiple interaction of
order к and corresponding to the transition between degenerate states is
given by the difference of the out-frequency v(JJ') and in-frequencies
v(JJ'k I JJ'k). Under dipole interaction к = 1 (see section 2.2) and for the
above quantity we have
Re[v(JJ') — v(JJ'l | JJ'l)] = (inelastic part) + 2[v(J./0 | JJO) +
v(J'J'O | J'J'O) - 2 Re v(JJ'l | JJ'l)].
(2.237)
By using the inequality (2.187) the part of the expression (2.237) in square
brackets is easily shown to be positive: it describes the line broadening
due to the phase shift of the atomic oscillator and disorientation (this
problem is studied in more detail in section 3.1).
Now let us turn to analysing relations between the characteristics of
direct and reverse transitions. It is known that the change of time sign and
complex conjugation leave Schrodinger equation unaltered if the Hamil-
tonian is Hermitian. Therefore, the amplitudes of direct and reverse
scattering are connected by the relation (reciprocal theorem; see, for
instance, ref. [1])
f(aflu | a^lUi) = C\f(a*fi* - и. | - «), (2.238)
68
Quantum kinetic equation for the density matrix
[Ch. 2
where a*. fl* designate states differing from a, fl by the sign changes of
the projections M and Mb of the moments J, Jb; if
a = aJM = nM: fl = bJbMb = nbMb
(a and b are other quantum numbers), then
a* = n-M; fl* = nb-Mb.
The factor Сг in relation (2.238) is
G = (-ir,
o, =J - M + л - + Jb - Mb + Jbl - Mbl. (2.239)
When states a and fl are characterized by several momenta, ox is equal to
the sum of terms analogous to eq. (2.239) over all momenta.
On some additional assumptions the reciprocal theorem makes it
possible to establish the connection between frequencies and kernels
corresponding to direct and reverse transitions. We write the expression
for the out-frequency, substituting in it the scattering amplitudes using the
reciprocal theorem
v(aa' | aia[, v) = -— С У I du pb(fl, v — u) x
ig p J
[fWfl* - и | a*fl* - u)da,a. -
baaif(a'i*fl* - и | a'*fl* - u)],
C = (-l)°, o=J — M + J'— M'+ Jx —Mx+—M\. (2.240)
If the distribution of perturbing particles pb(fl, v — u) depends on |v — u|
and in addition does not depend on the momenta projections so that in pb
the index fl can be substituted for fl*, then the equality
v(aa' | a, a',, v) = Cv(at*at[* | a*a'*, —v) (2.241)
easily follows from the formula (2.240), connecting the frequencies of
direct and reverse processes. In the Kq representation the formula (2.241)
is easily shown to take the following form:
v(nn'Kq | npt'tKtqi, v) = (—l)K~q+K,~q,v(nin'1K1 - qx | пп'к - q, -v).
(2.242)
Within the scope of the model of isotropic collisions the out-frequencies
are diagonal in J, J', к and q and are independent of Kq and v (see eq.
(2.234)) so in this case the reciprocal theorem gives no new information.
Attempts to apply the reciprocal theorem to in-frequencies and kernels
§2.5]
Frequencies and kernels of the collision integral
69
of the collision integral can be successful only on more detailed assump-
tions since we deal with the relation for the frequencies of direct and
reverse transitions between states with different energies. Under an
equilibrium distribution of perturbing particles (temperature Tb) and when
the conditions of applicability of the model of isotropic collisions
(и « й = vb) are fulfilled, the relation
v(aa' | »!»]) = С exp[(Ea, - Ea)/Tb]v(a*a\* | a* a'*). (2.243)
is valid*. The Boltzmann factor** in relation (2.243) containing the
temperature Tb of the perturbing particles reflects the rate differences of
direct and reverse transitions and is closely connected with the energy
level distribution of perturbing particles; a non-equilibrium distribution
would lead to quite a different form of this factor.
The relation (2.243) holds good for the model of isotropic collisions
within which the in-frequencies are diagonal in the Kq representation (in к
and q\ see eq. (2.235)) and are independent of q. Then the relation
(2.243) can be written as
v(nn'K | п^к) = exp[(E„, - Е^/Т^п^к | пп'к), (2.244)
i.e. it is a relation between the direct and reverse processes in the literal
sense of the word. The same holds, of course, for the JM representation
as well as can be readily shown using the relation (2.244):
v(aa' | »]»]) = exp[(Eai - Ecr)/7;]v(a1a; | aa'). (2.244a)
In the case of elastic collisions (n = nx, n' = nJ) the reciprocal theorem
can be applied to in-frequencies without assuming that the distribution of
perturbing particles is at equilibrium but the assumption about collision
isotropy is necessary since the latter leads to diagonality of the in-
frequency (in Kq) in the Kq representation; the reciprocal theorem will
add nothing new with respect to the model of isotropic perturbation (see
eq. (2.244), putting nT = n and n] = n').
As far as kernels of integrals are concerned, we can obtain simple
corollaries of the reciprocal theorem on the additional assumption that the
distribution of perturbing particles is at equilibrium. If the identity
"ib[(v1 - «О2 - (v - и)2] <5[v - Vi - g(u - uj/m]
— [д(и? - и2) - m(vj - v2)] <5[v - Vi - g(« - «i)/m] (2.245)
* Proof of the relation (2.243) and the subsequent formulae (2.245) and (2.246) is in problem (7). The
procedure is analogous to that for obtaining the relation (2.241).
* * It should be recalled that within the scope of applicability of the expression (2.215) it should be
assumed that Ea - Ett| == Ea, - Ea. (see the discussion of formula (2.129)). Therefore the asymmetry
of Boltzmann’s factor in eq. (2.24^) with respect to primed quantities is only apparent.
70
Quantum kinetic equation for the density matrix
[Ch. 2
is allowed for, application of the reciprocal theorem to eq. (2.216) yields
the relation
A(aa'v | (Xiat'iVt) = CA(a*a'!* — V] | a*a'* - v) x
exp[(Ee, - Ea)/Tb + (v? - v2)/v2]; = 2Tb/m,
(2.246)
which could be useful for treating a number of particular problems. It
must be stressed that both Maxwellian and Boltzmann factors in relation
(2.246) contain the temperature of perturbing particles Tb.
The relation (2.246) takes a simple symmetric form in the case of elastic
scattering:
A(nMn'M'v | nMxn'M\vi)W{vi) = CA(n — Мрг' — M{ — v1\n —
Mn'-M'- v)W(y) (2.246a)
where W(v) is the Maxwellian function.
With the help of the above relations the collision integral may be shown
to vanish under conditions of thermodynamic equilibrium. Substituting
the equilibrium distribution (as in the formula (2.101)) into the expression
(2.146) for the in-term S(2)(aa'v) and taking into account relations
(2.215) and (2.246), we obtain
5(2)(arar'v) = p(txv) У v(a*a{* | ar* ar'*, -v). (2.247)
On application of the generalized optical theorem and reciprocal theorem
to the sum over a*, eq. (2.247) can be reduced to the form
S^(aa'v) = p(av)(f(a | a') - f*(a' | ar)). (2.248)
In contrast, the out-term of the collision integral (2.144) is by Virtue of the
diagonality of the equilibrium distribution given by
Sw(aac'v) = (f(ac | ar'))p(ar'v) — p(arv)(/*(ar' | ar)). (2.249)
Since Ea = Ea. (otherwise the collision integral oscillates rapidly and must
be truncated), the states ar and ar' differ in degeneration indices, i.e.
under equilibrium conditions we have p(ar'v) = p(arv). Therefore in-
terms (2.248) and out-terms (2.249) are equal and the collision integral is
zero.
The discussed properties of the kernel and the frequencies of the
collision integral were due to general features of the scattering ampli-
tudes, independent of the form of the interaction potential, and to general
§2.5]
Frequencies and kernels of the collision integral
71
features of the ensemble of perturbing particles. From the standpoint of
applying non-linear resonances spectroscopy to collision studies an oppo-
site statement of the problem is interesting, namely to what degree the
properties of differential cross-sections typical of a particular situation are
exhibited in the properties of the kernels.
In the simplest case of non-degenerate states and structureless per-
turbing particles, for the kernel of the elastic part of the in-term we
have
A(aa'v | aa'vi) = 2 J du duj d[v — ц — p(u — и^/т] x
д(и2 — u?)pb(vi — Ui)o(aa'u | aa'ut), (2.250)
a(aia'u I aa'ut)=f((xu | aui)f*(a'u | |u — uj). (2.251)
The differential cross-section a(aa'u | aa'tii) depends on the magnitude
of the velocity u and on the scattering angle (between u and ut). It is
convenient to take и and |u — ut| as variables, as was done in the equality
(2.251), where for simplicity the indices aa' are also omitted. The
velocity distribution of the perturbing particles will be assumed to be
Maxwellian:
. . Nb /Vi-иЛ2] 2Tb
Pb(v1-u1)= r . exp —I , Ub = — • (2.252)
(VJt ub) L \ vb ' -I mb
From the law of conservation of total momentum, velocity changes in
laboratory and centre-of-mass systems are proportional to each other, i.e.
M Z V f.
— (u-Uj) = V - Vj = £,
m
and the angular dependence of the differential cross-section (i.e. depend-
ence on |u — uj) is completely “transferred” to the kernel in the form of
the dependence on multiplied by т/ц. The vector £ will later be of
great importance. It follows from the law of conservation of energy that
- the bisector of the angle between u and ut is orthogonal to £ (fig. 2.3). On
using 8 functions in expression (2.250) it remains only to integrate with
respect to u0 orthogonal to
N г / > A(v | vj = — C(v | vj4 ct( u, - vb J \ -if) exp — (-——) du0. (2.253) u / rL \ vb / J
72
Quantum kinetic equation for the density matrix
[Ch. 2
Fig. 2.3. Relative and absolute velocities under elastic collision.
Here the following notation is introduced:
U2 = ul + (?m/2M)2; § = v - ?(? • v)/£2 = v, - £(£ • v,)/?2;
Ди f Г / 2u \12 I 1
c<*’1 / «М
2ц _ 2mb l2Tb 2\/mblm [2Tb
Ди= — vb =--------\— = --------— л/ —.
m m + mb X mb 1 + mblm X m
(2.254)
(2.255)
The vector § is a velocity component (v or vj orthogonal to £ (fig- 2.3).
In some cases the differential cross-section a(u,m£/ju) is independent
of u. This is, for instance, the situation when the cross-section is estimated
in Born’s approximation, (see, for example, ref. [1]) and also for
quasi-classical scattering through large angles in the model of the
impenetrable sphere. In such cases the expression (2.253) takes an
especially simple form:
A(v | vO = Nbvb4jt<j(™ (fjc(v | vj. (2.256)
§2.5]
Frequencies and kernels of the collision integral
73
According to the expression (2.256) the kernel contains three factors of
different kinds. The factor Nbvb, which represents the collision frequency
for unit cross-section, may be called kinetic. The quantity 4лсг(т£/д) is
obtained on solving the problem of collision between two particles and
may be named a quantum mechanical factor. Finally, the function
C(v | V!) results from differential cross-section averaging over the velo-
cities of the perturbing particles and can be naturally called a statistical
factor.
The statistical factor, generally speaking, is asymmetric with respect to
the transposition of v and vt which means that atoms a are effectively
decelerated as a result of collisions with a Maxwellian gas of b particles.
One can find a symmetric multiplier of C(v | Vj):
Av
C(v | vO = — exp[|7v£]AK_s(v - y^),
AK-s(v “ yfi) = (Vn Au)-3 exp{—[(v - y^)/Av]2}, (2.257)
у = 1 - 2ц/т = (m — mb)/(m + mb).
The function AK_s(v ~ yi^) is called a Keilson-Storer kernel [28,29] and
is the only asymmetric (with respect to transposition of v and vt)
multiplier in the kernel of the collision integral.
Sometimes the function AK_s(v — yvj is used as a model approximation
of a kernel [28,29]:
A(v | Vj) = vAK-s(v ~ yfi)- (2.258)
Such a kernel contains only one parameter у which characterizes both the
width of Av and its asymmetry. By way of illustration write
{v } = yv15 Av = vjLplm = Vl - y2 vVTb/T.
According to these relations the first moment of the Keilson-Storer
kernel {v} (mean value of the velocity v after collision; see eqs (2.190)
and (2.191)) is proportional to the velocity Vj before the collision but in
magnitude {v} is less than v1? because |y| < 1. Therefore, у is the
persistence ratio for an atom of velocity V! and 1 — у = 2ц/т characterizes
the deceleration of atoms a by a Maxwellian gas of b particles:
2m
«1 - {«} = (1 “ У>1 = (2p./m)v1 = ——--• V!.
m + mb
If /n==mb, then y«l and deceleration is almost complete; if mb«m,
then 1 —y«l, deceleration during one collision is very small and for
74
Quantum kinetic equation for the density matrix
[Ch. 2
practically complete deceleration a great number of collisions is necessary
(«(1 - y)-1).
The width Ди of the Keilson-Storer kernel is also dependent on the
mass ratio mjm. When mb and m differ greatly (m » mb or m « mb) the
width Ди is considerably less than the thermal velocity v. Since actually
the root У/т/тъ serves as the real parameter the ratio Ди/D for collisions
of atoms in practice cannot be less than 0.1 and in many cases Ди/D ~ 1.
Considerable narrowing of the function AK_s(v — yi»i) could occur during
atom-electron collisions (mblm ~ 10-4-10-5). However, the electron
temperature is usually greater by an order of magnitude (and sometimes
more) than the atomic temperature so in this case too Ди/v э* 0.1.
Note that with m » mb the deceleration effect is of a greater order of
smallness (1 — у == 2mb/m) than kernel narrowing (Ди/й = 2\/mJm).
Therefore, when particular problems are being solved, deceleration can
sometimes be neglected, when the finite width of the kernel Ди is
retained.
It follows from the expressions (2.253) and (2.256) and the above
discussion of the factor Ди/D that the Keilson-Storer model (2.258) is to
some extent justified in the case of almost isotropic scattering in the
centre-of-mass system when weakly depends on t,. This is the
case, for instance, under a classical description of scattering by an
impenetrable sphere (o(m£//x) = (d/4)2 where d/2 is the radius of the
sphere) as well as in the Bethe approximation for inelastic scattering of
atoms by electrons for the transitions forbidden in the dipole
approximation.
If m « mb, then у = — 1 and the Keilson-Storer kernel has the form
AK_s(v - уц) = <5(v + vj. .
On the contrary, the inequality m « mb corresponds to the Lorentz model
and the kernel of the collision integral is proportional to ct(v | Vj)d(v2 —
v?) (see eq. (2.212)). Thus the Keilson-Storer kernel cannot be employed
as a model when m « mb. Nevertheless, under atom-atom collisions the
differential cross-section always includes a component much sharper than
the statistical factor and the velocity dependence of the corresponding
part of the kernel is almost completely determined by the quantum
mechanical factor. Note in this respect that the factor т/ц in the
differential cross-section argument narrows it in the velocity scale (in the
laboratory system) to a greater extent than the statistical factor. Let Ди be
the differential cross-section width in the centre-of-mass system deter-
mined by the interaction type of the colliding particles. Then the width of
§2.5]
Frequencies and kernels of the collision integral
75
the differential cross-section as a function of £ (in the velocity scale,
laboratory system) is
Д£ = Ди ц/т.
If, for example, Au is due to diffraction phenomena, then
Au ~ uKIpw = h/ppw, &£, = h/mp*
is an interaction radius), i.e. Д£ is inversely proportional to the mass
rather than to the square root of it, as it was in the statistical factor.
It is qualitatively quite obvious that the differential cross-section
dependence on £ = |v — vj leads to a decreasing kernel A(v | vt) width
and degree of deceleration compared with the Keilson-Storer kernel. By
way of illustration take a model expression
A(v | v,)a AK_s(v - yvO exp[—(£/A£)2],
where the exponent acts as a quantum mechanical factor. It can be easily
shown for such a model that the effective width Avet of the kernel and
degree of deceleration 1 — yef are given by
/ Д£* Д£2
AUef = V Д£2 + Ди2 ДV; 1 “ 7ef = Д£2 + Ди2 (1 “ У)’
Therefore ДиеГ«Ди and in the limit A£«Au we have Avef== A£. The
effective deceleration decreases to a still greater extent (proportional to
(A£/Av)2) which is analogous to the properties of the Keilson-Storer
kernel. This fact justifies application of a difference kernel model to the
solution of a number of particular problems.
One must keep in mind the universal integrated singularity of the kernel
(2.253) (the factor l/£ in C(v | vj) which is closely connected with the
conservation of energy. This fact is of great importance and must be
allowed for in model approximations of the kernel.
If unlike the expression (2.256) the differential cross-section is u
dependent then without the assumption of explicit u and £ dependence of
a integration in eq. (2.253) may be performed only with respect to the
angle between the planes u,ux and v,vt (see fig. 2.3), as a result of
which we find
A(v | V!) = 8nJVbvbC(u | vj x
f”uoduo / m \ / £2 + ug\ / £u0\
I ~^“аГ’7^ехрк—W2v); (2259)
Vb \ fl / \ Vb / \ Ub /
U2 = ul + (m£/2g)2,
76
Quantum kinetic equation for the density matrix
[Ch. 2
where 4>(z) is a modified Bessel function of zeroth order. Therefore the
kernel may be obtained by a single integration. As in the case (2.256)
kernel asymmetry with respect to the transposition of v and V! is
concentrated in the factor C(v | vj (the vector according to eq. (2.254),
is symmetric). If a(u, is a sufficiently sharp function of £ = |v — vj
then for the kernel specified by the relation (2.259) as well, the main
dependence on v and Vj is due to the properties of the differential
cross-section. Consequently, when Au « й the kernel may be assumed to
be dependent on the difference v — Vj (characteristic scale Au plm) and
also there exists a weak dependence on vt (characteristic scale is u).
There cannot be a difference kernel, in the strict sense of the word, as
in this case equilibrium velocity distribution is not necessarily retained,
i.e. the second thermodynamics principle is violated. Therefore, in
analysing the evolution of a distribution possessing a non-equilibrium
component with a width of the order of the thermal velocity difference,
the kernel model is not applicable. On the contrary, when studying the
problem of the broadening of a non-equilibrium structure which is sharp
on the scale of the thermal velocity the kernel asymmetry cannot be taken
into account and the kernel may be assumed to be dependent on the
velocity difference v - Vj.
As has already been noted, the function AK_s(v — yrj is the only factor
which is asymmetric with respect to transposition of v and This
asymmetry can be represented in a different manner. The identity
exp
/v ~yvi
\ Av
/ v2 - v2\
eXp(-^)
is easily verified; in its right-hand side only the second exponent is
asymmetric. Thus a universal and very simple asymmetric factor exists in
an arbitrary kernel. This factor contains no specific parameters and is
entirely determined by the thermal velocity v. It is readily seen that the
characteristic scale of asymmetry manifestation is V2 v. This illustrates
the statement that sharp non-equilibrium structures may be analysed by
means of a difference symmetric kernel. Furthermore, a simple substitu-
tion of variables makes it possible to do away with an asymmetric factor in
the kernel. For the function
Pi(»i) = exp(v2/2v2)p(v)
§2.5]
Frequencies and kernels of the collision integral
77
the collision integral
Si = -vp/v) + J K(y | dv,
has a symmetric kernel*
Zv2 — v?\
K(v | ц) = A(v | Vj) exp^ ~2_T J = K(vj | v).
Apart from general physics considerations, the symmetrization of the
kernel may be interesting from the practical point of view, since some
important properties of integral equations with symmetric kernels simplify
the solution of particular problems (see, for example, ref. [30]).
So far the kernel has been discussed by describing elastic scattering for
non-degenerate states a and a’. Otherwise (states a and a' are
degenerate) it must be taken into account that the differential cross-
section <j(aa'u (aa'»]) depends not only on the scattering angle but also
on the angle <p between the planes u,U] and v,vt (see fig. 2.3). It is
convenient to study the kernel in a special coordinate system whose polar
axis is directed along the vector $ = v —v^ and the axis Oy is orthogonal to
the plane v, Vj (v, V! system). In the Kq representation (see section 2.4)
we have
I M
A(Kqv | Ki^iVj) = 21 du du! 8 v — Vj------(u — uj x
J L m
8(u2 - utypblvt - Ui)a(Kqu | щм). (2.260)
The differential cross-section a{Kqu | Kxqxu^ is conveniently expressed in
terms of its value in a coordinate system connected with the collision
plane u, Ui (u, u, system) since in this coordinate system it depends only
on the scattering angle. The transformation of the differential cross-
section when passing from one coordinate system to another is given by
the formula (see Appendix II)
a(Kqu | KxqxUi) = £ DKqq(afrY)d(Kq' | Kxq\, u, |u - вДО^а/Зу),
ч'ч\
(2.261)
where Dqq{a(5y) is the Wigner matrix, and аДу denotes the set of
Eulerian angles connecting two coordinate systems. In the u,u} system the
* We may come to this conclusion using the formula (2.246a).
78
Quantum kinetic equation for the density matrix
[Ch. 2
vector ? wiH also be chosen as the polar axis. Thus, since both of the
considered coordinate systems have a common polar axis, only the third
Eulerian angle coinciding with the angle <p between the planes v, and
u, ut (see fig. 2.3) is non-zero. It is known that
Р^-(00ф)=д„«-^,
and consequently
a(Kqu | | Kxqu u, |u - uj), (2.262)
where a is a differential cross-section in the collision plane depending only
on the scattering angle and the magnitude of the velocity u. Therefore, in
the case of degenerate states the differential cross-sections in the v and
u15 u systems differ by the factor exp[i(<j! — <?)<p]. Consequently, as can be
readily seen, integration with respect to the angle <p leads to the
appearance of a modified Bessel function not of zero order but of order
equal to Д = — q\:
Ги i
~T^o[Kq
Vb '
Kfau-UX
M >
expl
£2 + “o\r ЛЛ«о\
.-.2 ИМ2 г.г I
(2.263)
Comparison of eqs (2.263) and (2.259) shows that the general structure of
the kernel of the elastic part of the collision integral in the case of
degenerate systems and allowing for collision disorientation is quite
analogous to that discussed above. If the differential cross-section
| /сi<71, u, is independent of u, it follows from eq. (2.263) that
A(Kqv | Kxqxvx) = Nbvb4n,a<Kq | к^, т£/ц) x
C(v | vj
Г(1 + Д/2)
Г(1 + Л)
Ф -, 1 + д-,
\2
(2.264)
which is analogous to eq. (2.256) and differing from it only by a statistical
factor (Ф(а, у; z) is a confluent hypergeometric function).
Differential cross-sections of processes taking place under atom-atom
collisions are characterized by some very peculiar properties. The
interaction radius pw of colliding particles as a rule greatly exceeds the De
Broglie wavelength Л owing to their translational motion. Indeed,
______ti h 4.9 x 10~8
pu цй \/2jif VMr Cm
§2.5]
Frequencies and kernels of the collision integral
79
where M is the atomic weight corresponding to the reduced mass and T°
is the temperature in kelvins. For T° = 300 К and M = 10 we have
Л = 10-9cm, typical values of pw being 10-8-10-7cm. So the condition
Л « pw (2.265)
is fulfilled.
In this case scattering may be described using an approximate method
similar to the Kirchhoff method of solving the problems of optical
diffraction. Denote the set of quantum numbers of initial and final states
of colliding particles by indices i and f. Equation (2.99) for the Moller
operator in the energy representation for internal degrees of freedom, in
the coordinate representation for relative translational motion and under
condition that before collision it had been characterized by a certain value
of the velocity u, may be written as follows
[Л + (p«f/ft)2]£2fl = (2р/й2) S WA, (2.266)
/
u? = u2 + 2(Ei — Ef)/p,
where the Wf/ are the matrix elements of the interaction potential. When
the condition (2.265) is fulfilled the system of eqs (2.266) may be solved in
two steps. In a limited volume with characteristic dimensions pw
diffraction effects are negligible and eq. (2.266) may be solved in an
approximation of classical trajectories. If in addition to eq. (2.265) the
inequality
|Wf/|« T (2.267)
is satisfied, the eikonal approximation,
= Sfl exp(ipufr • Uj/ft), u, = ujui,
iW \ i г u
Ki • V + —2)sfi = - — S ЖДч exp i £ (и, -
nllf / nUf L fl
Uf)r • Uj
(2.268)
can be applied. Let Sfi(p) denote solutions of the system (2.268) in a
certain plane, given by a radius vector p and located at a distance
pw « r • «i« pw/Л from the interaction region, where already Wtj = 0 but
diffraction effects are not yet important. With large r • u, the system of eqs
(2.266) may be solved by putting Wf, = 0 and taking Sfi(p) as boundary
conditions. This results in the following expressions for the scattering
80
Quantum kinetic equation for the density matrix
[Ch. 2
amplitudes:
[<5fi - Sfl(p)] exp[i/xuf(£i - fif) • p/й] dp
(2.269)
or, in more detailed notation,
f(apu | a^1u1) = i^
Znn
J ехр[1дм(й1 — й) • p/h] x
[Mb. “ Sfafiu I «1Д1И1, p)] dp. (2.270)
Therefore, when conditions (2.265)-(2.267) are satisfied, scattering
amplitudes can be represented as standard Fraunhofer diffraction in-
tegrals. In the absence of transitions between the states а, аг and /?,/?!
formula (2.270) takes a particularly simple form
f(u | u,) =
f {1 - exp[i<5(p)]} exp[ip(n1 - u) • p/h] dp,
ZJln J
8(p) = ^-tw^r)d(r-ul),
11U J
(2.271)
where <5(p) is a phase change of the wavefunction due to the interaction
potential and corresponding to the impact parameter p.
The characteristic dependence of W^r) (spherically symmetric poten-
tial) and 8(p) is shown in fig. 2.4. Let pw denote the greatest of the
impact distances for which the equality
|d(p)| = l (2.272)
is satisfied. The value pw is called the Weisskopf radius. In the region
p < pw the values of the phase 8(p) and its derivative are large and under
such impact parameters scattering takes place through the angles given by
the laws of classical mechanics (see problem (8)):
2 sinful = Л^. (2.273)
\2/ dp
The range of large values of impact parameters (p > pw) where
|d(p)| < 1 contributes to the diffraction part of scattering*. It is of no
importance for the latter in what manner the trajectory is curved inside
* The interval of impact parameter values in the vicinity of the extremum of the function 6{p)
contributes to a certain extent to the diffraction scattering (“glory” phenomenon). Nevertheless, the
area of the ring corresponding to such impact parameter values is small (see problem (8)) if |<5|» 1.
§2.5]
Frequencies and kernels of the collision integral
81
Fig. 2.4. Plot of the potential W(r) and phase <5(p) as a function of the distance r and the impact
parameter p.
the circle of the Weisskopf radius; therefore, the angular width of the
diffraction part of the differential cross-section is given by the standard
formula
0d ~ X/pw.
The origin of diffraction effects may be illustrated by scattering by an
impenetrable sphere: the classical part of scattering is due to the reflection
of particles from the sphere and the diffraction part is due to particles
“missing” the sphere.
Real potentials of the type shown in fig. 2.4 differ from the model of the
impenetrable sphere in classical scattering. The range of impact para-
meters p<a (see fig. 2.4) corresponding to the overlapping of electron
shells of colliding particles acts as an impenetrable sphere and leads to
practically isotropic scattering (in the centre-of-mass system). In the
intermediate region a <p< Pw the interaction energy is not large in
comparison with the kinetic energy of relative motion and, accordingly,
the deflection angle specified by the formula (2.273) is small:
Л ft Wp W ,
-d ----------r~----« 1
pup пи T
(2.275)
82
Quantum kinetic equation for the density matrix
[Ch. 2
Fig. 2.5. Differential cross-section of elastic scattering: (1) diffraction part; (2) classical part, small
angles; (3) isotropic part.
Thus the general structure of the differential cross-section may be
represented as a superposition of the sharpest diffraction part (angular
width of order classical scattering through small angles (0 ~
WIT» К/py/) and practically isotropic classical scattering (0~1). The
foregoing is schematically illustrated in fig. 2.5. It must be noted that
division into diffraction and small angle classical parts takes place
only provided that the values of the phase d(p) in the range p < pw
become rather large. In the opposite case all the scattering through
small angles becomes diffractive and is described within Born’s approxi-
mation.
It is useful to remember that the diffraction and classical scattering parts
contribute approximately equal values npw to the total cross-section (i.e.
to the integral of the differential cross-section with respect to the
scattering angles). Hence the total cross-section is ~2лр^.
In the overwhelming majority of problems of linear spectroscopy the
external electromagnetic field has the form of a plane wave. Non-linear
phenomena due to such fields directly cause non-equilibrium distributions
only for the projection vN of the velocity v on the wave vector к
(Vn = v • к/к). The distribution over orthogonal velocity projections v±
also becomes non-equilibrium owing to collisions. However, this effect is
weak and in many cases cannot be taken into consideration. If we assume
the distribution over v± to be at equilibrium (in the more general case
induced by an excitation process) it is better to consider so-called
§2.5]
Frequencies and kernels of the collision integral
83
one-dimensional kernels, i.e.
A(<x<x’v„ | a{v„0 = J A(aa'v | aialvi)Vy(v±1) dv± dv±1, (2.276)
W(vx) = (l/ли2) exp(-vl/v2), v2 = 2T/m,
and corresponding one-dimensional in-frequencies:
v(aa' | «1»], Гц) = I A(aa'V|n | а^а'^) dvm. (2.277)
In all particular problems of the broadening of non-linear resonances
treated later the notion of a one-dimensional kernel is used.
Let us determine the formulae (2.276) and (2.277) for small angle
scattering when the condition
Am«m (2.278)
is fulfilled, where Am is a characteristic width of the differential cross-
section as a function of |m - uj (width in the centre-of-mass system). As
already noted, the kernel under such conditions depends mainly on the
difference g = v -vt (characteristic scale is Am ц/т) and is a much slower
function of v (characteristic scale is v). Therefore it is convenient to take
£п = «и — Um and Um as independent variables of the one-dimensional
kernel. The width of the differential cross-section should be introduced in
the explicit form which will be written as
(Ш \ и
и, — £) = ot(u, G/if), ij= — Au. (2.279)
(i / m
For small values of «ц («щ « м) it follows from eq. (2.276) that
v « vb: 4л /Ам\2 A(^ll,vl|1) = 7VbM— — x Г) \ и /
v » vb: If" / м2\ M dM f“ /- 1 expl _2I _2 1 at(u,z)dz; vJt •'0 ' M / M •'|C|||/n (2.280) 4Z. . ж 4л /Ам\2 A(£n,vnl) = Nbu — (-г-) x - I exp( - ~ I \Jt2 + S^/rf2) dr. л Jq \ и / и Jq (2.281)
84
Quantum kinetic equation for the density matrix
[Ch. 2
These expressions correspond to the limiting values for the ratio v)vb of
thermal velocities of colliding particles (under the same temperatures they
correspond to light and heavy perturbing particles). It can be seen from
eqs (2.280) and (2.281) that in both cases and, consequently, at all the
intermediate values of v/vb the kernel width coincides in order of its
magnitude with the width of the differential cross-section ij. Since the
lower limit in eq. (2.280) contains |£J the derivative of the one-
dimensional kernel is characterized by a discontinuity at the point
|£J = |V|| — Vml = 0. This discontinuity decreases as the ratio v/vb in-
creases and vanishes at v/vb—»<». Note that this property of a one-
dimensional kernel is brought about by the singularity of the three-
dimensional kernel (2.253) which, in its turn, is due to the law of
conservation of energy. Therefore this property of the one-dimensional
kernel is universal and must be allowed for in its model approximations.
This property is typical, for instance, of the exponential difference kernel
A(Q = ^exp(-|Cnl/s),
Zu
(2.282)
which will be used subsequently in seeking solutions of particular
problems.
Under isotropic scattering the v dependence of the kernel is given by a
statistical factor. Let us write down expressions for the one-dimensional
kernel at some characteristic values of the parameters:
А(^,0)=Ао[1-ф(д/^)
L \ У mb LV)
А0 = 4лоМь~;
2/x
— =1:
A(£|„0)=^
1 - + exp(-^/v2)
\ v /
— «1: A(§ll,0) = Aoexp(-^/v2)
mb
(2.283)
(0(z) is the probability integral). Direct calculation with these formulae
shows that the kernel width (at half-height) is changed by 20% in the
interval 0^m/znb^l and by a factor of about 1.5 in the interval
1 m/mb =e 5. Therefore, to an accuracy of a factor of the order of unity,
the kernel width under isotropic scattering is equal to v. At the same
time, the one-dimensional kernel is essentially asymmetric if vN1^0, as
can be readily seen from the following expression corresponding to
§2.5]
Frequencies and kernels of the collision integral
85
m/mb = 1:
\ 1.4 fl \ I гтпР"1 \11
A(^ll,vl|1)-2A0|l 0^_^J + exp^ _2 )[l + 0(-
(2.284)
The general relationships discussed above can be illustrated by the
model problem of scattering by an impenetrable sphere. In this case the
differential cross-section which determines the diagonal collision integral
is
2
a2 f
оХм, £/q) = —11 +
>M\T2Ji(g/q)~
miq) L .
ft
ma
(2.285)
where Л(х) is a first-order Bessel function and 2a is the diameter of the
sphere. The expression (2.285) holds good if the condition а/К = аци/к »
1 analogous to inequality (2.265) is fulfilled. According to eq. (2.285), the
differential cross-section contains an isotropic part (in braces) and a small
angle part whose halfwidth (at half-height) is equal to 1.616/ma«u.
Isotropic and selective scattering cross-sections are equal to ла2, i.e. the
total cross-section is 2ла2. Note that the ratio of the diffraction and
classical parts of scattering is much the same for arbitrary potentials
similar to that shown in fig. 2.4.
Substituting expression (2.285) in eq. (2.259) we obtain on integration
A(y | V!) = Jta2NbvbC(y | vj x
1 +
I
vb
~2Л(5/д)121
L ^/7? J J
(2.286)
The contribution of the isotropic part of the scattering to the one-
dimensional kernel has already been discussed. For the selective part the
following relations can be obtained in the limiting cases of light and heavy
perturbing particles:
A(C„, vH1) = Ао{1 - у + /?) - V, -
A(^ll,vl|1) = Ao^H1(2r)/t2;
io
8а2 й
Jo — Л>(0> Л ~-A(0> Ao= у— Аь— j
v
— «1:
Vb
V
— »1:
Vb
(2.287)
' = l£nl/*i;
where H\(2t) is a first-order Struve function. Plots of kernels specified by
86
Quantum kinetic equation for the density matrix
[Ch. 2
Fig. 2.6. The shape of the one-dimensional kernel of selective scattering by an impenetrable sphere:
(1) v/vb«l; (2) v/vb»l.
the relation (2.287) are shown in fig. 2.6; their halfwidths at half-height
are equal respectively to 0.910т/ and 1.573т/« v. It is seen from fig. 2.6
that the increase in the ratio v/vb removes the discontinuity of the
derivative at the point £„ = 0 and somewhat changes the halfwidth.
The cases discussed above were comparatively simple and corresponded
to the model of non-degenerate states and real differential cross-sections.
Differential cross-sections which determine non-diagonal kernels, gener-
ally speaking, are complex and are oscillating functions of the scattering
angle. Such kernels as well as collision integrals for degenerate states have
not been adequately investigated*.
For the analysis of particular problems of non-linear spectroscopy
model approximations for collision integral kernels are often employed.
One such kernel has already been mentioned in connection with selective
* This problem is treated, for instance, in ref. [31].
§2.5]
Frequencies and kernels of the collision integral
87
scattering (see eq. (2.282)). In the case of almost isotropic scattering the
expression
A(y | Vi) = vIV(v); W(v) = (у/л v)~3 exp(-v2/v2), (2.288)
sometimes called the model of strong collisions, may act as a good
approximation of the kernel. The velocity distribution of atoms after
collision in this case does not depend on the velocity before collisions. The
one-dimensional kernel corresponding to expression (2.288) has similar
properties:
A(v„ I M = vW(Vn); VK(V||) = (Vn v)-1 exp(-v^/v2). (2.289)
On comparison of relations (2.289) and (2.283) we may conclude that
the model of strong collisions can be a good approximation for real
one-dimensional kernels if scattering in the centre-of-mass system is
almost isotropic and particles b are not too heavy. As for the three-
dimensional kernels, the model (2.288) fails to allow for the singularity of
real kernels (see eq. (2.255)) and must therefore be employed with care.
The properties of the model of strong collisions are most pronounced in
the kernel of the collision integral due to the dipole-dipole interaction of
identical atoms [32, 33].
If isotropic scattering is accompanied by almost complete disorientation
of particles (model of strong collisions with respect to velocities and M
projections) the following approximation is employed:
A(JMJM’v I ладм^) = VK(v)<5MM.<5M1M1 (2.290)
or in the Kq representation
A(JJKqv | = vW(v)dKOdKiOdqOdqiO. (2.291)
In the model of relaxation constants the notion of strong collisions with
respect to magnetic sublevels can be introduced. In this case the
expressions for the kernel will be
A(JMJM'v | ЩМ'м) = <5(v - V1) (2.292)
A(JJKqv | = vd(v - vj dKO8KiOdqOdqiO. (2.293)
In most commonly used model kernels (model of strong collisions and
Keilson-Storer’s model, difference kernel model) the collision frequency
v does not depend on velocity. In contrast, frequencies v of real kernels
undoubtedly vary with velocity, which is most pronounced in the Lorentz
88
Quantum kinetic equation for the density matrix
[Ch. 2
model. Therefore a very important question is how essential the v
dependence of v(v) is and to what extent it may influence the choice of
model and final results.
As an example, consider elastic scattering by a power potential
W(r) = a/r", n?2, assuming the eikonal approximation (2.271) to be
applicable. Since under elastic scattering v = v, formula (2.153) may be
used and simple, if awkward, manipulations yield
v(v) = v(v) = у(0)ф(—-z2); (2.294)
2 / 1 \
z = v/vb; v(0) = Nbvbo(vb)-7=r(2---).
x Д1 1 /
Here cr(vb) is the total cross-section of scattering with the relative velocity
value being и = vb, Г(х) is the Г function, and ф(а, у; у) is a confluent
hypergeometric function. The frequency v(v) depends only on |v| = v
which is a consequence of the isotropy of the potential and the buffer
particles’ distribution (in the laboratory system). The thermal velocity vb
of buffer particles is a characteristic variation scale.
Formula (2.294) takes its simplest form with n = 3:
v(v) = v(0) = Nbvb(j(yb), (2.295)
i.e. the collision frequency is entirely independent of v. Under other
values of n the frequency v(v) can be both an increasing and a decreasing
function of v/vb. With small v/vb an approximate relation
v(v) «= v(0)
n — 3 v2'
"ЗХ(л1-1) vd’
V2 « vb
(2.296)
may be obtained from eq. (2.294). The asymptotic value of v(v) under
v » vb has the form
v(v) — Nbvba(vb)(yIvbYn 3V(n n = Nbva(v), v»vb. (2.297)
Plots of v(v) for some values of n are shown in fig. 2.7. Scattering by an
impenetrable sphere formally corresponds to n —»» and n = 2 corresponds
to Born’s approximation. As can be seen from fig. 2.7, v(v) is a
monotonic function increasing at n > 3 and decreasing when n < 3.
Variation of v(u) in the range 0 v < D = vby/mblm is of practical
interest (for simplicity the temperatures T and Tb are assumed equal). If
mb/m «1, the variation in v(v) within this range is of order mb/m, i.e. it
is not large. The variability of v(v) is the largest in the case of
comparatively heavy buffer particles, mb!m »1.
§2.5]
Frequencies and kernels of the collision integral
89
Fig. 2.7. Frequency of elastic collisions v(v) versus the velocity v: (1) n—»'»; (2) n = 6; (3) n = 4; (4)
z» = 3; (5) n = 2.
As was noted previously, one-dimensional kernels (2.276) and cor-
responding one-dimensional collision frequencies (2.277) are of special
interest for the spectroscopy of non-linear resonances. For the case under
consideration the following relation can be obtained from the formula
(2.294):
v(vn) = I v(v)VK(v±) dv±
v(0) Hr) у (-1)*Г(« + *) / p \*
(1 + 0)“Г(а)£о к\ Г(у + *)к1 + 0/
а = у = 1, 0 = (v/vb)2,
И — 1 2
(2.298)
where F(...) is the hypergeometric function. The fact that v and not vb
serves as the variation scale is of great importance. It implies that within
the range 0 v the frequency variation v(vN) is relatively small even
for heavy buffer particles (mb»m). Plots of v(vN) shown in fig. 2.8
illustrate this statement.
90
Quantum kinetic equation for the density matrix
[Ch. 2
Fig. 2.8. One-dimensional frequency of elastic collisions versus the projection of the velocity
V|l(v° = v(V|| =0)): (1) п-»», (Г) 0 = 1; (2) n = 6, (2') n = 6, fJ = 1; (3) n =4,
(3') n = 4, ft = 1; (4) n = 3; (5) n = 2, 0-> «; (5') n = 2, 0 = 1; (6) 0 = 0.
Thus effects caused by the velocity dependence of the collision
frequency must be considerably reduced when we deal with velocity
distributions having one-dimensional sharp structures. Therefore model
kernels with constant collision frequencies prove to be useful in the
analysis of collision-induced processes.
Qualitative conclusions drawn for elastic collisions are also valid when
allowing for inelastic processes as well as for non-diagonal collision
frequencies characterizing impact broadening of spectral lines. In our
opinion, these conclusions hold true when the model potentials are
replaced by the real potentials. Clearly, there may be such statements of
problems and conditions when the v dependence of collision frequencies is
specially emphasized and in these cases additional analysis is required.
However, such problems seem to be the exception rather than the rule.
2.6. Transport frequency of collisions
Analysis of the velocity persistence and diffusion approximation in section
2.4 involved the introduction of the notion of a transport collision
§2.6]
Transport frequency of collisions
91
frequency v,r(v). The quantity vtr(v) is of fundamental importance in all
transport processes, particularly in light-induced phenomena of gas
kinetics. Specifically, v,r(v) arises on calculation of the macroscopic force
of internal friction Fa due to collisions of atoms in the state a with a buffer
gas. By definition (see, for instance, ref. [34])
Fa = mj vS(aav) dv. (2.299)
Under elastic collisions the friction force is connected with the kernel of
the collision integral in the following way (for brevity, the index a is
omitted):
F = -mj (v - Vi)A(vi | v)p(v) dvt dv. (2.300)
The physical meaning of the relation (2.300) is quite clear, i.e.
m(v — V]) is an atomic momentum change per collision and the kernel
specifies the frequency of such collisions per unit time defined for a unit
velocity range. Integration with respect to v and vt gives the total
momentum change of the entire ensemble of atoms in state a per unit
time and unit volume, i.e. the friction force, acting on a unit volume of
gas and caused by collisions with buffer particles.
In the case of a spherically symmetric potential and an isotropic velocity
distribution of buffer particles may be represented as
F = Vtr(v)vp(v) dv,
vtr(v) = \ [ v • (v - Vj)A(vi I v) dvb (2.301)
v J
where vtr(v) is the transport frequency of collisions connected with the
transport cross-section by the relation (2.193). The quantity v,r(v) is
strictly positive and characterizes the deceleration effect. The product
vp(v) is the density (in v space) of the flow of atoms with velocity v and
mvtr(v)vp(v) is the density of the friction force acting on the flow density.
In the general case vtr(v) depends on v and there is a simple relation only
between the densities of force and flow.
If for some reason the v dependence of vtr(v) may be neglected* we
* Note that in all commonly used models of collision integrals (strong collisions, Keilson-Storer
model, model of the difference kernel) the diffusion approximation transport frequency is independ-
ent of velocity.
92
Quantum kinetic equation for the density matrix
[Ch. 2
obtain a simple relation also for the force F:
F = —mvtrJ, J = vp(v) dv,
(2.302)
where J is the particle flow. The force of friction is therefore proportional
to the flow of particles and decelerates it. The proportionality factor
explicitly contains the transport frequency of collisions, which proves to
be an adequate characteristic of the gas kinetics regardless of the
particular cause of the non-equilibrium velocity distribution p(v) and its
type.
If the v dependence of vtr(v) is essential, the relation of the force F
with the macroscopic particle flow is not so simple. Formally, we may
write a relation of the type of eq. (2.302) [35]:
F = -m{vtI)j,
< vtr> = | v,r(v)e • vp(v) dv / [e • vp(v) dv, (2.303)
where e is a unit vector in the j direction. The coefficient (vtr) may be
treated as an effective mean transport frequency of collisions. However,
(vtr) depends on a particular form of p(v). To calculate (vtr), strictly
speaking it is necessary to obtain the solution of the kinetic equations,
which is known to be fundamentally difficult. Therefore it is of major
importance to obtain the velocity dependence of the transport frequency.
The transport cross-section o,r(u) (2.194) which determines v,r(v) will
be calculated by the classical formula
otr(u) = 2л [1 - cos 0(p)]p dp,
Jo
(2.304)
where p is an impact parameter and 0 is the classical scattering angle:
p dr/r2
в |л 2<p|, <p £ _ p2/r2 _ 2W(r)/pu2 ’
(2.305)
Here W(r) is an interaction potential; the distance of closest approach r0 is
found from the condition of the radicand’s vanishing.
As for the collision frequency v(v) (section 2.5), consider the power
potential W(r) = <x/r", when the v dependence of v,r(v) is explicit. By
§2.6]
Transport frequency of collisions
93
formulae (2.304), (2.305) and (2.193) we obtain
otr(u) = Otr(Vb)(M/vb)1_4/",
/2 1 5 v2\
(2.306)
u 4 / 2\
Vtr(0) = — Nbvbotr( vb) Г 3 - -).
m 3y3 \ n)
There is an interesting similarity of the formal structure vtr(f) to that of
v(v) (compare with eq. (2.294)) which except for some minor details leads
to some common qualitative conclusions, e.g. vb is a characteristic
variation scale of vtr(v). In the vicinity of zero velocities we have
Vtr(v)~ v«r(0)(l + ^z-“), v2«vt (2.307)
\ 5n vl/
The asymptotic value of v,r(v) at v » vb is
u / v \1-4/"
Vtr(v) “ — WtrW —)
m \vb/
= — Nbv<jtI(y), v»vb. (2.308)
Fig. 2.9. Transport frequency of elastic collisions vtr(v) versus the velocity v: (1) n—(2) n =6; (3)
n = 4; (4) n = 3; (5) n = 2.
94
Quantum kinetic equation for the density matrix
[Ch. 2
When n = 4 the transport frequency is independent of the velocity. In
the model of impenetrable spheres which corresponds to the case n—»<»
we have Vtr(v)« v. Examples of vtr(v) are plotted in fig. 2.9.
When a non-equilibrium distribution is created for only one coordinate
(this is a frequent situation in non-linear spectroscopy) the distribution in
transverse velocity projections is only slightly different from the equi-
librium distribution. Then transport phenomena are described by the
one-dimensional transport frequency:
Vtr(Vn)= Vtr(v)W(v±)di»±
**(0) ЛГ)Х
(1 + Д)“Г(а)
+ ft V
*=o kl Г(у + к)\1 + р)
f(<x + к, у - 1 + к1, у + к; л V-
\ 1 1 l + p/\v
2 1
n 2
Y =2, 0 = (v/vb)2.
(2.309)
Fig. 2.10. One-dimensional transport frequency of elastic collisions vtr(V||) versus velocity projection
V|| (v“ = vtr(V|l = 0)): (1) (Г) „-»□=, 0 = 1; (2) n = 6, 0->«; (2') n = 6, 0 = 1; (3)
л = 4; (4)n = 3, 0-»oo; (4') л =3, 0 = 1; (5) л = 2, 0—°=; (5') n = 2, 0 = 1; (6) 0 = 0.
§2.7]
Macroscopic equations of gas kinetics
95
This expression differs from eq. (2.298) only by the values of the
arguments a and у of the hypergeometric functions. In the case of light
buffer particles (Д «1) in the velocity range 0 v„ < v the dependence of
vtr(vn) may be neglected. Figure 2.10 shows plots of v,r(Vn) for some
characteristic values of the parameters. The strongest v dependence of
vtr(vb) is due to impenetrable spheres («—»<») and a 1/r2 potential. In all
cases (with the exception of the potential with n = 4) the dependence of
vtr(vii) on f|| increases with increasing buffer particle mass.
The relation (2.302) between F and J proves valid under weakly
non-equilibrium conditions, when p(v) may be represented as [36]
p(v) = NW(v) + -^v-JW(v). (2.310)
In this case
F = -mv„j, (2.311)
where the following notation is introduced:
2 f
Vtr = vtr(V||)v]]W(vll) dv„. (2.312)
v J
Here Vn is the velocity v projection on the direction of j. If vtr(un) is
independent of vN we have vtr = v,r.
The above considerations show that the model of constant transport
frequencies may provide a good approximation.
2.7. Macroscopic equations of gas kinetics
involving interaction with laser radiation
With the help of a given density matrix, the variation in the parameters of
radiation propagating through a medium may be calculated as well as the
characteristics of the medium itself. However, it is rather a complicated
problem to find the density matrix as a solution of a kinetic equation.
Therefore various simplified methods exist. One of those widely used in
statistical theories consists in describing the medium and its interaction
with radiation by means of not the density matrix itself but its velocity
moinents. Density matrix moments are basically macroscopic characteris-
tics of the medium and obey macroscopic equations (hydrodynamic
96
Quantum kinetic equation for the density matrix
[Ch. 2
equations; see, for instance, refs [34,37-39]). In the present section such
equations are derived for gas interacting with radiation.
Radiation causes transitions between quantum states of the atoms of the
absorbing gas. Particles in different quantum states are characterized,
generally speaking, by different values of the gas kinetics parameters
(coefficients of diffusion, viscosity etc.). Therefore a radiation-excited gas
can be conveniently treated as a mixture of two gases, i.e. the gas of
particles in the ground state and that of particles in excited states. Thus
the gas medium in the field of intense resonance radiation is fundamen-
tally a multicomponent medium. In addition, the buffer component not
interacting with the radiation will be also allowed for.
Consider one of the simplest situations, when the gaseous medium is a
mixture of the buffer component and the component interacting with the
resonance radiation, the latter component being described by the model
of two-level atoms (states m and n). Radiation induces transitions
between the ground state n and the first excited state m (absorption
conditions). Therefore, we may speak of a three-component gas.
Buffer particles are assumed structureless and are described by the
kinetic Boltzmann equation
(|z+vv)pbb(v) = Sb(v), (2.313)
where pbb(v) is the density matrix of buffer particles. The kinetic
equations for absorbing particles are (see eq. (2.60) and section 2.3)
/3 \
I —+ v • V + Гт\ртт(у) = Sm(y) + Np(y)-,
\at /
Э
— + v
.dt
• v)p„„(v) = 5„(v) + Fmpmm(v) - Np(y);
/ д Гт\
I — + v • V + —)pm„(v) = 5(v) - iVm„[p„„(v) - pmm(v)];
\dt 2 /
Np(v) = 2 ReliVX.pUv)]- (2-314)
Here Гт = 2ут; hVmn is the matrix elements of the Hamiltonian of
interaction with the field, N represents the total number of atoms per
cubic centimetre, and p(v) denotes the number of transitions due to
interaction with radiation per unit time per atom in a unit range of
velocities v. The state n is the ground state, so y„ = 0 and the term
2ynp„„(v) in the left-hand side of the equation for p„„(y) is absent. It is
§2.7]
Macroscopic equations of gas kinetics
97
assumed that » T, as a result of which the state m is excited only by
radiation.
Collision integrals S,(v) are sums of partial collision integrals
5,(v)= 2 •$/,(v)> i,j = m,n,b, (2.315)
j
where 5,;(v) describes the collision of the particle of type i with a gas of
particles of type j.
It must be emphasized that in eqs (2.314) the recoil effect (light
pressure) is not taken into consideration. In other words, the momentum
of a particle is not changed when a photon is absorbed or emitted. For
simplicity, the time and space arguments of the density matrix are
omitted.
Let us introduce macroscopic quantities characterizing a gas of particles
of the type i, namely concentration Nt, flow of particles / , tensor of
momentum flow I? with components P“p (pressure tensor), energy of
translational motion flow of translation energy <7, and flow of tensor
m,v“vpp„. The above-mentioned quantities are proportional to different
moments of the density matrix as a function of velocity* [34,37-39]:
Ni, = j Pu(v) dv, (2.316)
/ = f vp„(v) dv, (2.317)
P“p = J m,v“vpp„(v) dv, (2.318)
= / V2pii(v) dV = * ТГ(Р,)’ (2.319)
= / ~^vv“v PP“(V ) ’ (2- 320)
4‘ = f^ w2p,,(v) dv. (2.321)
Here m, is the mass of a particle of type i and va, vp are Cartesian
components of the velocity. Similar macroscopic quantities may be
* Strictly speaking, the quantities introduced are the density of particle flow jf, the density tensor of
particle flow Pt, translational energy density W, and the flow density of the translational energy qt.
However, for simplicity the word “density” is usually omitted.
98
Quantum kinetic equation for the density matrix
[Ch. 2
introduced also for the description of the gas of absorbing particles as a
whole:
N = Nm+Nn, j=Jm+jn, P=Pm + Pn,
W = Wm + Wn, q = qm + qn, qap = < + qanp. (2.322)
Equations for the macroscopic quantities defined by formulae (2.316)-
(2.321) are derived from eqs (2.313) and (2.314) for the density matrix by
multiplying them by the required powers of the velocity and integrating
over v. Obviously, these calculations will lead to the moments of the
collision integrals
Js0(v)dv, (2.323)
Fv = j mtvSv(y) dv
= J m,(vt - v | v )pu(v ) dv, dv, (2.324)
C“p = j miVavpSij(y) dv
= Jm,(v“vf - vavp)Aij(vA | v)p„(v) dv, dv, (2.325)
Wu = | у v2Stf(v) dv
= ^TrQ.
= | у (V1 - v2)Atf(v, | v )p„(v) dv, dv, (2.326)
Rii = / v2vS‘i(V) dv
= J у (ViV' - v2v )Aу (v, I v )p„(v) dv, dv. (2.327)
The relations (2.324)-(2.327) define quantities which characterize the
relaxation of momentum, of the flow of momentum (tensor Q with
components С$р), the relaxation of energy (%) and of energy flux (l?0)
of the type i gas due to collisions with the type j gas. Subsequently the
§2.7]
Macroscopic equations of gas kinetics
99
following notation will be used:
Ft = ^Ftj, J F = Fm+Fn, (2.328)
j c=C. + q„ (2.329)
Wi j w = wm + wn, (2.330)
R^Rv, R = Rm + Rn- (2.331)
the collision integrals [38,39]:
JVv)dv=0,
[ v[miS0(v) + m;5/,(v)] = 0.
+ mXfvll dv = 0.
We shall confine ourselves to the treatment of elastic collisions only.
Under such conditions the laws of conservation of the number of particles,
of momentum and of energy are reflected in the following properties of
(2.332)
(2.333)
(2.334)
As is traditional for hydrodynamics, we shall consider the state of the
absorbing gas as a whole. Therefore, in order to describe the gas the pairs
of quantities Nm and N,Jm and J, Pm and P, Wm and W, qm and q, q„p and
qap will be used (see relations (2.322)).
Multiplying the equations (2.313) and (2.314) by various powers of v“,
integrating over v and allowing for the laws of conservation (2.332)-
(2.334), we obtain a system of equations for the density matrix moments:
/3 \
I — + rm)Nm + divjm = Np,
\dt /
дм QfJ
—+ divj = 0, —b + divyb = 0, (2.335)
dt dt
/3 \
— + Гт jjm + V • Pm = Fm + mNQ,
\at /
m^+V'P = F, mb^ + V-Pb=-F, (2.336)
100
Quantum kinetic equation for the density matrix
[Ch. 2
(Э \
- + rm )Pamp + div q^ =
at /
C°p + mNQSp,
— PaP + div qaP = Cap, Pgp + div qap = C£p,
dt at
(d \ m
( + Гт ) Wm + dlV 9m = Wm + V NQ2,
\dt / Z
3W 3W^A-
— + dwq = w, —- + divtfb= -w,
Crt dt
/ 3 \ _ _ „ m
I ~ + Гт \qm + V • Dm= Rm + — NQ3,
\at / Z
^ + V D = R, ^ + ^-Db=Rb.
at at
(2.337)
(2.338)
(2.339)
In eqs (2.339) moments Dm, D and Ц, of a higher (fourth) order
appeared:
D“p = — f
2 J
v2vavppu(v) dv.
(2.340)
The operations of differentiating the tensors f? and Ц with respect to
space variables are designated by V • /? and V • Q , which have the
following meanings:
тп d
^•Pi = llea—P?p,
d
^’P = ^ea—Pap,
ap OXp
where ea is a unit vector of the a axis.
The right-hand sides of eqs (2.335)-(2.339) contain the terms due to the
absorbed electromagnetic radiative power:
(2.341)
d
V'D = ^ea—D“p,
afi VXp
(2.342)
(2.343)
Q = vp(v)dv,
(2.344)
Q“p= v“v^p(v)dv,
(2.345)
§2.7]
Macroscopic equations of gas kinetics
101
Q2 = j v2p(v) dv = Tr Q2, (2.346)
Q3 = J vv2p(v) dv. (2.347)
Consider the general structure of the obtained equations. Equations
(2.335) for the absorbing gas as a whole and for the buffer gas have the
form of ordinary continuity equations. The right-hand side of the equation
for Nm contains the “source” Np which gives the number of transitions to
an excited state m (due to radiation absorption) per second.
According to eqs (2.336), relaxation of the flows jm, j and jb is caused
by the internal friction forces Fm and F. By the law of conservation of
momentum (2.333), the forces Fmm and the sum of forces Fmn + Fnm are
equal to zero. Furthermore, the frictional forces acting on the absorbing
as well as on the buffer gas have equal values and opposite signs.
Therefore the total momentum of the entire gas mixture (absorbing and
buffer gases) remains unaltered. The quantity Q in the right-hand side of
eq. (2.336) for jm obviously gives the rate of creation of the flow of excited
particles owing to the fact that the radiation, generally speaking, creates
asymmetric velocity distributions of atoms at an excited level (the Bennett
structure; see ch. 4). The equations (2.335) and (2.336) were first derived
in ref. [36].
Equations (2.338) describe the balance of translational energy. The
terms wm and w in the right-hand sides of eqs (2.338) characterize the
relaxation rate of translational energy due to collisions (see formula
(2.326)). The law of conservation of energy under collisions expressed by
the relation (2.334) manifested itself by the fact that the quantity w is
given only by collisions of the absorbing and the buffer particles, i.e.
w = wm + wn = wmb + wnb, (2.348)
the total translational energy of the gas W 4- Wb being unaltered by the
interaction with the radiation.
The relaxation of the momentum flow tensors f* and P, as can be seen
from the eqs (2.337), is determined by the tensors Ct and C. The law of
energy conservation (2.334) refers only to the trace of the tensors Ci and
C. Therefore the right-hand sides of the equations (2.337) contain the
exchange of momentum flows under collisions between similar particles as
well as between those of different types.
The equations (2.339) for heat flows q^ and q include the third moments
Rt, R and Q3 of collision integrals and absorption probability integrals.
102
Quantum kinetic equation for the. density matrix
[Ch. 2
The vector describes the relaxation of heat flow of the gas i resulting
from collisions with the j type of gas. In the general case all the vectors Rv
are non-zero and unequal to each other.
In the absence of radiation the system of equations (2.335)-(2.339)
describes macroscopic non-equilibrium processes in a two-component
medium according to the traditional approach. That is, in this case the
medium’s being non-equilibrium is due to non-equilibrium initial and
boundary conditions. Medium characteristics are changed by collisions
which tend to bring the medium to the equilibrium state. The picture
changes drastically when the medium is irradiated by quasi-
monochromatic radiation which is known not to be in thermal equilibrium
with the medium. Such radiation acts as an additional source of
equilibrium state disturbance. The radiation can create non-equilibrium
distributions both in some components of the mixture and in the gas as a
whole, including the case when the initial state of the medium in the
absence of radiation is completely at equilibrium.
The hydrodynamic problems of gases in the presence of a radiation field
have been treated by different fields of physics, e.g. by astrophysics, the
physics of shock waves and many others. Usually the radiation spectrum
was assumed to be wide enough and the radiation was supposed to bring
about a non-equilibrium distribution of atoms in energy levels as well as a
non-equilibrium space distribution. The interaction with monochromatic
laser radiation is distinguished by the fact that radiative transitions lead to
a non-equilibrium velocity distribution of atoms. This important cir-
cumstance is formally reflected by the v dependence of p(v) (see formula
(2.314)). Physically, the velocity dependence of the probability of
transitions leads to a wide range of hydrodynamic light-induced phenom-
ena which will be discussed in ch. 6 and ch. 7.
The system of equations (2.335)-(2.339) is not closed for at least three
reasons. First, in an equation for some density matrix moment there exists
a term with space derivatives of a moment of a higher order. Therefore in
spatially inhomogeneous problems it is necessary either to break off the
chain of equations for moments or to use the models where the higher
moments are expressed in terms of the moments of lower orders.
Secondly, the equations (2.335)-(2.339) contain components proportional
to p, Q, Q“p and Q3 (formulae (2.343)-(2.347)). These terms reflect the
gas-radiation interaction and act as the source of the non-equilibrium
state. However, in the general case they are not prescribed functions, i.e.
they depend on the radiation characteristics, the state of the gas and the
properties of collisions, and to calculate them the solutions of the original
§2.7]
Macroscopic equations of gas kinetics
103
kinetic equations (2.313) and (2.314) must be known. Therefore, for these
terms the problem in a strict statement must be self-consistent.
The most complicated case is that of collisional integral moments
(2.323)-(2.327) which are functionals of the density matrix and, generally
speaking, cannot be expressed in terms of its moments. Earlier, the
necessary simplification of relaxation terms was successful only under a
weak deviation from the equilibrium state (Chapmen-Enskog method,
Grad’s method etc. [34, 37]). It is universally accepted that macroscopic
equations are unsuitable for analysing essentially non-equilibrium condi-
tions and effective methods for the solution of kinetic equations must be
sought.
Nevertheless, there prove to exist a wide range of problems of gas
hydrodynamics in the field of laser radiation for the solution of which the
macroscopic equations are very useful and quite adequate even when the
conditions are essentially non-equilibrium. The point is, radiation im-
mediately creates a non-equilibrium state of absorbing particles only and
their distribution can be non-equilibrium to a great extent; in particular,
the distribution of absorbing particles can have a sharp and deep structure
(Bennett structure; see ch. 4). The buffer particles do not directly interact
with radiation and their state may be non-equilibrium exclusively owing to
the collisions with absorbing particles. Collisions do not extend the sharp
structure of the velocity distribution to the buffer gas and “smooth” it
considerably. Therefore the velocity distribution of the buffer particles
may experience slow deformation and we may speak about its weakly
non-equilibrium state. If, again, the concentration of buffer gas con-
siderably exceeds that of the absorbing gas, the deviation of the buffer gas
state from the equilibrium state will obviously be small.
In the relations presented in this section no concrete coordinate system
was given. Usually the laboratory system of coordinates is associated with
the walls of a containing vessel, equipment etc. It is convenient for our
problems to select such a laboratory system where the buffer gas as a
whole is at rest (b system).
Subsequently the velocity distribution of the buffer gas in the b system
will be considered isotropic. Using the relations between the kernels and
differential scattering cross-sections (see section 2.4) we may present the
quantities Fib, w,b, Ctf and Rib in the b system as follows:
Fib = —m vib(v)vp„(v) dv,
(2.349)
104
Quantum kinetic equation for the density matrix
[Ch. 2
w,b = -2
2 -i
tnv
Vib(v) — - v'zVXjPiXv) dv,
(2.350)
C“bp= —2m [vib(v) - у'зь(и)]и“и^р„(г) dv +
i = m, n;
' f EbV'2b(v) - v5>b(v) p„(v ) dv,
3v!b(u) - £bVib(v) vp„(v) dv;
_ f wbv2 . . ,
Nb£b = j y-Pbb(v)dv,
(2.351)
(2.352)
where ёь is the mean thermal energy of a buffer particle. Here transport
frequencies of collisions are introduced which are determined by transport
scattering cross-sections
и f v • и
— —r- oi (M)Pbb(v - u)u do,
mJ v
(2.353)
v?(v) = —^~[ cr*b(u)pbb(v - u)u3 du,
m2sb J
/ \ 2
(2.354)
aib(“) + 7 <^b(“)
D
\m
pbb(v - u)u du,
vib(v)=-^TT- f (2<(и)Г1 + 21
/П J v.
3(v • u)2 —
v2u2
4 -----x
V4
(2.355)
V • и
vu -
Г- . , 7 -»
^(u) i_3 +2МЦ?
L \ vu / m v J
pbb(v - u)u3 du.
p V • и
2
-2
m v
x
(2.356)
The transport cross-sections c^b(u) and o'2b(u) are defined by the formulae
(compare with eqs (2.194) and (2.203))
о[ь(и) = 2л I (1 — cos 0)o,b(u, 0) sin 0 d0,
Jo
(2.357)
olb(u) = 2л I (1 — cos20)a,b(u, 0) sin в d0,
Jo
(2.358)
§2.7]
Macroscopic equations of gas kinetics
105
where в is the scattering angle, and o,b(u, 0) denotes the differential
scattering cross-section.
Let the i-type particles and buffer particles be in a state of thermal
equilibrium. Then the collision integrals are equal to zero and as a
consequence all their moments are zero too. From the equality w,b = 0 we
find
f [vib(v)v2 — V2b(u)lv2] exp(—v2/v2) dv = 0. (2.359)
This strict integral relationship between v'ib(v) and V2b(v) can be easily
checked by means of the formulae (2.333) and (2.334) and physically
implies the absence of heat exchange between the particles z and b when
they are in a state of thermal equilibrium. The relation (2.359) must be
taken into account when model dependences on v for transport frequencies
are selected and justified.
As in the general theory of kinetic equations an important role in the
hydrodynamic description is played by the limiting case of light buffer
particles (mb«m, vb» v). It will also be assumed that the velocities of
flows are small with respect to thermal velocities. Under such conditions
transport frequencies are weakly dependent on v. If we neglect this
dependence and assume pb(v) to be a Maxwellian function, then
vib= vib = — v‘4b = —7= — Nbvb o‘ib(u)exp(-u2/vg)-^-, (2.360)
1U 771 •'O Vb
47 /m \2 f°° Г 1 1 fi7 cifi
v? = (~) A/bDb + a 02 ехР(-м2/йь) -гг" •
(2.361)
Note that for the model of constant transport frequencies the relation
(2.359) is satisfied. The frequency v'3b is of a higher order of smallness with
respect to mblm and one can put v'3b = 0.
The relaxation quantities Fib, w,b, Qb and Rib in the model (2.360) and
(2.361) may be written as
Fib=-v\bmj„ (2.362)
(W ЦГ \
24 "Я/74, (2'363)
/IV w \
C^= -2(v'b- vib)P^-Ш(2.364)
\ IN; INb'
106
Quantum kinetic equation for the density matrix
[Ch. 2
IV .. ГГь
Яь = -Зу',4. + - v\b -fj,. (2.365)
Thus in this case the collection integral moments are expressed in terms of
density matrix moments of the same or lower orders.
Therefore, under typical conditions when monochromatic radiation
causes a sharply non-equilibrium velocity distribution for one of the gas
mixture components, the macroscopic equations prove to be a convenient
and adequate formalism for the analysis of hydrodynamic phenomena.
This approach can be successful owing to the introduction of transport
frequencies of collisions which in many cases may be considered weak
functions of velocity (see section 2.6).
In the present section the simplest model of an atom with two
non-degenerate states is considered. The introduction into consideration
of degeneration of levels, radiation polarization, a large number of levels,
inelastic processes and other circumstances certainly complicates the
system of hydrodynamic equations. Nevertheless, the general idea that
this approach is useful for the analysis of gas hydrodynamics in the laser
radiation field holds true.
References
[1] L.D. Landau and E.M. Lifshitz, Kvantovaya Mekhanika. Nerelyativistskaya Teoriya, 3rd edn
(Nauka, Moscow, 1974) [Quantum Mechanics. Nonrelativistic Theory (Pergamon, Oxford,
1978)].
[2] LI. Sobel’man, Atomic Spectra and Radiative Transitions (Springer, Berlin, 1979).
[3] Yu. I. Heller and A.K. Popov, Lazemoe Induztirovanie Nelineinykh Rezonansov v Sploshnykh
Spektrakh (Nauka, Moscow, 1981) [Laser-induced Nonlinear Resonances in Continuous Spectra
(Plenum, New York, 1985)].
[4] J.A. Armstrong and J.J. Wynne, Spettroscopia non Lineare (North-Holland, Amsterdam, 1977),
p. 192.
[5] V.G. Arkhipkin and A.K. Popov, Usp. Fiz. Nauk. 153 (1987) 423 [Sov. Phys. Usp. 30 (1987) 952].
[6] E. Wigner, Phys. Rev. 40 (1932) 749.
[7] S.R. de Groot and L.G. Suttorp, Foundations of Electrodynamics (North-Holland, Amsterdam,
1972).
[8] V.I. Tatarsky, Usp. Fiz. Nauk 139 (1983) 587 [Sov. Phys. Usp. 26 (1983) 311].
[9] A.P. Kol’chenko, S.G. Rautian and R.I. Sokolovsky, Zh. Eksp. Teor. Fiz. 55 (1968) 1864 [Sov.
Phys. JETP 28 (1969) 986].
[10] A.P. Kol’chenko and G.I. Smirnov, Zh. Eksp. Teor. Fiz. 71 (1976) 925 [Sov. Phys. JETP 44
(1976) 486].
[11] S.G. Rautian and G.I. Smirnov, Zh. Eksp. Teor. Fiz. 74 (1978) 1295 [Sov. Phys. JETP 47 (1978)
678].
References
107
[12] M.I. Dyakonov, Zh. Eksp. Teor. Fiz. 51 (1966) 612 [Sov. Phys. JETP 24 (1966) 412].
[13] G.I. Smirnov and D.A. Shapiro, Zh. Eksp. Teor. Fiz. 87 (1984) 1639 [Sov. Phys. JETP, 60
(1984) 940].
[14] K. Blum, Density Matrix. Theory and Applications (Plenum, New York, London, 1981).
[15] A. Omont, Prog. Quantum Electron. 5 (1977) 69.
[16] M.I. Dyakonov and V.I. Perel, Opt. Spektrosk. 20 (1966) 472 [Opt. Spectrosc. 20 (1966) 257].
[17] R.F. Snider, J. Chem. Phys. 32 (1960) 1051.
[18] E.G. Pestov and S.G. Rautian, Zh. Eksp. Teor. Fiz. 64 (1973) 2032 [Sov. Phys. JETP 37 (1973)
1025].
[19] S.I. Yakovlenko, Kvantovaya Elektron. 5 (1978) 259 [Sov. J. Quantum Electron. 8 (1978) 151].
[20] N.R. Rahman and C. Guidotti, eds, Photon-assisted Collisions and Related Topics (Harwood
Academic, Chur, Switzerland, 1982).
[21] M.I. Dyakonov and V.I. Perel, in: Proc. 6th Int. Conf, on Atomic Physics (Zinatne, Riga;
Plenum, New York, London, 1979), p. 410.
[22] R.W. Anderson, Phys. Rev. 76 (1949) 647.
[23] M.I. Dyakonov and V.I. Perel, Zh. Eksp. Teor. Fiz. 47 (1964) 1483 [Sov. Phys. JETP 20 (1965)
997].
[24] S. Chapman and T.G. Cowling, The Mathematical Theory of Non-uniform Gases, 3rd edn
(Cambridge University Press, Cambridge, 1970).
[25] 1.1. Sobel’man, Vvedenie v Teoriju Atomnykh Spektrov (Fizmatgiz, Moscow, 1963) [An
Introduction to the Theory of Atomic Spectra (Pergamon, Oxford, 1972)].
L.A. Vainshtein, LI. Sobel’man and E.A. Yukov, Vozbuzhdenie Atomov i Ushirenie
Spektralnykh Linii (Nauka, Moscow, 1979) [Excitation of Atoms and Broadening of Spectral
Lines (Springer, Berlin, 1981)].
[26] P.R. Berman, Appl. Phys. 6 (1975) 283.
[27] S. Stenholm, Perturbation of atoms, in: Progress in Atomic Spectroscopy, eds W. Hanle and H.
Kleinpoppen (Plenum, New York, 1978), ch. 3.
[28] J. Keilson and J.E. Storer, Q. Appl. Math. 10 (1952) 243.
[29] S.G. Rautian and 1.1. Sobel’man, Usp. Fiz. Nauk 90 (1966) 209 [Sov. Phys. Usp. 9 (1967) 701].
[30] V.A. Alekseyev and A.V. Malyugin, Zh. Eksp. Teor. Fiz. 80 (1981) 897 [Sov. Phys. JETP 53
(1981) 478].
[31] P.R. Berman, T.W. Mossberg and S.R. Hartmann, Phys. Rev. A 25 (1982) 2550.
[32] A.P. Kazantsev, Zh. Eksp. Teor. Fiz. 51 (1966) 1751 [Sov. Phys. JETP 24 (1967) 1183].
[33] Yu. A. Vdovin and V.M. Galitsky, Zh. Eksp. Teor. Fiz. 52 (1967) 1345 [Sov. Phys. JETP 25
(1967) 894].
[34] V.M. Zhdanov, Yavleniya Perenosa v Mnogokomponentnoi Plazme (Energoizdat, Moscow,
1982).
[35] F.Kh. Gel’mukhanov, L.V. Il’ichov and A.M. Shalagin, Physica A 137 (1986) 502.
[36] F.Kh. Gel’mukhanov and A.M. Shalagin, Zh. Eksp. Teor. Fiz. 78 (1980) 1674 [Sov. Phys. JETP
51 (1980) 839].
[37] J.H. Ferziger and H.G. Kaper, Mathematical Theory of Transport Processes in Gases
(North-Holland, Amsterdam, 1972).
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York, 1975).
[39] C.V. Heer, Statistical Mechanics, Kinetic Theory and Stochastic Processes (Academic Press,
New York, London, 1972).
3
Resonance radiation processes
3.1. Spectral line broadening in the absence of
non-linear phenomena
The shape of the spectral lines in the absorption and emission spectra of
ratified gases is due to the Doppler effect, which is connected with the
thermal motion of atoms and random perturbations of internal degrees of
freedom induced by spontaneous transitions and collisions. The role of the
Doppler effect can be qualitatively explained in the following way. Let the
atomic oscillator moving with velocity v emit a wave whose frequency is
(i)0 in an atomic coordinate system. In the laboratory coordinate system
the frequency of this wave will be <o0 + к • v where к is the wavevector.
The spectral density of radiation emitted by atoms moving with velocity v
is described by the function <5(ct> — <o0 — к • v) (if we ignore oscillation
damping). The line shape of radiation emitted by an ensemble of atoms is
obviously given by
where VK(v) is the distribution of atoms in projections v = k’v/k of
velocities v on the direction of observation k/k. For example, with a
Maxwellian distribution the line shape is Gaussian (or Doppler):
/<ffl) = VSwexp
(O — (O0
kv
v2 = 2T!m.
(3-2)
According to the quantum theory of emission, the line shape whose
width is entirely due to spontaneous transitions is Lorentzian [1], i.e.
Ут + Уп
л: (<o — <o0)2 + (ym + y„)2’
(3.3)
109
110
Resonance radiation processes
[Ch. 3
and its halfwidth is equal to the half-sum of the decay probabilities of the
combining levels.
The simplest models of the collision disturbance of the emission process
allow for quenching of combining states (Lorentz mechanism), oscillation
phase shift of the atomic oscillator (Weisskopf mechanism) and change of
its orientation (Anderson mechanism) [2-5]. Each of these causes
individually also brings about a Lorentzian shape of the spectral line with
a certain characteristic width determined by the damping (correlation)
time of the dipole moment. Oscillation phase changes under collisions
lead also to a certain shift of the frequency at which 7(<o) attains its
maximum value.
According to the correlation theory of spectral line broadening based
on the classical model of an emitting (absorbing) oscillator a slow
envelope of the dipole moment or the field emitted by it is treated as a
random stationary quantity characterized by its correlation function
ф(т) = (е-^>), (3.4)
where <р(т) is the oscillation phase change over the time т and the angular
brackets denote averaging over all random parameters which influence
<р(т). The line shape according to the Wiener-Khintchine theorem is
given by the formula
1 Г
I(oj)= -Re I 0(r)ei("_"o)Tdr. (3.5)
Л Jo
The Lorentzian shape of the isolated spectral line profile (if Doppler
broadening is negligibly small) is determined by the impact approxima-
tion. Indeed, according to this approximation the process of wave
emission by an atomic oscillator is disturbed over a short time interval tc
and the radiation has the form of more or less extended wave trains,
differing from one another by a phase or by an amplitude. Simple
calculations show (see problem (9)) that the correlation function in this
case is exponentially dependent on т and the line contour has the
Lorentzian shape
ф(т) = e_(r+i4)T, 7(<o) = —-——----------—. (3.6)
7 v ’ jt Г2 + (a> — co0 — A) 7
Therefore the shape of the line contour proves to give little information
and most interesting are the values of the linewidths Г and shifts A which
§3.1]
Spectral line broadening in the absence of non-linear phenomena
111
in the simplest case of purely phase modulation may be calculated from
Г + iA = J P(g)[l - e-ivte)] dg, (3.7)
where g denotes the set of parameters which determine the value of the
phase shifts <p(g), and P(g) dg is the number of collisions per unit time for
which g are in the interval g, g 4- dg.
In the general case the theoretical analysis of the line shape turns out to
be far more complicated. This is due to the fact that the above-mentioned
mechanisms of broadening are not statistically independent even if only
because the processes corresponding to them take place in one and the
same impact. The situation becomes still more complicated if Doppler
broadening must be included. In this case the correlation function has the
form
ф(т) = (exp[—i<p(r) — ik • г(т)]), r(r) = f v(t) dt, (3.8)
Jo
where r(r) is the change of the atom’s coordinates over time r, and v(t) is
the atom’s velocity, which depends on collisions. Since velocity changes
are simultaneous with the oscillation phase change of the oscillator, line
broadening due to interaction and Doppler broadening are also statisti-
cally dependent. Finally, non-linear phenomena bring about the emer-
gence of a non-equilibrium velocity distribution of atoms as well as an
orientation and level distribution which necessitates simultaneous analyses
of dipole moment relaxation and state distributions of atomic numbers.
Thus a simple physical pattern based on the concept of parameter
(phase, amplitude, frequency) modulation of emitted waves and quite
useful for qualitative understanding of the very fact of spectral line
broadening proves to be not very suitable for its quantitative description.
The apparatus of the quantum kinetic equation, treated in chapter 2,
enables one to analyse all the enumerated causes of spectral line
broadening from the unified point of view separately as well as in
combination. In the present section the case of comparatively weak
electromagnetic fields is considered; non-linear phenomena may be
neglected and attention may be concentrated on the relaxation of the
dipole moment determined by non-diagonal elements of the density
matrix.
Consider the case of a single spectral line corresponding to a transition
between two stationary states m and n with energies Em> En. We call the
112
Resonance radiation processes
[Ch. 3
line a “single line”, implying that the external monochromatic field
E(r, t) = exp[—i(cttf -k-r)]+%* exp[i(<ot — к • r)]}
interacts with the transition m-n and interacts with no other if its
frequency oj is close enough to the transition frequency o)mn. Let the
discussion be within the limits of the model of non-degenerate states and
consider transitions allowed in the dipole approximation. The matrix
element of the atom-field interaction Hamiltonian is given by
Vmn(r, t) = E(r, t)dmn e\p(uomnt) =-G exp[-i(i2r - к • г)], (3.9)
G = dmn%/2h, Q= 03- 03mn,
where dmn is a matrix element of the dipole moment for the transition
m-n. In the relation (3.9) only the resonance term exp(—iQt) is retained
and the term with a high oscillation frequency, exp[i(ct> + is
dropped. The system of equations for the necessary elements of the
density matrix is (see sections 2.2 and 2.4)
/ d \
— + v • V + ym + y„ pm„(vrt)
\at /
= S(vr) - iG exp[—i(Qt - к • r)][pmm(vrt) - p„„(vrt)];
S(vr) = -v(v)pmn(vrt) 4- J A(y | v^p^^rt) di^; (3.10)
/ d
(— + v • V 4- 2y.
\dt ’
= Sj(yr) T 2 Re{iG* exp[i(£2r - к • r)]pm„(urt)}, / = m, л;
Sj(vr) = -Vjiyyp^vrt) 4- J A,(v | vjp^rt) dvt. (3.11)
Indices in the collision integral for the non-diagonal elements can be
omitted without causing ambiguity.
The “spectral line contour” is understood as the dependence of the
work performed by the field per unit time on its frequency:
P(a>) = — 2h(i) Re i V^„(rt)pm„(vrt)drdv . (3.12)
Consequently, to calculate P(co) it is necessary to know pmn(yrt). The
term of eq. (3.11) proportional to G* determines the change of level and
§3.1]
Spectral line broadening in the absence of non-linear phenomena
113
velocity distribution of atoms as a result of their interaction with an
external field, i.e. it describes non-linear phenomena which in this section
will not be taken into consideration. In this case the determination of the
non-diagonal element pmn(vrt) reduces to the solution of eq. (3.10) with
the known right-hand side; the population difference — pnn which is its
factor is determined by excitation and relaxation processes reflected in the
collision integrals 5y(vr) and in the spontaneous relaxation terms. Let us
suppose that these processes give rise to a population difference inde-
pendent of time and coordinates, i.e.
Pmm(vri) - pnn(vrt) = N(y) = NW(v), (3.13)
where N indicates the total (integrated over velocities) difference of
populations of the upper (m) and lower (n) levels. When carrying out
concrete calculations we shall assume the velocity distribution W(u) to be
Maxwellian:
W(v) = (Vn v) 3 exp
v2 = 2T/m.
(3.14)
If the aforementioned conditions are fulfilled pmn(yrt) can evidently be
represented as
Pmniyrt) = p(vkQ) exp[—i(Qt - к • r)] (3.15)
and from eq. (3.10) we can obtain the following equation for p(vkQf.
[y - i(Q - к • u)]p(iM) - 5(v) = -iGNW(v), (3.16)
S(v) = - v(v)p(vkQ) + J A(v | vx)p(yikQ) dvb у = ym 4- y„.
The solution of eq. (3.16) is expressed in terms of its right-hand side by
an integral relation
p(vkQ) = —iGN I F(vkQ | v ')W(y') dv(3.17)
here F(ykQ\v'), which is called the Green’s function, is the solution of
the equation
[y 4- v(v) — i(Q — к • u)]F(u££2 | v') — J A(v | vt)F(ytkQ | v') dU)
= <5(u-v') (3.18)
114
Resonance radiation processes
[Ch. 3
The physical meaning of the Green’s function is clear from eq. (3.18):
F(vfc£2|v') is the amplitude of a non-diagonal element of the density
matrix when it is excited with a certain velocity equal to v'. The total
amplitude p(yk£2) according to the relation (3.17) equals the sum of such
partial amplitudes taken with the weights — iGNW(y'). At the same time
F(ykQ | v') is a Fourier transform of the space-time Green’s function
F(ykS21 v') = f exp[i(£2t — к • r)]F(vrt | v') dr dt, (3.19)
which satisfies the equation
ГЭ
— 4- v
Lar
• V 4- у 4- v(v) F(yrt | v') - J A(v | v^F^rt | v') dV]
= <5(0<5(r)<5(v -v') (3.20)
and implies physically that it describes the evolution of a non-diagonal
density matrix element when the latter is excited at the point r = 0 with a
velocity v' at the time t = 0.
The power emitted (absorbed) by atoms can be expressed in terms of
the Green’s function:
P(co) = 2tio) |G|2 NVI(co), j 1(a)) dfl> = 1,
/(<o)= —Re ( F(vkQ I v')W(v') dv dv'
it J
= —Re [ <p(t)e'“'dt,
Jt Jq
ф(г) = | e-i* rF(vrt | v')W(v') dv dv' dr.
(3-21)
(3.22)
Thus the line contour is given by the average (over v, v') value of the
Green’s function F(vit£2|v'). The function whose area is normal-
ized to unit magnitude and describing the line shape, can be treated as a
Fourier transform of the function 0(t) which is a space Fourier harmonic
of the Green’s function F(vrt | v') averaged over velocities.
Formulae (3.21) and (3.22) establish the exact meaning of quantities
appearing in the correlation theory of spectral line broadening based on
the model of the classical oscillator. On comparing expressions (3.5), (3.8)
and (3.21), (3.22) we may conclude that the function ф(Г) specified by the
formula (3.22) is identical to the classical correlation function (3.8). By
§3.1]
Spectral line broadening in the absence of non-linear phenomena
115
way of better illustration of this analogy we may introduce into the
classical theory the distribution function of oscillators Л(ггфт|г') in
phase <p, coordinates r and velocities v and write the averaging procedure
in eq. (3.8) as [6]
ф(т) = J exp(-i<p — it • r)Fi(vr<pT | v') dv dv' dr d<p. (3.23)
For the variables <p and r to signify changes in phase and coordinates
over the time t, the distribution function must be subject to the initial
conditions
Fj(vr<pO | v') = <5(<p)<5(r)<5(v — v')W(v'). (3.24)
If the distribution function Fj(vr<pT | v') is now obtained as a solution of a
kinetic equation of the type (3.20), the classical theory of spectral line
shapes will be identical to the quantum theory.
The model of a classical oscillator, although providing the correct
intensity distribution in a spectral line, requires additional extraneous
assumptions when absolute values of the absorbed or emitted energy are
calculated. This is because radiation processes (see, for instance, eqs
(3.10) and (3.11)) are essentially dependent on the state distribution of
atoms, i.e. on values not appearing in a simple model of a classic
oscillator. The situation will become still more complicated when non-
linear phenomena are taken into account, with the external field not only
inducing a dipole moment in the atom but also affecting the state
distribution of atoms. Hence it is useless to “update” or add anything to
the classical theory as all such attempts will in the final analysis amount
merely to postulating a system of equations of the type (3.10) and (3.11).
Let us define concretely the general formula obtained above for some
collision integral kernels. First of all we consider the case when velocity
changes under collisions can be neglected (model of relaxation constants;
see section 2.4), i.e.
A(v | Vj) = v<5(v — v^.
Equation (3.16) for the Green’s function for such a kernel takes an
especially simple form:
[Г — i(£2 — А — к • v)]F(vit£2 | v') = <5(v — v'), (3.25)
where the following notation is used:
Г + izi = у + v - v = ym + y„ + v - v. (3.26)
116
Resonance radiation processes
[Ch. 3
The solution of eq. (3.25) is evident:
<5(u — v')
(3.27)
From eq. (3.27) we easily obtain
F(ykt | v') = <5(v — v') exp[—(Г + izi)t — \k • vr], (3.28)
F(yrt | v') = <5(v - v')8(r- vt)exp[-(r + izi)t]. (3.29)
Thus in this case the dipole moment relaxation is characterized by a
simple exponential law of damping. In the expression (3.29) one can
clearly see the absence of velocity changes under collision and uniform
motion of an atom. In the representation (3.28) the latter is reflected in
the factor exp(—i£ • vf).
The line contour of radiation emitted by atoms with a given velocity v'
is specified by the formula
Re ( F(vki2 | v') dv = Re ——————---------—, (3.30)
J v ’ Г-i(Q-Л-k-v'Y v ’
i.e. it has a Lorentzian shape with a halfwidth Г = ym 4- y„ 4- v' — v' and a
shift of maximum 4 + Lv'. For ensemble emission, by the general
relation (3.21) we obtain
1 f W(v')dv'
I(Q) = - Re I „ ;— -------, (3.31)
л J Г — i(Q— A — к • v ) '
i.e. averaging of the expression (3.30) takes place with the weight function
determined by the velocity distribution of atoms W(v').
For a Maxwellian distribution and provided that Г and A are independ-
ent of velocity v, the expression (3.31) is reduced to a probability integral
of a complex argument tabulated in detail in refs [7, 8]:
1 f W(v)dv 1
I(Q) = - Re -- '— ------- = -7=—- Re w(z), (3.32)
л J Г— i(£2 — A — к • v) ynkv
2 (z
w(z) = ez2[l — 0(z)], 0(z)=-t= e^dx,
Vn Jo
z = [T-i(Q- A)]/kv.
Figure 3.1 shows Z(£2) as a function of (S2 — A)/kv calculated from the
§3.1]
Spectral line broadening in the absence of non-linear phenomena
117
formula (3.32) for various ratios of the Lorentzian and Doppler halfwidths
Г/kv. With r/kv—>0 the profile approaches a Gaussian shape; when
r/kv »1 it becomes close to Lorentzian. The profile determined by the
relation (3.32) is sometimes referred to as the Voigt contour.
Consider the model of selective scattering corresponding to relatively
small velocity changes when the kernel may be assumed dependent on the
difference v — v, (see section 2.5). In this case
/ Э \ f
I — + и • V + у + v )F(vrt | v') — I A(u — Vi)F(ihrt | v') duj
\at / J
= 6(t)d(r)d(v —v').
(3.33)
It is convenient to solve eq. (3.33) by a Fourier transform with respect to
the variables r and v [9], writing
F(xkt | u') = J exp[—i(Jk • r + к • u)]F(urt | u') dr du, (3.34)
F(vrt | и') = (2л)-6 | exp[i(& • r 4- к • u)]F(jdtt | v') dk dir. (3.35)
118
Resonance radiation processes
[Ch. 3
From eq. (3.33) follows the equation
' Э 1
— — A-VK+r + iZl + v — A(k) F(idct | v') = d(t) exp(-i« • v'),
.at J
(3.36)
where the designation
A(k) = A(v - v,) exp[-iir • (y - v,)] d(v - v0 (3.37)
is introduced for the Fourier transform (with respect to u) of the kernel.
Equation (3.36) is readily solved and Ffydct | v') may be represented as
F(idb| v')
= exp}—(Г4- izi + ik • v')t — iir • v' — J [v-A(x + h)] dr}.
(3.38)
The inverse Fourier transform yields the formula
F(ykt | v') = (2л)-3 exp[—(Г 4- izi 4- ik • v')t] x
Г f Г* 1
I dir exp] i(v — v') • к — I [v - A(k + Jtr)] dr к
(3.39)
The integral term of the exponential function in eq. (3.39) evidently
reflects the role of velocity changes under collisions: if A(y — i^) =
vd(v-Vj) then v — A(k) = 0, the integral over к equals (2л)3д(и — u')
and eq. (3.39) is transformed into eq. (3.28).
The influence of the selective scattering can be illustrated by the
expression for the Green’s function averaged over the “final” velocities v
(for a fixed velocity v' before collision):
| F(ykt | v') dv = exp! — [Г 4- i(zi 4- k • v')]t — | [v — A(Jtr)] dr к
J I Jo J
(3.40)
Within the correlation theory the integral term has a simple interpreta-
tion: by means of eq. (3.37) we may write
v - A(kx) = f [1 — exp(-i& • Ди т)]А(Дг) d(Av) (3-41)
§ 3.1] Spectral line broadening in the absence of non-linear phenomena 119
and compare eq. (3.41) with the expression (3.7) for the halfwidth Г and
the shift A of the line due to phase modulation. The phase shift in eq.
(3.7) is quite analogous to the quantity к • Av т that is the phase shift over
time т due to the velocity change and the Doppler effect. Therefore the
expressions (3.7) and (3.41) are equivalent and we can state that
v - A(kr) characterizes the effect of the frequency modulation due to the
elastic scattering of the atom.
The frequency modulation is different from the phase modulation
mainly because the equivalent phase variation к • Av т is time dependent,
which results in the law of Green’s function damping being different from
the simple exponential law. For example, for the case of a one-
dimensional kernel of the model type (2.282) we find
A(kx) = v/[l + (for)2], f [v - Л(Лт)] dr =
Jo L ks
With kst« 1 this formula yields exp[-(As)2vt3/3] in the expression (3.40).
Consider the expression (3.40) in simple limiting cases. Let S now
denote an effective width of the kernel A(v — vj (its halfwidth or the
square root of the second moment) and assume that
ks»T+ v' = ym + y„ + v', v' = Rev, v'= Re v. (3.42)
Under such conditions exp(-iJt • Av r) in eq. (3.41) oscillates rapidly;
A(kr) in eq. (3.40) can be omitted and the expression (3.40) appears
exponentially damping. Consequently, the line contour will be of a
Lorentzian shape:
Re f F(vkQ I v') dv = Re------------—---------, ks » у + v'.
J v 7 y +v-i(£?-Jt-v')
(3.43)
Its halfwidth is given by a spontaneous relaxation (y) and the real part of
the out-frequency v'.
The result obtained can be readily interpreted within the scope of
modulation ideas.
The collisions break the wave emitted by the atom into a succession of
wave trains whose average duration is l/(y + v'). Within the limits of
each train the frequency shift due to the velocity change is к |v - v'| ~ ks.
Generally speaking, a certain phase correlation exists between the trains,
i.e. they are not independent and are able to interfere. If, however, the
random velocity changes over the time l/(y + v') lead to a large phase
120
Resonance radiation processes
[Ch. 3
change (ksr ~ ks/(y + v')»1) which coincides with the condition (3.42),
wave trains turn out to be practically uncorrelated. Consequently, the
emission spectrum width must according to eq. (3.43) be determined by
the inverse duration of a single wave train l/(y + v'). Spontaneous
relaxation brings about the additional decrease of correlation between the
trains and у additively enters into the line halfwidth.
In the opposite limiting case kst«1 (the frequency modulation leads to
a small phase shift kst over the total time of relaxation of the dipole
moment) the wave trains may prove to be correlated if no phase changes
occur under collisions and the train interference must be allowed for. It
follows from the above that the difference (3.41) is small and may be
neglected compared with Г. Therefore, instead of the relation (3.43), we
obtain
<344>
r + i^ = y + v-v,
i.e. also a Lorentzian contour, but its halfwidth equals the smaller
quantity у + v' - v'. Thus the linewidth and its shift are non-linear
functions of v' and, consequently, non-linear functions of the perturbing
particles’ concentration Nb (fig. 3.2). If v'>0 and v'-v'<v', which
corresponds to the insignificant role of inelastic processes and phase
modulation, then the slopes under small and large Nb values may differ
greatly. If v'<0 (see section 2.5) there will be no great difference
Fig. 3.2. The qualitative dependence of the line width on v'.
§3.1]
Spectral line broadening in the absence of non-linear phenomena
121
between v' and v' — v' and the degree of non-linearity of the plot in fig.
3.2 will not be large.
The line contour (3.40) in the intermediate range of values of the ratio
ks/Г is analysed in section 4.3. There the applicability criterion of the
expression (3.44) is established which depends on the value of the
phase variation due to frequency modulation over the total time of dipole
moment relaxation and is determined by the finer properties of the kernel
?l(v — rj. In particular, if a kernel possesses a finite second moment the
expression (3.44) holds when the following condition is fulfilled:
r2»(ks)2v'/r. (3.45)
The above criterion is interpreted in the following way. Over the period of
time 1/Г which is of interest to us v'/Г elastic collisions take place,
resulting in a mean square velocity change which by the law of large
numbers is s2v'/Г. Hence the frequency deviation reaches the value
fcsVv'/Г and the corresponding phase shift will equal ksVv'//7Г. If this
phase shift is small, the frequency modulation due to the collision-induced
velocity changes cannot manifest itself in the line shape, which is
confirmed by the criterion (3.45).
It was emphasized in section 2.5 that the model of the difference kernel
can be applied under the conditions when its width s is considerably
smaller than mean thermal velocity v. This requirement is connected with
the neglected dynamic friction effect. When kinetic problems are being
solved a still stronger condition must be satisfied:
sVl + v'/Г« v, (3.46)
which implies that collisions over the lifetime 1/Г bring about a velocity
change essentially less than v. Therefore the averaging of the expression
(3.40) with Maxwellian weight is meaningless.
The above discussion implies that within the frame of applicability of
the difference kernel model a frequency modulation cannot appreciably
affect the line shape of the radiation emitted by an ensemble of atoms
with a thermal distribution in velocities. The considered phenomena can
be observed in experiments with atomic beams or when non-linear
resonances are investigated (see ch. 4).
Now we analyse the line shape within the strong collisions model,
according to which the velocity distribution of atoms after the collision
does not depend on the atomic velocity before the collision (see section
2.5):
Л(г | ц) = vW(y). (3.47)
122
Resonance radiation processes
[Ch. 3
Unlike selective scattering, in the model (3.47) it is more convenient to
deal not with the time function but with the spectral Green’s function
F(vk£2| v') obeying the equation (see eq. (3.18))
[y + v — i(Q — k • v)]F(vJt£? | v')
= iW(v) J FfytkQ | v') d«! + d(v - v'). (3.48)
Dividing eq. (3.48) by у + v - i(£? - к • v) and integrating it with respect
to v we obtain
J F(ykQ | v') dv
=__________1________ [1 - V (_____irtvjd»_____1-
у + v — i(£? — к • v') L J у + v — i(£? — к • v)J
In order to calculate the line contour of the atomic ensemble the obtained
expression must be averaged over v' with the weight W(v') (see relation
(3.22)):
1(0) = 1 Re [----И'(2<“’е n Ь ' » f----------J"'
л J y+ v — \(Q-k • v ) L J y+ v -i(Q — к • v)J
(3.50)
According to modulation ideas (see the discussion of formula (3.43))
the factor in square brackets in eqs (3.49) and (3.50) indicates the role of
the interference of wave trains. Indeed, the phase shift over the train due
to the random velocity change equals kv/(y + v') in the model con-
sidered. If this phase shift is large (kv/(y + v')»1), the trains are not
correlated and the interference must be absent. This is the conclusion that
can be drawn also from eqs (3.49) and (3.50) since under the aforemen-
tioned conditions the integral term in square brackets is of the order of
magnitude of v'/kv«l, it can be neglected, and eqs (3.49) and (3.50)
become expressions characteristic of the spectrum of a set of uncorrelated
trains with mean duration l/(y + v'):
J F(yk£2 | v') dv = l/[y + v - i(£?- к • v')],
1 f V¥(v)dv
I(Q) = - Re --------(3.51)
л J y+ v-i(Q — k-v)
§3.1]
Spectral line broadening in the absence of non-linear phenomena
123
If the collision frequency is high enough, then over the mean free time
l/(y + v') the random phase shift kvl(y + v') will be small and the
integral term in square brackets can be comparable with unity (inter-
ference is essential).
Let us consider in more detail the case when W(v) and W(v') are
Maxwellian functions and I(Q) takes the form
Z(£2) =
Re
и'(г)
1 — Vit (v/kv)w(z) ’
у + v — iQ
_
kv
(3.52)
where the function w(z) is defined by the relation (3.32). If |v| «kv the
formula (3.52) in accordance with the above goes over into eq. (3.51). If
the approximate equations
2/ 2
w(z)==e211 —7=z +...
\ 71
И «1;
. . / iv1
w(z)~[z+ —) ,
|г|»1,
(3.53)
(3-54)
are used, the limiting cases can be investigated:
/(£2)«^i— exp[-(G- v")2/(kv)2],
ул KV
у + v' « kv-,
1 Г (kv)2/2 I-1
Z(£2)«-Re y+v-v+ V -iO ,
v 7 л L y+v —1£? J
y+ v' »kv.
(3.55)
(3.56)
For small values of the spontaneous damping у and collision frequency v'
the line contour is described, as would be expected, by the Gauss
function. The above-mentioned interference of wave trains is most
pronounced under conditions when the inelastic processes and phase
modulation are absent (v = v) and when the spontaneous relaxation can
be neglected (y = 0):
(3.57)
The expression (3.57) describes the central interval of the line contour
where the greater part of the energy is concentrated. Thus with large
collision frequencies the central interval of the line has a Lorentzian shape
and its halfwidth yd, called the diffusional halfwidth, considerably smaller
than both the Doppler width kv and the collision frequency v.
124
Resonance radiation processes
[Ch. 3
Consequently, the elastic collisions bring about the narrowing of the
spectral line*.
Spontaneous damping (y^O), inelastic processes and the phase shift
(v =# v) can significantly reduce the effect of line narrowing and even
completely eliminate it. In the limit |v|«y+v' such a conclusion is
obvious both physically (the wave trains are not correlated) and formally,
since the integral term in square brackets of the expression (3.50) will
obviously be small. The role of the processes masking the elastic
scattering may be illustrated by the expression for I(Q) at Q = 0:
1 Г Jtv' — 2(v' + y)l
Z(£2)-^— 1 +----------; v',v',Y«kv. (3.58)
VJtfcvL VJth J
Since the integrated (over the frequency) intensity of the spectral line in
our normalization is equal to unity the line narrowing must lead to an
increase in its maximum value and vice versa. From the expression (3.58)
we can conclude that line narrowing is to be expected when the following
condition is satisfied:
v' л jiv' Jt
In the opposite case collisions with velocity changes lead to line
broadening although to a lesser degree than at v = 0.
The line contour determined by the relation (3.52) is, generally
speaking, asymmetrical. Indeed, the expression (3.52) may be written as
= 1_________________m(z) - \At(v7fcv) |w(z)|2_______________
\/jt kv 1 + Jt |v'/kv\2 |w(z)|2 — (2\/jt/kv)[v'u(z) — 2v"v(z)] ’
(3.60)
w(z) = u(z) + iv(z), z = (y + v — i£2)/kv,
where u(z) and v(z) are the real and imaginary parts of the function
w(z). It can easily be shown that u(z) and v(z) are respectively
symmetrical and antisymmetrical functions of Q - v". All the quantities
entering into eq. (3.60) are symmetrical functions except for the term
v"v(z). Thus, if v"#=0 then /(£?) is an asymmetrical function of
frequency.
Recall that in the limiting cases of large and small collision frequencies
the line contour turned out to be symmetrical (see eqs (3.55) and (3.57)).
* The effect of spectral line narrowing due to elastic collisions was predicted by Dicke [10] and is
named after him.
§3.1]
Spectral line broadening in the absence of non-linear phenomena
125
Therefore asymmetry is essential in the intermediate range of values
v' ~ kv.
Physically, the asymmetry of the line contour can be explained in the
following way. The fact that there is an imaginary part of the in-frequency
implies that a single collision is accompanied both by a phase shift and by
a velocity change of the atomic oscillator. The simultaneity of these events
makes broadening due to interaction and Doppler broadening statistically
dependent (in terms of correlation theory), which, in its turn, inevitably
leads to asymmetry [6]. Consequently, line asymmetry is caused by the
statistical dependence of these two mechanisms of broadening.
Line asymmetry may also be due to the dependence of v in eq. (3.50)
on the velocity v: since v"(v) may depend on the velocity’s magnitude,
the equation (3.50) will evidently be an asymmetric function of frequency.
In this case we may also speak of the statistical dependence of two
mechanisms of broadening: if v is velocity dependent, then the phase shift
under collision is determined by the velocity imparted to the atom by the
previous collision, i.e. there is an explicit statistical dependence between
phase modulation and elastic scattering*.
The considered cases of the effect of collisions on spectral line shape
(models of selective and strong collisions) are good examples illustrating
the general statement of statistical modulation theory about the elimina-
tion of frequency modulation when the sign of the frequency deviation’s
instantaneous value is changed rapidly. General qualitative results are not
very sensitive to the particular law of modulation and depend basically on
effective frequency deviation and mean modulation period (see, for
example, ref. [12]).
In this connection an abstract case of two-position modulation is of
particular interest when the velocity v can have two values Vi and v2:
F(vkQ | v') = F^kO | v')d(v - vj + F2(y2kQ | v')d(v - v2),
A(v | Vj) = vn<5(v - v,) + v2i^(v - V2),
(3.61)
Л(г I v2) = Vi2^(v - vt) + v22d(v - v2),
W(v') = Wi^(v' — rj + W2d(v' - ц>), Wi + W2 = 1.
For such conditions the integral equation (3.18) reduces to the system of
* For more detailed analysis of these phenomena see refs [6, 11].
126
Resonance radiation processes
[Ch. 3
two coupled linear algebraic equations
(y + vt - vn - iQ^F^VikQ I v,) - vl2F2(v2kQ | ц) = 1,
-v^F^kQ | Vj) + (y + v2 - v22 - i^F^VikQ | rj = 0, (3.62)
Vj = v(vt), v2=v(v2), Qx = Q-k-vu Q2=Q-k'V2.
The right-hand side in eqs (3.62) corresponds to v' = vb and to the
second possible value v' = corresponds a value of zero in the right-hand
side of the first equation and a value of unity in the second equation.
Solution of the system (3.62) when the relation between parameters is
arbitrary proves to be too awkward (see problem (10)) and subsequently
the simplest case is considered:
Wj = W2 = 2, —k’V1 = k‘V2 = kv,
v, = v2=v, Vn = V21 = V12 = v22= v/2,
The spectrum then takes a comparatively simple shape:
Z(£2) = -Re 2 WiFf(vJkQ\vi)
Л U=l,2
- 1 Rc[ 1 + i5>/2d + 1 ~ iv/2d 1
2л Чу + v — v/2 — i(£? — d) y+ v — v/2 — i(£? + d)J’
(3.63)
d = V(kv)2 — (v/2)2.
If the in-frequency is significantly smaller than the deviation (v « kv) then
two isolated lines correspond to the expression (3.63); these are centred
at Q= ±kv and possess equal halfwidths у + v - v/2 and equal inten-
sities. As v increases, the value of the radical d = V(^VT “ (v/2)2
decreases and the two spectral lines crowd together. With v>2kv the
lines completely overlap, their centres are at the frequency £? = 0 (the
radical V(kv)2 — (v/2)2 is imaginary) but they differ in halfwidths and
intensities. In the limit v » kv from the relation (3.63) we obtain
/(O)=iRe[----------. l—_ . --ВД. (3.64)
л Ly + v - v + (kv)2/v -1£? y + v-i£?J
The spectrum described by the expression (3.64) consists of a sharp
component whose structure coincides with the shapes already discussed
(compare the first term in eq. (3.64) with eq. (3.56)) and of a less intense,
wider “negative” component decreasing the spectral intensity in its
“wings”. Thus in this case the elastic collisions also lead to the narrowing
§3.1]
Spectral line broadening in the absence of non-linear phenomena
of the spectrum whose halfwidth under the most favourable conditions
(v = v, у = 0) is given by the diffusion parameter (kv)2/v.
The aforementioned example of two-position frequency modulation
may be treated as an analogue of the broadening of two lines correspond-
ing to two different transitions with almost the same Bohr frequencies (om„
and (Oji (fig. 3.3). If the collisions are able to induce polarization transfer
Fig. 3.3. Doublet of overlapping lines: (a) scheme of the transitions; (b, c, d) the spectral change with
the increase in pressure.
128
Resonance radiation processes
[Ch. 3
between the transitions m-n and j-l the atom emits radiation whose
frequency alternatively takes on the values o)m„ and cd,7, i.e. two-position
frequency modulation will take place here as well. With large enough
values of the collision frequencies v(mn\jl) exceeding |cdw„ — cd7/| the
doublet components will overlap and spectrum narrowing may take place.
This phenomenon, identical to the Dicke effect from the viewpoint of the
general theory of stochastic frequency modulation, has been called the
collapse of the spectral structure [3,13,14].
Let us consider the broadening of two close spectral lines without
allowing for the motion of atoms (see fig. 3.3). Substitution of variables
Pm„(0 = Pmn exp[-i(w - p;7(t) = Ph exp[-i(cu - <ty)r]
(3.65)
reduces the kinetic equation (2.142) to the following system of algebraic
equations with respect to pm„ and p;7:
[y„,„ + Vmn - vmn - i(tu - wmn)]pm„ - v(mn | jl)pn = -iGmnNmn,
-v(jl | mn)pmn + [y;/ + Vn - Vn - i(tu - <Оц}]рц = -iG7/2Vy/,
У,* = У/ + У*, vik = v(ik | ik), Nik = Nt-Nk. (3.66)
For the sake of simplicity we assume that
Gmn = Gjt = G, Nmn = Nji = N, ym„ = Yj, = у,
vm„ = Уц = v, vmn = v, = v(mn | jl) = v(jl | mn) = v/2.
In this case for the work done by the field we can by general rules obtain
the following expression:
1(0} = - Rc[ 1+и>/2Д' + ‘~W2ai 1
2л Ly + v — v/2 — i(£? — dt) у + v — v/2 — i(£? + djj’
t________________ (3-67)
£? = CD - (w„,„ + cd,7)/2, dj = W(<«m„ - <w7/)2 - V2.
From the general arguments presented above the structures of the
formulae (3.63) and (3-67) are the same.
The effect of collisions on the spectral line shape was interpreted
according to the notion of stochastically modulating parameters of the
wave emitted by atoms. Collision-induced velocity changes and polariza-
tion transfer were interpreted in this case as frequency modulation.
Another kind of interpretation is also possible based on the notion of
§3.1]
Spectral line broadening in the absence of non-linear phenomena
129
spectral exchange, i.e. formulated using not the time language but the
spectrum language. The discussed case of doublet broadening can be the
simplest example of spectral exchange: according to the system (3.66) the
transitions m-n and j-l “exchange” their polarizations with frequencies
v(mn | //) and v(/7 | mn). The Dicke effect may also be treated as a result
of spectral exchange between polarizations pmn(ykQ) with different
velocities which can be converted to the frequency scale owing to the
Doppler effect. The modulation picture can be described by the space-
time Green’s function F(ykt | v') whereas to the notion of spectral
exchange corresponds the Green’s function F(vkQ | v').
The general theory of the broadening of arbitrary spectral structure (of
overlapping and non-overlapping lines) not allowing for non-linear
phenomena is based on the system of integral equations
[Ymn + v(mn | mn, v) - i(<u -штп-к- v)]pm„(v) -
2 A(mnv | du,
ji J
= -iGmnNmn(y). (3.68)
The population differences Nm„(y) in the linear theory are taken to be
given functions of the velocity. Summation in eq. (3.68) is performed over
the transitions j-l. Therefore the pair of quantum numbers j, I
enumerating the transitions serves as the summation index and the system
of equations (3.68) may be written in a compact matrix form
(y + v - i£2)p(v) - ( A(v | uOp^i) du, = -iN(u)G, (3.69)
which resembles eq. (3.16) for an isolated spectral line. In eq. (3.69) the
quantities p(v) and G denote the columns made up of matrix elements
pm„(y) and G„,„; the matrices y, v, Й and N(u) are diagonal:
(v)mnji = v(mn | mn,
(ff)mnjl — (Уm + Ynj^mnjb .
(Q)mnJl =(o)-a)mn-k- v)^,; (J
(N(u
A(mnv | /Zvt) are the elements of the matrix A(v | u,).
The solution of eq. (3.69) may be expressed with the help of the
Green’s matrix F(u | и')
p(u) = —i f F(u | и ')N(u') du' G, (3.71)
130
Resonance radiation processes
[Ch. 3
which satisfies the equation
(y + v — i£2)F(v | v') - J A(v | Vi)F(vi | v') dv! = l<5(v — v '), (3.72)
where 1 is a unit matrix. Finally, the work done by the field is given by the
relation
P = —2йю Reflr j dv dv’ GGF(v | v')N(v')}. (3.73)
In the model of relaxation constants
A(v | v0 = vd(v - v0, F(v | v') = F(v)<5(v - v'), (3.74)
eqs (3.69) and (3.72) are reduced to the algebraic equations
h(v)p(v) = -iN(v)G; h(v)F(v) = l; (3.75)
h(v) = у + v - v - iS2 (3.76)
and the Green’s matrix is equal to the matrix inverse to h(v):
F(v) = [h(v)]"1. (3.77)
Thus, in order to calculate the work of the field within the scope of
applicability of the model of relaxation constants one must invert the
matrix h and average it according to the relation
P = -2йю RefrrlGG* f [h(v)] *N(v) dvl). (3.78)
The quantities Gmn are proportional to the matrix elements of the dipole
moment dmn and contain ? as a common factor. Hence the expression
(3.78) may be written as
Р(ю) = -2h(o Re(Tr{d<r J [h(v)]'N(v) dv}), (3.79)
(d)mnj/ dmndmnJi.
If non-diagonal elements v(mn | jl) of the matrix h are small enough in
comparison with the real and imaginary parts of its diagonal elements
Ym + Yn + Vm„, k-v, and to — tOmn, then h(v) can be assumed diagonal
and its inversion reduces to calculating reciprocal values of diagonal
elements. This case evidently corresponds to non-overlapping spectral
lines and the contour of each of them is independently broadened by
collisions. On the contrary, large values of v(mn | jl) (as compared with
§3.1]
Spectral line broadening in the absence of non-linear phenomena
131
kv, (Dmn — o)ji) imply the existence of intense spectral exchange, the matrix
h(v) is essentially non-diagonal and its accurate inversion is necessary. In
this case collapses of spectral structures are possible.
By way of specific examples where it is necessary to take into account
the complex nature of the broadening of a spectral line, we can suggest
the Q branches of rotation-vibration infrared absorption and combination
scattering, Zeeman and Stark structures of spectral lines, and the edges of
electronic-rotation bands. The main problem to be solved in analysing
any kind of collapse consists in treating specific collision processes leading
to polarization exchange between collisions*.
So far the discussion has been confined to the model of non-degenerate
states. The general theory of spectral line broadening formulated by the
relations (3.68)-(3.79) may be extended also to degenerate states if in
these formulae radiation and collision-induced transitions are interpreted
as transitions between magnetic sublevels. In other words, the indices m,
n, j and I in eq. (3.68) must be substituted by the rules
(3.80)
and the set of quantum numbers nMn'M' which determine the transition
between magnetic sublevels M and M' of the states n and n' must be
considered as the summation index. It is emphasized that such extension
of theory implies that the applicability conditions of isotropic collisions are
satisfied (see section 2.4), as in the opposite case the out-term of the
collision integral may have a more complicated form. Finally, the matrix
element of the interaction Hamiltonian Gm„ in eq. (3.68) must be replaced
in accordance with eq. (2.76) (dipole approximation):
Gmn^G(nMn'M') = £ - M' | la)G„„-(a), (3.81)
0
<7 = 0, ±1; G„„-(o) = d„M2h-, dm.= (n|| </||n')M,
where the are the circular components of the electric field, and
(n|| d ||n') is the reduced matrix element of the dipole moment.
Consider the broadening of the spectral line corresponding to the
transition between two degenerate states n and n' using the model of
relaxation constants. In accordance with the aforementioned rule (3.80),
* For detailed treatments of the collapses of spectral structures see refs [13-15].
132
Resonance radiation processes
[Ch. 3
instead of eq. (3.68) we have
[/„„• + v(nn') - i(<y - <ym- - к • v)]p(nMn'M’)
= v(nMn'M' | пМргМ'^р(пМрг'M'^ - iG(nMn'M')Nn„-(v).
(3.82)
The sum over in the right-hand side describes the spectral exchange
between the transitions nM-n'M' and nM-n'M' illustrated in fig. 3.4.
The vertical wavy arrows denote the radiation processes determined by
the term -iG(nMn' M')Nnn-, the solid semicircular arrows designate the
polarization exchange corresponding to one component of the sum in eq.
(3.82) (in fig. 3.4 it is implied that the field polarization is linear and the
quantization axis is chosen along the direction of its electric vector). Since
collisions are isotropic the arguments of the in-frequencies are subject to
the condition M - M' = Mt - M'i (see eq. (2.180)) as shown in fig. 3.4.
From the viewpoint of the modulation concept the effect of disorienting
collisions on the spectral line shape can be interpreted as a consequence of
the amplitude modulation. Indeed, the dependence of the quantity
G(nMn'M') in eq. (3.82) on MM' implies that with equal populations of
magnetic sublevels the wave amplitudes emitted by an atomic oscillator
are unequal for different transitions. Therefore the collisional polarization
transfer from the transition Mx-M{ to the transition M-M' will result in a
change in the emitted wave amplitude. Consequently, the disorienting
collisions bring about the emission break-up into a succession of wave
trains with different amplitudes, i.e. the amplitude modulation of the
wave. According to classical ideas, the role of disorienting collisions may
Fig. 3.4. Scheme of the radiation and collision transitions.
§3.1]
Spectral line broadening in the absence of non-linear phenomena
133
be accounted for in the following way. The wave amplitude emitted by a
linear .iscillator in a certain direction depends on the angle of this
d\ vdon with the oscillator axis. The collisions change the orientation of
«.he oscillator and as a result the wave amplitude in a certain direction is
random. Consequently, stochastic modulation of the wave amplitude
takes place which is accompanied by a spectral line broadening.
Within the scope of the model of isotropic collisions the kq
representation is rather convenient as it diagonalizes the relaxation matrix
in eq. (3.82). Indeed, the standard transformation (2.61) leads to the
following form of the system of equations (3.82):
[уЛЛ' + v(nn') - v(nn'K | пп'к) и)]р„„(к<?)
= -iA„„(u)G„„ (q)dKl, Q = w- a)nn.. (3.83)
Thus, for the broadening of spectral lines connected with the dipole
atom-field interaction, only the value к = 1 is essential. From eq. (3.83)
we obtain:
n x i4„ (v)G„„ (^)
PnnXq) rnn.-i(Q-Ann.-k-vy
Гпп. + iAnn. = Y„ + /„ + v(nn') - v(nn'l | nn'l) (3.84)
and making use of the relations (2.63) and (2.77) we estimate the work
done by the field as
P = -2ha>Nnn,V 2 |GU?)№),
4
Ца) = 1 Re [ JWd” , (3.85)
JC J A„„. -k-v)
where W(u) is the velocity distribution for the population difference of
the states n and n'.
Therefore the spectral line due to the set of radiation transitions
between the magnetic sublevels of degenerate states is described by a
single Lorentzian contour with a halfwidth Г„„ and a shift A„„. indepen-
dent of the field polarization and atom’s velocity. It differs from the results
obtained for the model of non-degenerate states only in the parameter
Г„„ + According to the relation (2.237), the collisional component
134
Resonance radiation processes
[Ch. 3
of the width Гпп may be written as:
2(1^.- - ) = 2 Re[v(nn') - v(nn'l | nn'l)]
= S v(»i»iO | nnO) + v(n[n[0 | n'n'O) +
n\^n n\+n*
v(nnO | nnO) + v(n'n'O | n'n'O) —
2 Re v(nn'l | nn'l). (3.86)
Sums over nt and n[ in the relation (3.86) determine the contribution of
the quenching of the combining levels n and n' into Гпп., i.e. of all the
inelastic processes n-Mj and If there is no quenching, Г„„. is
determined only by the phase shift of an atomic oscillator and its
disorientation. By virtue of the inequality (2.187) we may write
v(nnO | nnO) + v(n'n'O | n’n'O) — 2 Re vfnn'l | nn'l)
> [}/v(nnO | nnO) - y/v(n'n'O | n'n'O)]2>0. (3.87)
When the phase modulation is also absent the only physical cause of the
broadening is the amplitude modulation due to the disorienting collisions,
as has already been shown.
The problem of the spectrum of overlapping lines can also be
formulated in the Kq representation..Taking into account the rules (3.80)
and transformation (2.61), one can derive from eq. (3.68) the following
system of equations:
- i(<u - a)„„. - Ann. - k • и)]р„„.(к<?)
= v(nn'к \ nrn'^p^Kq) -
щл'нМм'
iNnn(v)Gnn.(q)dK1. (3.88)
In this case too the elements of the density matrix differ from zero only
for к = 1. It will be recalled that there exists a dependence between the
frequencies of direct and reverse transitions given by the equality (2.244).
Within the scope of the impact approximation we may consider the
overlapping of lines corresponding to very close Bohr frequencies.
Specifically, they may correspond to radiation transitions between groups
of closely located levels, so that |E„ — E„,| « T. According to eq. (2.244),
under these conditions we have
v(nnrK | п^к) = vfntn’iK | пп'к), (3.89)
i.e. the relaxation matrix in eqs (3.88) is symmetrical.
§3.1]
Spectral line broadening in the absence of non-linear phenomena
135
The general structure of the equations (3.88) describing the intensity
distribution in the spectrum of overlapping lines with degeneration taken
into account is similar to that of the model of non-degenerate states
(compare with eq. (3.75)). They differ only in specific values of the
relaxation matrix elements. Therefore everything that has been said about
the collapse of the overlapping lines may be directly applied to the
transitions between the degenerate levels.
All the theoretical results of this section, strictly speaking, hold true
only within the scope of the model of isotropic collisions. In particular,
the comparatively simple shape of a spectral line which is specified by the
relations (3.84) and (3.85) has been obtained under the assumption that
the relaxation matrix v - v is diagonal in Kq and that it is independent of
q, v. The model of isotropic collisions in its turn is based on the
assumption that the atomic velocity v is small compared with the velocity
ub of the perturbing particles (see eq. (2.169)), which is known not to be
fulfilled in many cases. Under the conditions of a considerable velocity of
the emitting (absorbing) particle, its perturbation by the buffer gas is
characterized not by a spherical but by an axial symmetry (wind effect;
section 2.4), the matrix v —v becomes non-diagonal, and the spectral line
contour is more complicated.
If the collisional disorientation is absent or negligibly small then the
frequencies v and v can be readily shown not to depend on Kq but to be
functions of the velocity magnitude |v| = u. The shape Z(<u) of the spectral
line is in this case given by the expression (3.35); however, we must take
into account the dependence of the width Гп„ and the shift on v:
1 f W(v) dv
Z(<y) = - Re I ,, z 4 4 ,-----7, (3.90)
л J rnn (v) - i[i2 - Am.(u) - к • v]
Q= (о-ыпп..
Now, let the collisional disorientation be non-negligible and the
matrices v and v be non-diagonal. Then instead of the equations (3.83) we
have
{^{Kq | Kq, v) - i[$2 - Д(к<? | Kq, v) - к • v]}p„„(Kq, v)
= S [rnn (Kq I *144, V) + i^nn’^q I Kiqi, v)] x
Kiqi^xq
pnn (Kiqi, v) - iNnn (v)Gnn (q)dKi, (3.91)
i.e. the density matrix element p„„(lq, v) is obtained from a linear system
of several coupled algebraic equations. Therefore, as can readily be
136
Resonance radiation processes
[Ch. 3
shown, the spectral line contour will contain not one but several
Lorentzians with different widths, shifts and amplitudes, the number of
Lorentzians being dependent on the momenta J and J' of the combining
levels and on the interaction types of the particles a and b*.
Calculations carried out for a van der Waals’ potential showed that the
influence of the splitting of the spectral line due to the “wind effect” was
comparatively small, i.e. the correction for anisotropy for the total
contour of all the Lorentzians amounts to a few per cent although at the
same time the widths and amplitudes of components may differ by a few
tens of a per cent.
Therefore the main manifestation of the “wind effect” amounts to the
width and shift dependence on the velocity; we can make use of the
formula (3.90), with Гпп.(у) + iAnn.(v) implying the mean value:
Глл.(и) + izMv) = 1E [ГЛД1<71v) + 1«)]• (3.92)
4
So far the only known exception to the formulated rule (3.92) is spectral
line broadening under the charge-dipole interaction. In this case the
“wind effect” at m «mb may lead to a significant change in the line
profile as compared with formulae (3.90) and (3.92) [21].
3.2. Interaction of atoms with a strong resonance
field
In non-linear spectroscopy the main problem is that of the interaction of
an atom with an electromagnetic field whose spectral components are
close enough to the frequencies <ym„ of the transitions between the atomic
levels m and n. With the assumption Em > En, this may be expressed as
the inequalities
I<y - « <y + <ymn, I<y + <y;J, (3.93)
where <u is the field frequency and the (ojk are the Bohr frequencies of the
other transitions. The condition (3.93) defines the so-called resonance
approximation within which the external field induces transitions only
* The effect of collision anisotropy on the contours of spectral lines was first considered in refs [6, 16,
17]. For detailed analysis and references see refs [18-21]. The role of the wind effect in Dicke
nanowing of the Doppler shape of scattering lines was studied in ref. [22].
§3.2]
Interaction of atoms with a strong resonance field
137
between the states m and n. Other states are influenced by the external
field indirectly; for example, as a result of the spontaneous cascade
transitions m—>l, n^j from the field-perturbed levels.
First consider the simplest case of stationary atoms with non-degenerate
states, making use of the formalism of probability amplitudes. In the
expansion (2.3) of the wavefunction we separate the terms corresponding
to the states m and n:
4\t) = am(t)Wm + an(t)4'n + 2 (394)
,n
and in the system of equations (2.5) for the probability amplitudes we
take into consideration the fact that the matrix element of the interaction
Hamiltonian differs from zero only for the transition m-n (resonance
approximation):
dm + Ymam = -iVmnan; an + y„a„ = a; + y,a; = 0.
(3.95)
The case of a monochromatic field and the dipole approximation,
vm„= -1 dmnEe“-', E = + £*e'“), (3.96)
is of the greatest practical importance. The condition (3.93) enables one
to retain in the expression (3.96) only one of the two terms
Vm„ = —Ge~'a‘, Q = a)-(omn, G = dmn%/2ti; (3.97)
dm + Ymam = iGe-,a,an, dn + Yan = iG*eia,am. (3.98)
The system of equations (3.98) can be readily solved and its matrix S(t, t0)
(evolution operator), which makes it possible to find solutions under
arbitrary initial conditions, is given by the formula
/ iGe-il2(0 \
/Л1е_“1Т + Л2е_“2Т ------(e~“,T —e_“2T) \
a2—ocx I
S(t, to)” iG!*eiQr I’
-------(e-“1T - e~“2T) + Л2е-“,т)е1От /
\ <x2 - a i /
= M/m + y„ + i*2 ± V(y„-ym + ii2)2-4|G|2],
т = t —10, А = Ут—2, A2 = l-Al = Ym~<X-, (3.99)
a, — a2 a2 —
138
Resonance radiation processes
[Ch. 3
where oc2 and A12 are the roots of the characteristic equation and
integration constants respectively.
The matrix S(r, t0) may be represented as
S(t, t0) = e“toS1(T)e_“too,
(fi)v = QA, Qm = 0, Qn = Q. (3.100)
The matrix exp(iQt) is used to carry out the unitary transform from the
system (3.98) to the equations with constant coefficients; the matrix S^r)
is the evolution operator of these equations depending only on т = t —10.
The explicit expression for S^r) is given by the formula (3.99) if the
multipliers exp(i£2r), exp(-i£2r0) and exp(i£2r) are omitted from it.
The elements Smm and Snm of the first column are the solutions of the
system of equations (3.98) for the initial conditions am(t0) = 1, an(t0) = 0
(the excitation of the upper level m), and the elements S„„ and Smn of the
second column are the solutions for the initial conditions, am(t0) = 0,
a„(t0) = 1- After the upper level m has been excited the interaction with
the electromagnetic field gives rise to the induced emission of a photon;
after the lower level has been excited a photon is absorbed. Substituting
SJm and Sjn from eq. (3.99) into the formulae (2.20) and (2.23) we can
obtain the emitted ($%m) and absorbed (£Й„) energies per excitation:
Я,=
ym+yn___________ha) |G|2_________
ym I22 + (ym + y„)2[l + |G|2/ymy„]’
^п= — ?Лт. (З.Ю1)
Yn
If the field has a low intensity
(ym + y„)|G|2
m ym422 + (ym + y„)2’
|G|2
YmYn
«1 +
Q \2
Ym + YJ
(3.102)
i.e. the emitted (absorbed) energy is proportional to the field intensity
|G|2. Such a dependence is characteristic of linear spectroscopy. As |G|2
increases the term of the denominator omitted in formula (3.102) becomes
essential; the 3lm dependence on |G|2 becomes non-linear and in the limit
of very strong fields it reaches “saturation”:
-^»1 + (—£—V (3.103)
Ym + Yn Ук + Yn YmYn 'Ym + Yn'
The relation (3.103) can be easily interpreted. After the atom has been
excited to the level m under the influence of the internal field the atom
can either make a transition to the state n, being induced to emit a
§3.2]
Interaction of atoms with a strong resonance field
139
photon, or relax to other states j^n as a result of spontaneous transitions
m—>j. For the emitted photon to remain in the field, the atom after the
induced transition m—>n must relax from the state n, since in the opposite
case the external field can return it to the state m, causing the atom to
regain the energy of the photon. The quantity yn(ym + y„) is obviously
equal to the ratio between the rate of decay contributing to the induced
emission and the total decay rate from both of the combining states. If
Ym « Yn the probability of photon emission is near unity. The factor
ym/(ym + Yn) in the expression (3.103) for the absorbed energy 9ln may be
similarly interpreted, as the relaxation from the upper state m (after the
excitation of the atom’s lower level n) contributes to the absorption.
The kinetics of the atom’s transitions and n-*m has a simple
form under equal decay rates of combining states:
Ym = Yn, a1.2=ym+i(i2±2i21), £2t = V|G|2 + £2^4,
Г i£2 1
t0) = exp[—(ym + i£2/2)r] cos^r) + — sin^r) ,
L J
S„m(t, to) = i 77 exp[i£2t0 - (ym - i£2/2)r] sin^r). (3.104)
The probability amplitudes Smm(t,t0) and S„m(t,t0) turn out to be
oscillating and damped functions of time. Oscillations obviously reflect the
fact that after the m—*n transition the field induces the reverse transition
n->m, then again the transition etc. Roughly speaking, part of the
time the atom is in the state m; in the other part it is in the n state. Since
Ym = Yn the atom relaxes from both of the states with the same rate and
the damping of Sjm does not depend on ]G|. If the oscillation amplitude is
large enough (4|G|2ȣ22) and during the damping time l/ym several
oscillations occur (|G|-1«l/ym), then, on average, the atom stays in the
states m and n for approximately the same times, and it “gives up” to the
field the maximum possible proportion of the energy hco accumulated in it
according to eq. (3.103).
К Ym^Yn, which is typical, for instance, of the electronic states of an
atom, the evolution in the external field takes a different form. Let £2 = 0
(exact resonance) and 4 |G|2<(y„ — ym)2. Then the roots аг and a2 are
real, i.e.
а1Д = 1(ум + уй)±Х, % = V(yn - ym)2/4 - ]G|2, (3.105)
140
Resonance radiation processes
[Ch. 3
and the probability amplitudes do not oscillate. For instance,
Smm = exp[—i(ym + y„)r] cosh(/r) +
Yn~Ym
2%
sinh(/r) ,
G*
S„m = i — exp[-i(ym + y„)r] sinh(/r).
X
(3.106)
It is easy to explain the absence of oscillations in this case. Assume
y„ » Ym- Then after the atom has been excited to the level m and the
induced transition m—>n the atom rapidly relaxes from the state n, so that
the field “has no time” to cause the reverse transition n —> m (since it has
been assumed that y„ — ym==y„>2 |G|). Such evolution is similar to the
well-known mode of “aperiodic” evolution of a strongly damped pen-
dulum. If the inverse inequality is fulfilled, y„ « ym, the oscillations are
suppressed by a rapid relaxation from the initial state.
Remember that the degree to which non-linear phenomena are mani-
fested is determined by the value of the parameter |G|2/ymyn. If y„»ym
or Yn « Ym the following inequalities may hold true:
Yn»\G\2»YmYn, Ym»\G\2»YmYn,
which means that the emitted (S?m) and absorbed (!%„) energies may reach
saturation (as a function of |G|2) under the conditions of aperiodic
evolution described by the formulae (3.106).
When the external field intensity is high enough the roots a1>2 become
complex,
aM = i(r». + r»)±i|xl, 4 |G|2 > (y„ — ym)2, Q = 0, (3.107)
the hyperbolic functions become trigonometric functions and the evolution
of the atom takes the form of damped oscillations. In the limiting case
4|G|2»(y„ - Ym )2 the formulae (3.106) go over into (3.104) (if in the latter
we assume £2 = 0) with the difference that the damping is given by the half-
sum (ym + y„)/2. Such a damping rate is in conformity with the notion of
approximate equality of probabilities of an atom’s being in states m and n.
The increase of |G|2 and |£2| brings about an increase in the oscillation
frequency of amplitudes but the conclusions with respect to emitted
(absorbed) energy prove to be different. Indeed, according to the
relations (2.20) and (3.104)
P(t) = -2h(o Re[iVnmSmmS*m]
= -2ЙШ ^VQ22^ 4^GP 1 + 4 |G|2t)’ Ут = y-
V ie T 4 | (j |
(3.108)
§3.2]
Interaction of atoms with a strong resonance field
141
The sign changes of the field work can be interpreted in accordance with
the aforegoing as a result of successive transitions of the atom, i.e. m—*n,
n—etc., which are accompanied by the emission and absorption of a
photon. The increase of |G|2 leads to an increase in the oscillation
frequencies as well as the coefficient of the sine so that ultimately the
emitted energy summed over the total evolution time also increases. As
for the increase of Q, this increases the oscillation frequency and
decreases the coefficient of the sine in eq. (3.108), so that with respect to
the emitted energy the result will be the opposite.
Physically, it is quite clear that an external field with any spectrum
induces the transitions between atomic states. Therefore the qualitative
properties of the evolution are far from being typical of the monochroma-
tic field. Let, for example,
Vmn = -С(0е-^, ym = y„, (3.109)
where <p0 is a constant quantity and G(r) is an arbitrary real function of
time. This is the situation when the field is bichromatic:
E = 2^[cos(cl»iZ + tpj + cos(<w2t + <p2)], (3.110)
if the frequencies and <w2 are symmetric with respect to <wmn:
mn
dmn% Шу-ш2 _i<r
—— cos —-— + —-— e ,ф,
2й X 2 2 /
(3.111)
2
co, + <w2 = 2<um„, <px + cp2 = 2<p.
In more general cases, the assumption (3.109) also requires some
symmetry of the field spectrum with respect to the frequency of the
transition wm„. Under the conditions (3.109) the matrix S(Z, t0) of the
system (3.98) is obtained in the explicit form
cos/(t)
.iei’’°sin/(r)
ie i’’"sin/(t)
cos/(t)
f(t)= Г G(Mdt,.
•'/O
(3.112)
It is seen from eq. (3.112) that with ym = y„ and sufficiently general
assumptions about the form of an external field, the probability ampli-
tudes prove to be damping and oscillating functions of time, and to the
large values of G (as compared with ym) there correspond approximately
equal mean probabilities of finding the atom in the states m and n.
In contrast to this result, if the relaxation constants ym and y„ differ
significantly and the external field is not too strong, i.e.
max I Vmn|2 «(yn-ym)2/4, y„»ym, (3.113)
142
Resonance radiation processes
[Ch. 3
the evolution possesses the properties of an aperiodic mode. It follows
from the eq. (3.95) that if the conditions (3.113) are fulfilled the
amplitude an will be changing much more rapidly than am, so approxi-
mately we can put
t0) = —i Г VX„(tO exp[—r„(t - t0) dr,
•'to
« —iSmm(t, t0) f VXn(ti) exp[—y„(t - t,)] dtp (3.114)
Ло
Substituting this expression into the equation for Smm(t, t0) we obtain the
formula
Smm(t, t0)
«=exp{-ym(t-t0)- f dt, [ dt2 VmB(t1)VX.(t2)exp[-yl,(t1-t2)]),
Ло Ло '
(3.115)
which reflects the main peculiarities of the “aperiodic evolution”.
From general considerations it must be clear that under a high enough
intensity of the external field the mode of “damping oscillations” will exist
in spite of the difference in relaxation constants: whatever the dependence
of Vmn on t might be, the external field will of necessity cause multiple
direct and reverse the transitions m —> n and n -* m during the lifetimes of
the levels m and n. Indeed, it can be readily shown that under these
conditions the matrix S(t, t0) can be represented as (see ref. [11])
S(t, t0) = exp[—(ym + y„)r/2] x
(exp{—MO “ ф(*о)]} cos/(t) i exp|— [<p(t) + <p(t0)]} sin/(t)\
i exp{^ MO + V(<o)]} sin/(t) exp|i [<p(t) - <p(t0)]j cos/(t) j
(3.116)
where the following designations are introduced:
f(t) = Г G(t,) dt„ G(t) - e2**'» = Vi,(t)/U).
JtQ
(3.117)
With <p(t) = constant the expressions (3.116) pass into the exact solution
§3.2]
Interaction of atoms with a strong resonance field
143
(3.112). It must be emphasized that the damping rate of probability
amplitudes in the expressions (3.116) is equal to the arithmetic mean
value of the damping constant of the states m and n of an isolated atom,
which is in accordance with the ideas presented above*.
When some problems of non-linear spectroscopy (e.g. of the emission
spectrum due to the spontaneous decay of atomic states perturbed by an
external field) are being analysed, another description of the kinetics of
transitions is of interest. This is based on the expression of probability
amplitudes in terms of exponential functions. Substitute the expressions
(3.99) for Smm(t, t0) and S„m(f, t0) into the formula (3.94):
^(f) = (Aie-e,T + A2e_“2T)V’me-i£'”'/* +
iG*
-------(e_“,T - e-aiT)eiatb„e~iE'',/,‘ +
a2 — <*i
S ау(/о)^ехр(-уут-1Е//й),-
j^m,n
<*1,2 = <*1,2 + »<*T,2- (3.118)
It can be seen from the relation (3.118) that the atom’s evolution in the
external monochromatic field is described by a wavefunction consisting of
the terms characteristic of a system with quasi-stationary states with the
energies (fig. 3.5)
Eml = Em + ha", Em2 = Em + ha2,
Enl = En + й(а? - £2), En2 = En + й(а'' - «)• (3.119)
The damping constants of the sublevels ml, nl and m2, n2 are equal to
respectively a\ and a2. We may, consequently, say that the monochroma-
tic field causes the splitting of isolated atom levels Em and E„ into the pairs
of sublevels Eml, Em2 and Enl, En2.
The'values of the “quasi-energies” of the sublevels, their relaxation
rates and the probability amplitudes of finding an atom on these sublevels
depend on |G|, £2, ym and y„. Figure 3.6a,b shows the quantities
I a'1,2 - (Ym + y„)/2|, |aT,2 - £2/2|
versus |G|2 for different values of |y„ — ym| and |£2|. When the resonance
is exact and the values of |G| are small (4 |G|2<(y„ - ym)2) there is no
energy splitting and the sublevels differ only in damping rate (a'!,2 = 0,
* More detailed analysis of the kinetics of induced transitions for some particular cases is given in refs
[11, 23].
144
Resonance radiation processes
[Ch. 3
♦n-E,
♦ , Em
m' m
Fig. 3.5. Level splitting of atom interacting with a monochromatic held.
al #= al); if 4 |G|2 > (y„ - ym)2 the situation is quite different: the damping
rates are equal (a{ = al) and the quasi-energies of the sublevels are
different (af^aS)- If £2=^0, then always aj^al and for large
enough values of |G|2 depending on |£2| and |y„ - ym| the damping rates
asymptotically approach (ym + y„)/2 and the corrections to the quasi-
energies ajf and approach ±|G| + £2/2.
The splitting of atomic levels by a monochromatic field can be
interpreted in terms of the stationary theory of perturbations as a removal
of degeneracy in the “atom + field” system. For the analogy to be
complete let us assume y„ = ym and consider two states of such a system:
(1) an atom on the level m, in the field there are N photons й<м; (2) an
atom on the level n, there are N + 1 photons in the field. The energies of
these states in the absence of the atom-field interaction are given by
Ej = Em + tuoN, E2 = En + ttco(N + 1) (3.120)
respectively. As a result of the atom-field interaction the energies’ values
will change, the corrections to the expressions (3.120) being obtained
from the secular equation
Em - En - й<м + AE hVmn =
«VL AE
the solutions of which are
AE1j2 = ^[<o- <om„± V(<w-<wTO,)2 + 4|Vmn|2].
44
§3.2]
Interaction of atoms with a strong resonance field
145
Fig. 3.6. Characteristic roots 2 versus |G|: (a) |or' - (ym + y„)/2|/|y„
(b)|<?-fi/2|/|y„-ym|.
ym|;
146
Resonance radiation processes
[Ch. 3
The expression obtained for АЕ1>2/й coincides with the imaginary part of
the roots determined by the relation (3.104). Thus from this viewpoint the
level splitting is interpreted as a result of degeneracy removal in the
“atom + field” system owing to their interaction.
If the field is not monochromatic, level splitting continues to take place,
but it has a more complicated form. Assume that the matrix element
is a periodic (with period T) complex function of time; then the
evolution operator of the system of equations (3.95) can be expressed as
8(/,/о) = Ф(0е-% (3.121)
where R and ф(£) are a constant matrix and a periodic matrix (with period
T) respectively (see, for example, ref. [23]). The solutions (3.121) can be
expanded into a Fourier series and then
^(0 = exp(—iEmt/ft) У As exp(—yst - is A<w t) +
s
ipn exp(—iE„t/ft) У Bs exp(~rst — is A<w f) +
s
У ipjOj exp(-Yjt — iEjt/h); Aw = 2л/T, (3.122)
j
i.e. in this case the levels m and n split into an infinite number of sublevels
rather than into two as for the case of the monochromatic field. The
interaction with a bichromatic field and the solution (3.112) may be a
simple illustration of the expressions (3.121) and (3.122).
The level splitting effect is characteristic not only of the two-level case
of the resonance approximation. Assume that the external field spectrum
contains N — 1 monochromatic components, each of them being resonant
to one of the N — 1 adjacent transitions (fig. 3.7), so that N states are
involved in the interaction. Probability amplitudes under such conditions
Fig. 3.7. Scheme of levels and transitions in a polychromatic field.
§3.2]
Interaction of atoms with a strong resonance field
147
satisfy a linear system of N equations which by means of an appropriate
substitution is reduced to a system of equations with constant coefficients,
and the evolution operator in the new representation depends on t —10:
a(t) = ein'a1(0> («)iy = «A,
S(t, t0) = e^'S^t - t0)e-in'°. (3.123)
The characteristic equation is of order N and has the same number of
roots. If the roots of the characteristic equation are not multiple the
probability amplitude of each state is a linear combination of N
exponential functions. Therefore, in this case too, level splitting into
sublevels takes place, the number of sublevels being equal to the number
of interacting states.
Thus the splitting of atomic levels by an external electromagnetic field is
a universal effect of non-linear spectroscopy.
It should be kept in mind that there is an essential difference between
the notion “stationary state” and the state characterized by a quasi-
energy. Each stationary state of a closed system possesses a certain energy
(in the strict sense of the word) and can be excited separately, independ-
ent of the other stationary states. The latter is especially important from
the viewpoint of problems of spectroscopy. In contrast, the states
characterized by the values of the quasi-energy (Eml, Em2 and Enl, En2 in
the two-level system) cannot be excited independently. Therefore the
quantities such as Eml have a meaning different from that of the energy of
stationary states of a closed system. Nevertheless, the idea of level
splitting of an isolated atom proves to be very useful when analysing
problems of spectroscopy, because the transitions from the states m and n
perturbed by the external field may be reduced to the set of transitions
from the sublevels ml, m2, nl and n2. These ideas will be discussed in
more detail in section 3.3.
The difference between the field level splitting and the level splitting in
a steady external field, e.g. an electric field (Stark effect), must be
emphasized. In the latter case the electric field removes the orientation
state degeneration, and therefore the states of different values of the M
projection of the angular momentum have different energies. In other
words, the Stark effect consists in the shift and not in the splitting of M
sublevels. The resonance electromagnetic field at the same time causes, as
has already been stated, the splitting of each magnetic sublevel into a
couple of components, if we deal with a monochromatic field, and into an
infinite number of sublevels in more complicated cases.
148
Resonance radiation processes
[Ch. 3
The second difference, very important for spectroscopy, is as follows.
According to relation (3.118) and fig. 3.5, the sublevels with close values
of the quasi-energy (Eml, Em2 or Ея1, En2) are characterized by one of two
wavefunctions, either ipm or ipn. The situation for the steady field is
different. Putting w=0, штп»ут,уп from the expression (3.118) we
obtain
^(0 = (A2ym - Bip„) exp
[A,ipm + Bip„] exp
JGI\1
\ й com„/ J
+
(3.124)
The energy levels* corresponding to eq. (3.124) are shown in fig. 3.8.
Unlike the four sublevels in fig. 3.5 here there are two levels, each of
them corresponding to a linear combination of both wavefunctions of an
isolated atom. It is quite obvious that the aforegoing difference between
the wavefunctions (3.118) and (3.124) is fundamental to the spectroscopy
of transitions from the states m and n (the line structure, selection rules
etc.).
Despite the above, the splitting of atomic levels in an electromagnetic
field is sometimes called the dynamic Stark effect. We do not think that
Fig. 3.8. The levels of an atom in a steady electric field.
* It must be taken into account that under ш = 0 both terms must be retained in the expression (3.96)
for Vm„. The multiplier 4 of |G|2 in eq. (3.124) is a result of this.
§3.2]
Interaction of atoms with a strong resonance field
149
this term is good although it dates back to the pioneer work of Autler and
Townes [25] and subsequently throughout the book the term “field
splitting” will be used.
The interpretation of non-linear phenomena revealed the essential role
of relaxation processes. At the same time, until now a simplified scheme
has been used that does not allow for the collisions and spontaneous
cascade transitions, only including the quenching of levels due to
spontaneous decay. Still within the scope of the model of non-degenerate
states we shall now treat the interaction of an atom with the external field
making use of the kinetic equation by means of which relaxation can be
most fully taken into account.
In the resonance approximation it suffices to know diagonal and
non-diagonal density matrix elements relating to the pair of levels m and
n. According to eq. (2.60)
/ d \
\dt+4Pii
±2 Re(iV*„pm„) + AmnPmm&jn + Qj,
(3.125)
I} = 2y; + Vj- vh T + iA = ym + y„ + vm„-vm„,
where 7J and Г+ 1Д are relaxation constants incorporating both spon-
taneous and collisional processes; the term Am„pmmdy„ describes the
spontaneous cascade transition m—Qj is the in-term of the collision
integral giving the number of excitation acts of the level per time unit; the
signs ± in the equation for p„ correspond to j = m and j = n respectively.
Let the external field be monochromatic, so that the matrix element Vmn is
given by the formula (3.97); the substitution
pmn = pe~ia>, Q = (o -ытп (3.126)
reduces eqs (3.125) to a system of equations with constant coefficients:
/ Э \
I + f}/ )Pjj = T2 Re(iG*p) + Amnpmmdjn + Qj,
\at /
д 1
- + r- i(Q-A) p = -iG(pmn - pnn).
Lot J
(3.127)
Consider first the kinetics of a transition which occurs after the
instantaneous excitation of levels (Q; °c <5(t)). Such a statement of the
problem is equivalent to the solution of the system (3.127) under some
150
Resonance radiation processes
[Ch. 3
initial conditions. Taking
Pa = p = Ae~u, p* = Ae~u
we may come to a uniform system of linear equations in A;, A and A,
whose characteristic equation has the form:
(Л - Гт)(Л - Д)[(Л - Г)2 + (Я - A)2] +
4 |G|2(Л - Г)[Л - КД. + Гп - A™)] = 0. (3.128)
As is well known, the roots of a fourth-order equation can be obtained in
an explicit form. However, in the general case the expressions for the
roots are too cumbersome and practically useless. Because the coefficients
in eq. (3.128) are real the roots are either real or pairwise complex
conjugate. It can easily be shown that the sum of the roots equals
Гт + Г„ + 2Г.
If the external field has a low enough intensity, the term with |G|2 in eq.
(3.128) can be neglected and its roots
Л1 = Гт, Л2 = Г„, A3,4 = r±i(£2-A) (3.129)
describe the evolution of diagonal and non-diagonal elements of the
density matrix of an isolated atom. In the opposite limiting case when |G|
significantly exceeds all the relaxation constants and |£2—A| it follows
from eq. (3.128) that
Ai = 2(Гт + Гп — Amn), A2 = Г,
Лз.4 = 1(Д. + Д -+ 2Г + А™) ± 2i |G|. (3.130)
The root Aj evidently corresponds to the damping of the atom on both
levels; the term — Amn reflects the fact that the spontaneous transition
m—>n, taking place at the rate Amn, does not decrease the number of
atoms on the levels m and n. The root Л2 characterizes the damping of
non-diagonal density matrix elements and A3 and A4 describe the prob-
ability oscillations due to the induced transitions m—>n, n—>m. The
Einstein coefficient increases the oscillation quenching rate, which may be
interpreted as the absence of coherence between the spontaneous and
induced transitions.
Now consider the case of a stationary excitation (<2; is independent of
time). Here p„ and p do not depend on t and the equations (3.125) change
over into the linear system of algebraic equations
= А^р^д^ T 2 Re(iG*p) + Q„
[Г - i(£2 - A)]p = —iG(pmm - pnn), (3.131)
§3.2]
Interaction of atoms with a strong resonance field
151
whose solution can be written as
= 2T|G|2 Nm — Nn
Pmm m Гт (Q-Ay + I?
= N 2Г|С|2/ Amn\ Nm-Nn
pnn 741 + гп V Гт/(Й-Л)2 + П’
p = —iG
Pmm Pnn
Г-Цв-ДУ
Pmm Pnn
Nm~Nn
1 + Гк/[Г + (Q- A)2] ’
П = П1 + ^),
2 |G|2 / 1 ! 1 Am„\
г \гт Гп ГтГп/
^m = Qm/rm, N„ — QJ Г„ + AmnQm! ГтГ„.
(3.132)
(3.133)
(3.134)
(3.135)
Here Nm and Nn are the stationary populations of the levels m and n at
|G|=0, which are determined by the excitation rates and inelastic
relaxation, spontaneous as well as collisional. From the expression (3.134)
it can be seen that induced transitions decrease the population difference
of levels:
I Pmm Рил I < NJ-
The extent of the decrease in the population difference is determined by
the value of the factor
ГЪ 2|G|2F / 1 _1_ Л-Д „
Г + (О-4)2 Г + (О-4)2\Г_ Г. ÄÄà ' ’
in which characteristics of all the relaxation processes are taken into
account. If the relaxation takes place only as a result of the spontaneous
transitions, 2Г = Гт + Гп = 2ym + 2y„ and the probability of spontaneous
transition Amn is negligibly small (Атп«Гт + Гп), then the expression
(3.136) goes over into an expression discussed above (see eq. (3.101)). In
the general case the deviation from the spontaneous relaxation model can
be fairly substantial.
The combination 1/Гт + 1/Гп is the total lifetime of an atom on the
levels m and n during which the atom is interacting with the external field;
the influence of the field is stronger for longer time of interaction.
According to eqs (3.132) and (3.133) the ratio of the differences pnn - N„
152
Resonance radiation processes
[Ch. 3
(3.137)
and pmm - Nm due to the field effect is given by
Pnn Nn Im /1
Pmm Nm I~n ' Im
The appearance of the factor Гт/Гп in this relation is in agreement with
the above interpretation. The multiplier 1 — Атп/Гт which appears in eq.
(3.137) and in the denominator of the expressions (3.132)-(3.134)
effectively decreases the decay probability of the state n. If the level
relaxes as a result of transitions only to the state n (Amn = Гт) the latter’s
population does not change at all under the action of the external field.
This fact can be easily explained. The system’s evolution after it has been
excited to the state m is a result of several processes, i.e. induced and
spontaneous transitions m-n and spontaneous transitions to other states
j Ф n. However, if the latter are absent (Am„ = Гт) then all the excitations
of the upper level m, regardless of the external field intensity, result in the
population of the lower state as well, so that the number of atoms in the
state cannot depend on |G|, which is proved also by eq. (3.133).
Similarly to the above analysis of the role of the atom’s lifetime on the
levels m and n, the quantity
Г _ 1
F+tQ-A)2 Rer-i(Q-A)
can be interpreted as the effective time of the atom’s coherent interaction
with the field due to the relaxation of the non-diagonal density matrix
element. Therefore the degree to which non-linear phenomena are
manifested is determined by the product of populations’ lifetimes with the
dipole moment lifetime.
The work performed by the external field per unit time can be
calculated by making use of the relation (3.134):
P = 2Ha)Re(iVZnpmn)
=2йшГ|С|2(Ч~^.) (3 138)
(Г2-Л)2+И1 + 2(|С|2/Г)(1/Гт + 1/Гп-Ати/ГтГй)]
Depending on the sign of the difference Nn — Nm the value of P is positive
(N„ > Nm) or negative (Nn < Nm). The former corresponds to the absorp-
tion of the radiation by the medium and the latter corresponds to
emission.
The non-linear dependence of P on |G|2 is due to a decreasing
population difference of the levels m and n as a result of induced
§3.2]
Interaction of atoms with a strong resonance field
153
transitions. The functional relationship between P and |G|2 retains the
form it had under exclusively spontaneous relaxation. The specific
properties of the relaxation scheme manifested themselves only in the
above-discussed term in the denominator of the expression (3.138). For
large enough values of |G|2 according to eq. (3.138) we have
ttw(Nn-Nm)
1/Гт + 1/Г„-Ат1ГтГп
Qn-
(3.139)
By the first of equalities (3.139) the limiting value of the power is
determined by the non-saturated difference of populations Nn — Nm and
the effective lifetime 1/Гт + 1/Гп — Атп/ГтГп of atoms on both levels m
and n. The probabilities of absorption and emission per excitation of the
states n and m are equal respectively to
Г Г — A
лт лп ^*mn
Г 4- Г — А ’ Г 4- Г — А
*т ' *п **тп лт 1 лп ™тп
If Р is treated as a function of the frequency of the external
monochromatic field co, then eq. (3.138) gives the Lorentzian profile of
the emission (or absorption) line. However, its width, in contrast to the
linear theory presented in section 3.1 (compare relations (3.138) and
(3.6)), proves to be dependent on the field intensity |G|2. The quantity
|G|2 / 1 ! 1 Amw\
Г \Гт Г„ ГтГ„/
1/2
(3.140)
is called the saturated halfwidth or the saturation halfwidth. The
expression (3.134) shows that the line halfwidth for the given population
difference p„„ — pmm is equal to Г. The field broadening of the line profile
described by the formula (3.140) is due to the dependence of the
population difference on |G|2 and co. The smaller Q=w — wm„ is, the
more pronounced is the saturation effect, i.e. the smaller is the population
difference p„„ — pmm. As the interval between co and ытп becomes larger
(line wings) the saturation takes place to a lesser degree and the
absorption (or emission) coefficient approaches the value which it would
have had in a weak field. Therefore the line contour recorded when the
external field frequency is scanned appears broadened compared with Г
[26].
154
Resonance radiation processes
[Ch. 3
3.3. The method of a probe field
In the previous sections it has been established that the external field
(either mono- or polychromatic) induces transitions between the atomic
levels characterized by a more or less complicated oscillator time
dependence of the probability amplitudes. Induced transitions manifest
themselves, in particular, by the fact that the work done by the field which
induced the transitions when summed over the total evolution time
appears as a non-linear function of the field intensity.
To obtain a more detailed picture of the kinetics of induced transitions
one may make use of the so-called probe field method which amounts to
the following. Assume that a strong external field (e.g. monochromatic) is
resonant with the transition m-n (fig. 3.9). As well as the states m and n,
the atom possesses other states j, I, g,... for which the transitions m-l,
n-g, m-j etc. can be allowed which are denoted in fig. 3.9 by wavy
arrows. Such transitions including one of the levels m and n are called
adjacent to m-n. Let comparatively weak radiation resonant with one of
the transitions adjacent to m-n, say m-l, interact with the system, and we
shall be interested in the absorption (or emission) coefficient of this field
as a function of its frequency <oM. If the probability amplitudes am and at of
the states m and / are damping exponentially, the profile of the spectral
line corresponding to the m-l transition has a Lorentzian shape as shown
in section 3.1. This will also be the case for an isolated atom. If, however,
Fig. 3.9. On the probe field method.
§3.3]
The method of a probe field
155
the atom interacts with a strong external field, then am(f) proves to be not
only a damping but also an oscillating function of time and the line profile
of the transition m-l must have a different non-Lorentzian shape,
depending on the intensity and the spectrum of the strong field.
The same conclusion can be drawn from different reasoning. According
to section 3.2 the external field causes the splitting of the level m, the
same wavefunction of an isolated atom corresponding to the sublevels ml,
m2, m3,.... Therefore the absorption (emission) of a weak field may be
considered as the result of a transition between the level I and sublevels
ml, m2,... of the state m. The line profile due to the set of these
transitions can be expected to have a complicated form which is
determined both by characteristics of the combining levels and by the
strong field parameters (its frequency, intensity and polarization).
Thus it can be said that a weak field which does not bring about
non-linear phenomena by itself “probes” the structure of atomic states
excited by a strong field, its parameters being considered unchanged when
the weak field’s frequency is being scanned. The above peculiarities of this
branch of non-linear spectroscopy account for its name—the method of
the probe field.
The probe field may be resonant with one of the adjacent transitions as
assumed in the above example, but it can also interact with the transition
m-n. The probe field method is applied also for the cases when a strong
polychromatic field simultaneously interacts with several adjacent transi-
tions (see fig. 3.7). The possibility of variations of the frequency,
polarization and propagation direction of the probe field makes this
method a powerful means of investigation of the atomic levels’ structure
and various relaxation processes.
Consider the absorption (emission) spectrum of the probe field for the
simplest case of a system of three non-degenerate states: the strong field is
resonant with the transition m-n (fig. 3.10), and the monochromatic
probe field
Ец = + ^*е‘"м') (3.141)
is resonant with the transition m-l. Assume that the applicability
conditions of the resonance approximation are fulfilled so that the matrix
elements of the interaction Hamiltonian (see eq. (3.96)) and
are determined only by one of spectral components of the total field. The
probability amplitudes am(t), an(t) and a,(t) of the states m, n and I obey
156
Resonance radiation processes
[Ch. 3
Fig. 3.10. Scheme of resonance scattering in a three-level system.
the system of equations resulting from eqs (2.4):
am + Ymam = + iG^-^a,,
an + Ynan = at + у/Д, = iG^e*^, (3.142)
= coM - a)ml, Glt = dml^/2h.
As has been noted, calculation of the work of the field (3.141) averaged
over the time of atom’s evolution must be carried out (see eq. (2.23)):
= 2ft<oM Re [ iGMe-iQ"'a/(t)aX(0 dt. (3.143)
Ло
Within the scope of applicability of the resonance approximation the
system of equations (3.142) describes the interaction of an atom with a
probe field of arbitrary intensity. The assumption of the small value of the
latter will lead to different methods for the approximate solution of the
system (3.142) depending on the initial conditions. If an atom is excited to
the state / (a/(t0) = 1) the right-hand side of the equation for a,(t) must be
neglected and then
at(t) = exp[-y,(t - to)] (3.144)
and the system of equations for am and an must be solved with a,
determined by the formula (3.144). Obviously, to such initial conditions
corresponds the probe field absorption. When level m or n is excited
(flm(^o) = 1 or a„(t0) = 1) in the equation for am the small term containing a,
must be discarded and after solution of the system of equations
am + Ymam = -iV^X, an + у„д„ = -iV^.(t)am (3.145)
its exact solution must be introduced into the right-hand side of the
equation for az. To such initial conditions corresponds the probe field
emission.
§3.3]
The method of a probe field
157
Consider in more detail the problem with the initial conditions
а„(Го) = 1 (3.146)
corresponding to the typical pattern of scattering: a photon of the
probe field is emitted and a photon ha) of the strong field is absorbed.
Performing the aforegoing calculations, one must use the element
to) of the matrix (3.99). As a result one can obtain the following
expression for the work done by the probe field:
. 2IGGmI2 n Г^ + аГГ-^+а?)-1 ,
|«i-a2| У/4-at+i^
(a2+a2*)-1-(a1 + a2*)-1~|
y/ + a2* + i£2lt J’ }
where and a2 are the characteristic roots of the system of equations
(3.145) determined by the relation (3.100)
«1.2 = lb + + ifi ± V(y» — Ут + i£2)2 — 4 |G|2]. (3.148)
According to the relation (3.147) the emission spectrum of the probe
field as a function of its frequency is described by two Lorentzian-type
terms with complex coefficients. Such a form of can be interpreted as a
consequence of the field splitting of the level m into two sublevels ml and
m2 (fig. 3.11): the imaginary part of the Lorentzian denominators
vanishes when the frequency of the probe field is equal to the
differences between the quasi-energies Eml and Em2 of the upper level and
the energy E, of the lower level divided by ft:
Q ~ (x"i = (Emi ~ Ei)/H = 0,
- a? = - (Em2 - E^/h = 0.
The real parts of the denominators (halfwidths of the Lorentzians) are
equal to the sum yz + a’1<2 of the damping rates of level / and sublevels ml
and m2. In other words, the Lorentzian denominators are characteristic
also of transitions ml-l, m2-l (the wavy arrows in fig. 3.11), which serves
as a basis for our interpretation.
If the external field is not very intense and the conditions
|G|2«|y„-ym + i£2|2, a1 = ym, a2 = Yn + iQ (3.149)
158
Resonance radiation processes
[Ch. 3
Fig. 3.11. The interpretation of a splitting of spectral lines.
are satisfied, the relation (3.147) takes the form*
2h^\GG,\2 r/J_
|уи - Ym + ii2|2 eL\2y,
1X1!
Ym + Yn + '&> Yt + Ym + 1Ц.
1
Ym + Yn -
_________1________
Yi + Yn - КЯ - £)-
(3.150)
The first term in square brackets of the expression (3.150) depends on the
frequency of the probe field in the same way as when the level m is
excited, i.e. it may be interpreted as a result of the successive absorption
of the photon ftto and the subsequent “independent” act of emission of
* The conditions specified in eq. (3.149) are necessary for the separate treatment of each term in eq.
(3.147). The conditions of applicability of perturbation theory to the expression p.147) are generally
different, i.e.
|G|2« ymy„[l + «^/(y» + y„)2],
|G|2« |y, + ym + iflM| |y, + y„ + i(i2M - O)|.
§3.3]
The method of a probe field
159
the photon йюм. Such processes are called stepwise or cascade and are
regarded as being associated with the transition through a “real” state
(sublevel ml). In this case (see fig. 3.11) the stepwise process is a
fluorescence. The second term in square brackets has a different structure:
the imaginary part of its Lorentzian denominator becomes zero under the
condition = Q or cuM = ы — ы1п, i.e. this term describes the combina-
tion scattering (Raman scattering) which is related to the transition
through the virtual intermediate state (the sublevel m2 at a distance ft |£|
from Em). Such processes are called two-photon or two-quantum proc-
esses because they are characterized by “simultaneous” disappearance of
the photon fto) and emergence of the photon йюм. The states n and I
appear as the initial and final states for the scattering and consequently
the position of its resonance is determined by the frequency ajtn and the
line halfwidth is given by the sum y„ + yz.
It should be borne in mind that to consider stepwise and two-photon
transitions independent processes will prove to be correct only when the
discussed terms of the expression (3.150) do not overlap, i.e. when they
take large values in different spectral regions. That is, under
|Я|»у„ + ут (3.151)
the terms (ym + y„ ± i£2)-1 in eq. (3.150) may be omitted and then
Ob = -t,( IGGm|2[ (У?+Ут)/Ут (К + Уп)/?*
* 1(у,+ уи)2 + й2/(у/ + ув)2 + (Я-й)4
(3.152)
The photon emission probability — £%м/йюм equals the sum of two positive
terms which may be interpreted as the fluorescence and combination
scattering probabilities, the probabilities of the stepwise and two-
quantum processes. Therefore they can be treated as independent
radiative processes. The condition (3.151) means that the distance
between the sublevels |а?— a^'| = l^l considerably exceeds the sum of
their widths a] + = ym + y„ (the states ml and m2 do not “overlap”).
In the opposite case, (|£2| < ym + y„) the stepwise and two-photon
processes are not independent, i.e. the interference of states ml and m2,
characterized by the terms l/(ym + y„ ± i£2) in eq. (3.150), cannot be
neglected. The role of interference is most pronounced under the
160
Resonance radiation processes
[Ch. 3
conditions of exact resonance (12 = 0):
g» = _h(o 2|GGM|2F 1 Yt + Ym 1 у, + y„ 1
" " Y2n~Ym L2ym(y, + ym)2+i2^ 2y„ (y, + y„)2 + i2^J’
(3.153)
when one of the terms is known to be positive and the other is known to
be negative. If ym < y„ the term with the width characteristic of a stepwise
process is negative, and for the inverse relation ym > y„ it is the other term
that will be negative. Since there is no physical sense in the notion of a
process with a negative probability, under resonant conditions the
two-photon and stepwise processes have no independent meaning and we
can speak only of the entire process of resonance scattering.
The emergence of the interference of the sublevels ml and m2 is quite
striking at 12 = 0, ym = y„ when each term in the expression (3.153) tends
to infinity, which emphasizes the fact that their separate interpretation is
quite meaningless; nevertheless, their difference remains finite (compare
the conditions (3.149) and see the footnote on p. 158):
|GGM|2 f y, (y, + y„)2
Уп(У/ + У„)2 l2y„ (y, + y„)2 + $2£
(У/ + Уп)2 121
-(У/ + ул)2 + Г22] J
(3.154)
In this case the line contour of the resonance scattering is the sum of a
Lorentzian and a Lorentzian squared, the latter having the dominant role
if У/« Yn-
Interference of real and virtual states surely manifests itself not only in
the spectrum of resonance scattering. Calculate, for example, the average
probability Wmn of an atom’s being at level m after the excitation of the
state n:
Jr00
|5mn(t,t0)|2dr. (3.155)
№
If the condition (3.149) is fulfilled it follows from the relation (3.99) that
to) =--------—77. {exp(-ymr) - exp[-(y„ + i£2)r]} exp(-ii2t0),
(3.156)
C = 1 (г“ + Л " 2 Re--------------- (3157)
|y„ - Ym + 1*2| \2ym 2y„ Ym + Yn +
§3.3]
The method of a probe field
161
The first two terms in the relation (3.157) may be regarded as the
probabilities of the atom’s being in real and virtual intermediate states.
Integrated intensities of fluoresence and combination scattering are
proportional to these terms under non-resonance conditions (see eq.
(3.152)). The cross-term in the relation (3.157), reflecting the interference
of the real and virtual states, is inessential when |i2| is large enough (the
condition (3.151)). However, in the general case the expression for Wm„
becomes absurd if the interference term is dropped:
|G|2 Ym + Yn
Yn (Yn~Ym)2+^2'
(3.158)
It becomes infinite under the exact resonance and under equal damping
rates of the states m and n. The correct expression can be obtained only
by allowing for interference:
w
rrmn
|G|2 Ym + Yn
Yn (Ym + Yn)2+^'
(3.159)
The very fact of the existence of interference phenomena is evident
from the general statements of quantum mechanics and the above
examples only show under what conditions and to what extent the
interference of sublevels arising from atom interaction with a strong field
manifests itself in non-linear spectroscopy.
Consider the relation (3.150) from a different viewpoint. Speaking of
the combination scattering term it can be said that at the stage of photon
emission the atom “remembers” with which photon it had interacted at
the absorption stage (£2M = Q or cuM = co - co/n). The linewidth yz + y„ of
the combination scattering also “remembers” that at the absorption stage
the atom’s transition from the state n took place. These properties of
combination scattering are called the frequency correlation properties. In
contrast to combination scattering, fluorescence is characterized by the
absence of frequency correlation, because the position of the maximum
(£2M = 0) and its linewidth ym + y, are independent of the absorbed photon
frequency co and of the characteristics of the initial state n.
The frequency correlation properties are closely connected with the
peculiarities of the intermediate state, whose evolution is described by
the relation (3.156). The oscillating term in eq. (3.156) (virtual intermedi-
ate state m2) contains information about the initial state (y„) and about
the absorbed quantum (Q) and it brings about the appearance of a line of
combination scattering. The time dependence of the other term in the
162
Resonance radiation processes
[Ch. 3
relation (3.156) gives no indication of the absorption act and is no
different from the dependence for the case when the state m is an initial
state. For this term it is the real atomic state which serves as the
intermediate and the fluorescence line is created by the transition through
the real state.
It should be noted that the frequency correlation properties retain sense
both under resonance and non-resonance conditions, i.e. also when the
interference of real and virtual intermediate states is allowed for and in
the case when the interference is not essential. Therefore the frequency
correlation properties may serve as a more general basis for classifying
processes under resonance scattering.
Now let the intensity |G|2 of a strong field be arbitrary. For initial
conditions (3.146) (the excitation of the level n) we have
* G
am{t) = Smn(t, t0) = —-----(e-“1T - e-“2T)e-ia,°. (3.160)
a2-
The two exponential terms are formally analogous to the virtual and real
intermediate states. Both indices at and a2, however, are now dependent
on the characteristics of the field (|G|, £2) and the two levels m and n
(ym, Yn)- As for it follows from the above that both of the resonances
provide information about the initial state and the absorbed photon.
The external field affects the imaginary as well as the real parts of
and a2 (see eq. (3.148) and fig. 3.6). Consider a simple case ym = y„ when
a1>2 = Ym + (Q ± V£22 + 4|G|2).
Figure 3.12 shows a?,2 versus £2. The asymptotic approach of the plots to
the £2 axis and to the straight dashed line corresponds to the real and
virtual states of the perturbation theory. As £2 changes from large positive
values to the negative values, for a? a smoothly passing transition from
the virtual state properties to those of the real state takes place, and vice
versa for a'i. In the interval |£2| < |G| the properties of the states differ
insignificantly.
As a quantitative measure of the memory about the absorbed photon
ftw we may take the quantity
d 1
M1-2"dfi<2"2
1 ± V£22 + 4 |G|2
(3.161)
which is called the memory factor or the correlation factor. The values of
§3.3]
The method of a probe field
163
Mit2 are within the range (0,1). To the limiting values of M = 0 and 1
there correspond the complete absence of memory (the stepwise transition
through the real state) and the absolute correlation of frequencies of the
emitted and absorbed photons (two-quantum transition through the
virtual intermediate state) respectively. Since th + a2 = У« + yn + i£2,
under arbitrary intensities of the external field |G|2 and any ratios between
ym and y„ the relation Mx + M2 = 1 holds true. This fact may be accounted
for in the following way: the intermediate state as a whole, without being
broken into sublevels ml and m2, retains information about the absorbed
photon. In the limiting case |G|2» |£2|2 we have M} = M2 = 1/2, i.e. the
memory is “shared equally” among two addends of am(t).
Under high intensities of the external field,
a1>2 = ib + y„ + i(G ± 2 |GI)], |G|2»|y„ - ym + ii2|2, (3.162)
and both of the terms in the expression (3.160) turn out to be oscillating.
In this respect they are analogous to the virtual state. This analogy is,
however, of no interest as the causes of oscillations under the weak and
strong fields are different: the weak field brings about oscillations only
because it is non-resonant, whereas oscillations under the strong field
describe the field-induced transitions between the levels m and n (see the
formula (3.155) and its discussion).
It can be seen from the relation (3.162) that the damping rates of the
two terms in the expression (3.160) are the same and are determined by
164
Resonance radiation processes
[Ch. 3
the mean value of the decay rates of the states m and n. This fact, as has
been noted in section 3.2, is a general property of the limiting case of
strong fields as the atom has almost the same probability of being in states
m and n. Under these conditions there is no longer any physical sense in
the problem of from which state (initial state n or intermediate state m)
the atom makes a transition to the state I, since the strong external field
“mixes” the states m and n, which were treated in perturbation theory as
initial and intermediate states, and the photon fttuM is emitted by this
mixed state which includes almost the same proportions of the states m
and n. Thus the notions of virtual and real intermediate states, stepwise
and two-photon transitions, are indissolubly related to the perturbation
theory and have no physical meaning beyond the scope of its applicability
when the intensity of the external field is large enough.
For the conditions given in the formula (3.162), it follows from the
expression (3.147) that
IGJ2 Г у___________
2(ym + Уп) Ly2 + (£2M — £2/2 — |G|)2
____________Y____________1 v = у + +
у2 + (£2M — £2/2 + |G|)2J’ У 7/ 2 ’
(3.163)
i.e. under large intensities of the external field the resonance scattering
spectrum consists of two Lorentzians located near the frequencies
= £2/2 ± |G| and having the same amplitudes and widths. Thus, as |G|
increases the following changes in the resonance scattering take place (see
fig. 3.13): the doublet separation increases, and as a consequence the role
Fig. 3.13. The contours of a resonance scattering doublet. Calculations are carried out for
Г/« У„ = 3y„, Q = 4(ym + y„).
§3.3]
The method of a probe field
165
of interference phenomena diminishes; the sharper component broadens,
the wider component narrows.
The above example of the resonance scattering of powerful monochro-
matic radiation, the influence of relaxation and induced processes has
been treated as well as the associated properties of the level splitting,
the role of sublevel interference, and the radical change in the frequency
correlation properties of resonance radiation processes, which reflects the
limitedness of our concept of real and virtual states of an atom in the
external field. Quite similar phenomena arise with all the transitions
involving the states m and n (see fig. 3.9), for such processes as
two-quantum absorption, two-quantum fluorescence and scattering.
Qualitatively the above-discussed effects take place under other, more
complicated, conditions. Let us consider, for example, a multilevel system
and a polychromatic external field each of whose spectral components is
resonant with one of the transitions (see fig. 3.7). Under such conditions
each of the levels splits into N sublevels where N is the number of states
interacting with the field. We apply the method of the probe field to the
m-l transition including one of the split levels m and the level I that is
unperturbed by the strong field (fig. 3.14). Under these conditions the
(a)
Fig. 3.14. (a) The transition scheme and (b) the emission spectrum of the probe field in a multilevel
system with a polychromatic strong field.
166
Resonance radiation processes
[Ch. 3
work of the probe field as a function of its frequency is a set of N
Lorentzians. Indeed, according to section 3.2 the probability amplitude
am(t) may be written as a linear combination of N exponential functions:
««(0 = S A exp(- asf). (3.164)
Each of the terms of the sum in eq. (3.164) will lead, as can be seen from
the relation (3.143), to the emergence of a Lorentzian
1 1
Yi + a,* + ii2M Yi + + i(I2M - a'')
in the expression for the work of the probe field
If a bichromatic field is resonant with the m-n transition then according
to the expansion (3.122) the levels m and n split into an infinite set of
sublevels. Consequently, in this case the emission (absorption) spectrum
of the probe field at an adjacent transition will consist of an infinite
number of Lorentzians. If the relaxation constants of the levels m and n
are the same, and the spectral components of the strong field w, and ш2
are symmetric about the transition frequency cumn, then the sublevels have
the same damping rates and are equidistant (the distance between the
neighbouring sublevels equals the difference of frequencies | <Wi — cu2|; see
the discussion of formula (3.122)). Therefore the Lorentzians in the
spectrum of the probe field are also equidistant and have the same widths
Y/ +
9L=ft(uJG„|2 £ --------------Сд ------------. (3.165)
. Yi + Ym + 4Я - s((o1 - w2)]
As has already been noted, the method of the probe field may be
applied for the same transition with which a strong external field is
resonant. In this case the probe field spectrum has a more complex form,
since both of the combining states are perturbed (split), whereas for the
adjacent transitions discussed above one level had not been perturbed by
the external field.
Equation (2.11) for the probability amplitudes can be written as
a + ya = -iV(t)a - iVM(t)a, (3.166)
explicitly separating the matrix VM(t) of the interaction with a probe field
resonant with one of the transitions, e.g. m-n:
VM(t) = -6ме-‘а"РЖ - GJe^'PX, (3-167)
§ 3.3] The method of a probe field 167
where py is the row where only the /th element is non-zero, i.e.
(₽/)* = «/*, = (3.168)
so that р*,рл specifies the element (VM)mn.
Let us represent a column of amplitudes a as
a = b + c, (3.169)
where b is the zeroth-order approximation with respect to the probe field
amplitude, and c is the first-order correction due to VM(t). Substituting the
expression (3.169) in eq. (3.166) and neglecting the term VM(t)c which is
of a higher order of smallness, we obtain the equations
b + yb = —iV(t)b, (3.170)
c + yc + iV(t)c = —iVM(t)b. (3.171)
The initial conditions for c are zero, so the solution sought must be
determined by the right-hand side of eq. (3.171):
c = c+ + c-, (3.172)
c+ = iG*e'Q"' ( e-iQ^-''>S(t, ti)PXb(ti) dt, (3.173)
A)
c- = iG^e-10"' Г eia^S(t, t1)PJ,P„b(t1) dt15 (3.174)
where the matrix S(t, ^) is the evolution operator of the system (3.170).
The addends c+ and c~ due to the two terms in the Hamiltonian (3.167)
of the interaction with the probe field respectively describe the emission
and absorption of the photon йюм. Indeed, it is clear from the system of
equations for c* and c~,
Cm 4" YmCm 4" 1 0, 175)
c+n + y„c; + i 2 V^cf =
j
that the probe field induces the transition from the upper level (bm) to the
lower level (c+). In the system of equations for c~ and c~, in contrast, the
right-hand side is proportional to GMb„ only in the equation for c~ which
corresponds to absorption of the photon йсом.
168
Resonance radiation processes
[Ch. 3
In the general expression for the work of the probe field
Я = Тг[ dEMa(0at(0 d',
*»0
aa+ = bb+ + bc+ + cb+ + cc+ (3.176)
the small term cc+ obviously must be dropped. The expression (3.176) in
addition to the stationary part contains terms oscillating with the
frequencies cuM - co and 2 (co — cuM). The point is that two monochromatic
field components (co and coM) induce in the atom dipole moment
components oscillating with the frequencies coM, co, = 2cu — cuM (see the
expression (3.222)), the latter’s phase being dependent on the phase
relations between the strong and probe fields. The frequencies coM and co
are symmetric with respect to co. The dipole moment component which
oscillates with the frequency co gives rise to the so-called parametric
phenomena*. Specifically, radiation with a frequency co may be generated
in the medium. The non-stationary part of the work in eq. (3.176)
corresponds to the work of the field with the parametric component of
the dipole moment and with the component oscillating with the frequency
of the strong field.
Retaining only the stationary part of the work done by the probe field
and taking into account the relations (3.167) and (3.172) we may obtain
the following formula:
Я = 2йюм ReliGM f e^'p„[b(f)c+(0 + с+(Г)ЬЧ0>т (3-177)
Ло '
Comparing eq. (3.177) with the expression (3.143) for the field work at
the adjacent transition we can conclude that b(f) and bf(t) in eq. (3.177)
are analogous to the factor am(t) in eq. (3.143) and the c± play the role of
the amplitude at(t). The above analogy is formal since the c± in contrast to
a, are perturbed by the strong field and contain information about the
kinetics of induced transitions (see eqs (3.173) and (3.174)). The second
significant distinction from eq. (3.143) consists in that fact that the
expression (3.177) contains two terms which, as already noted, describe
the emission (c+) and absorption (c_) of the probe field. Physically, this
distinction can be accounted for by the fact that the sign of the work of
* For a detailed discussion of parametric phenomena of non-linear optics and spectroscopy see ref.
[27].
§3.3]
The method of a probe field
169
the field at the adjacent transition is predetermined by the initial
conditions whereas in the case under consideration both absorption and
emission of the photon can take place under any initial conditions.
Indeed, let an atom at the initial time be excited to the state m. This may
be accompanied by the emission of the photon due to the transition
m—>n induced by the probe field. At the same time, the strong external
field, generally speaking, can also induce a transition to the state n, after
which the photon of the probe field can be absorbed followed by the
transition n —> m.
For the two-level system and monochromatic external field (fig. 3.15)
the amplitude b(r) consists of two exponentially time-dependent com-
ponents (splitting of the levels m and n into the sublevel pairs ml, m2 and
nl, n2). The elements of the matrices also consist of two components.
Consequently, the probe field spectrum at the transition m-n is a sum of
four Lorentzians:
| GJ2 Re
_____cn______
4- a* + ie
c22
<x2+ a* + if
C21।£12
a2 4- a* 4- ie 4- a* 4- is
e = Q^ — Q.
(3.178)
The first two terms are near the frequency Qfl = Q and their halfwidths
equal 2a{ and 2a2, the second two terms have the halfwidth a't + a2 =
ym 4- y„ independent of |G|2 and displaced from the frequency of the
strong field:
of - »2 = ImV(y„ - ym 4- ifi)2 - 4 |G|2.
These properties enable us, evidently, to treat individual terms in the
Fig. 3.15. Scheme of transitions induced by the probe field in a two-level system.
170
Resonance radiation processes
[Ch. 3
expression (3.178) as results of transitions between the sublevels of the
upper and lower states (see fig. 3.16)
ml —>nl, m2—>n2, ml —>n2, m2—>nl, (3.179)
the sequence of transitions (3.179) corresponding to the sequence of
Lorentzians in the expression (3.178).
In the case plotted in fig. 3.7, the polychromatic external field splits
each level into N sublevels. Therefore the probe field spectrum at the
transition between two perturbed levels will consist of N2 Lorentzians
which may be treated as a result of transitions between the sublevels:
ms—>nr; s,r = 1,2,... ,N,
where s, r enumerate the sublevels of the states m, n. The Lorentzians
corresponding to the transitions s —> r = s have the form
(as + a* + ie)"1, s = l,2,...,N,
i.e. they are located near the frequency = Q and have a halfwidth 2a's
determined by the damping rate of the sublevels ms, ns (Q = w — co
is the frequency of the strong field spectral component resonant with the
transition m-n). The other N(N — 1) Lorentzians are broken into pairs
[a, 4- a* 4- i(QM - Д)]-1, [a, 4- ar* 4- i(QM - O)]-1.
The components of such a pair have the same halfwidth a' 4- <x's and prove
to be symmetrically displaced from the strong field frequency:
The sublevel interference studied in the course of discussing the
formulae (3.147) and (3.150) occurs also under more complicated condi-
tions (formulae (3.165), (3.168) etc.).
The concept of the interference of sublevels is closely connected with
the evolution description by means of probability amplitudes. Within the
scope of the density matrix formalism non-linear phenomena are inter-
preted in a somewhat different manner since the interference of sublevels
is automatically taken into account in such quantities as the probability of
an atom’s being in a specific state. The latter has been found by
considering the two-level system as an example (see discussion of the
formula (3.157)).
Now let us return to the analysis of the simplest type of the probe field
method (see fig. 3.10). The external field mixes the states m and n, and
the probe field is resonant with the transition m-l adjacent to m-n. If the
§3.3]
The method of a probe field
171
levels m or n are excited, the amplitude az(t) obeying eq. (3.142) can be
represented as
az(0 = iG* exp(i£Mt) [ exp[-(yz 4- i£M)(t - t,)]am(ti) dt,. (3.180)
Ло
Substituting this expression in the relation (3.143) for the work of the field
we obtain (!%' is for emission)
9?; = -2ftwM |GM|2 Re Г dt [ dit exp[-(yz - i£M)(t - Г1)]ат(г)аХ(Л).
Ло J to
(3.181)
In this relation the product am(0aXGi) resembles the correlation function
whose Fourier transform determines the spectral line shape (see section
3.1). Nevertheless, the interaction of the atom with the external field
results in the fact that the values of the amplitudes am(t) and am(tt) at the
different instants t and tx differ not only for stochastic reasons (relaxation)
but also owing to the dynamic properties of the evolution due to the
induced transition to other states. Consequently, not only must correla-
tion between the relaxation processes taking place in the state m be
essential but also the correlation between the dynamic processes taking
place in the states m and n.
Making use of the evolution matrix S(r, rz), we can express the
amplitude am(t) at a later instant of time in terms of the amplitudes am(zt)
and a„(ti) at the time tT.
am(t) = ti)am(t,) 4- Smn{t, ti)a„(ti),
after which the expression (3.181) takes the form
-27tftwM^-Re[JZm(QM)<Tmm 4-Jz„(QM)<TBm],
2ym
where the following notations are introduced:
4ХЯ) = “ / 0/m(T)eiO"T dr,
<Mr) = e"y,TSmm(T),
Л.(Ч) = dr,
ФМ = e-^S^t, t1)eia'1,
(3.182)
(3.183)
(3.184)
(3.185)
172
Resonance radiation processes
[Ch. 3
(Jmm = 2yml |am(fi)|2 d/b
Ло
Jr°°
a„Gi)aXGi) drj. (3.186)
fo
According to the relation (3.183) the probe field emission spectrum can
be represented as the sum of two terms whose spectral properties are
determined by different correlation functions ф1т(т) and ф,„(т). The
function ф/т(г) is expressed in terms of the probability amplitudes of the
states I and m corresponding to the initial conditions az(t0) = 1 and
am(t0) = 1 respectively, i.e. it has the standard meaning of the correlation
function used in linear theory* discussed in section 3.1. The only
difference is in the explicit time dependence of those amplitudes which in
the case (3.184) contain information about the interaction with the
external field. The coefficient amm of JZm(i2M) corresponding to the
correlation function </>Zm(r) is obviously equal to the probability of the
atom’s being in the state m. As stated above, the interference of sublevels
ml and m2 is reflected in amm.
The second correlation function ф/л(т) appearing only in non-linear
spectroscopy is proportional to the amplitude of probability of the state m
under the initial conditions a„(r0) = l, which makes allowance for the
correlation of relaxation processes in the states m and n in conformity
with the aforementioned general considerations of the structure of the
expression (3.141) (the product Smn(t, tj) exp(i£?Z,) as is clear from eq.
(3.99) depends only on т = t — tx}. Obviously, the coefficient onm serves as
the measure of mixing of the states m and n by the external field. If its
intensity tends to zero, then crnm—>0 and Smn—>0, so that the expression
(3.183) becomes a conventional formula of the linear theory.
The integrated (with respect to £M) work of the field is determined
exclusively by the first term in eq. (3.183). Indeed, the integration over
involves only exp(i£2Mrj and results in the appearance of a 8 function of t,
since
Re I d4 = 1, I Jln(W d£M = 0, (3.187)
= [ 9% d£M = amm. (3.188)
J 2ym
* In section 3.1 theoretical studies were based on the analysis of the non-diagonal element of the
density matrix ptm. The line contour is determined by Fourier transform of evaluated under
initial conditions p,„(t0) = 1- In terms of the probability amplitudes such initial conditions comply with
those given above.
§3.3]
The method of a probe field
173
Thus the integral intensity of the probe field emission is given by the
probability omm of atom’s being in the state m, the upper state for the
considered transition m-n. The specific value of depends on the
external field energy (the saturation effect), but the physical meaning of
this quantity remains the same as in the linear theory.
It follows from the above that the second term in eq. (3.183) describes
the change of the spectral line shape due to correlation of the states m and
n owing to the external field, its integral being unchanged.
Because of these peculiarities the term Jz„(i2M)o„m is called the
non-linear interference term, and phenomena described by it are called
non-linear interference effects (NIEFs). It must be noted that in this case
we deal with the interference or mixing of the states m, n rather than the
interference of the sublevels ml, m2 or nl, n2.
NIEFs are general for non-linear spectroscopy, no less general and
universal than the splitting of spectral lines which is reflected in the
correlation functions 0Zm(r) and 0z„(t), and the saturation effect which
determines the value of amm.
If the state m which is the upper state for the transition m-n is mixed
by the external field not only with n, but also with the other states j, e.g.
as in fig. 3.14, then instead of the relation (3.182) we have
am(t) = Smm(t, tiMti) 4- Smj(t, (3.189)
and the work of the probe field contains N — 1 interference terms:
ip I2 Г
= -2лйюм Re JZm(£M)omm 4- £ Jzy(QM)a;m . (3.190)
Here ajm and таУ be written as in eqs (3.186) and (3.185) if we
substitute in these formulae n for j. No interference term in the sum over j
contributes to the integrated work of the probe field; all these terms
influence exclusively the profile of the emission spectrum and become zero
in the absence of the external field.
There are no non-linear interference phenomena in the processes
accompanied by the transitions from the level unperturbed by a strong
external field (e.g. I in fig. 3.14) to the perturbed level (ли). Indeed, to
these processes there correspond initial conditions az(t0) = 1 and in the
expression (3.143) for one must substitute the solution of the system of
equations
a 4- ya 4- iVa = az(t) = e_y'('_'(l), (3.191)
174
Resonance radiation processes
[Ch. 3
(3.192)
(3.193)
this solution being determined by the right-hand side:
am(t) = d/b
Ло
so that for the absorbed energy we have
= 2лйюм Re JZm(QM)
2/z
according to the above statement (JZm(£M) is still given by the formula
(3.184)).
The difference between perturbed and unperturbed states from the
viewpoint of the existence of NIEFs can easily be accounted for. NIEFs
arise as a result of dynamic transitions (induced by the external field, i.e.
not stochastic transitions) from the initial level in the probe field
processes. The level I is supposed not to interact with the external field
(the probe field is weak enough by definition and does not perturb the
evolution) and the dynamic transitions from the level I are absent, which
leads to the absence of NIEFs.
Within the scope of linear theory the general laws have been stated
which relate the spectra of induced emission and absorption. Let us
formulate these laws for the absorption coefficient which is defined as
the ratio of energy absorbed in a unit volume per unit time to the density
of the energy flux of the probe field. If Qz and Qm denote the number of
excitation acts of the levels I and m per unit time per unit volume, then
= (G,!»‘ + | VL (3.194)
The absorption coefficient has the dimension of inverse length. By
virtue of the above-mentioned general laws (see, for example, ref. [2]),
=ygmAmZ(-'-^)zZm(QM); = Re JZm(Qp); (3.195)
' Si Sm '
[ A2 A (N, Nm\ .. 1Q_.
«,= I au dflu = -^XI-------------), (3.196)
J 4 ' gz /
where g, and gm are the statistical weights of the states I and ли; N, and Nm
are the populations of the levels I and m per unit volume. Integral
Einstein coefficients for the induced emission (B„,z) and absorption (Z?Zm)
are proportional to each other
Л2
gtBtm = gmBml = — gmAml. (3.197)
§3.3]
The method of a probe field
175
The spectral densities of the Einstein coefficients are also proportional:
Л2
(3.198)
In contrast, allowing for formulae (3.187), (3.190) and (3.193) and the
relations
c3 О
|dmZ|2 = — tiAml, ри = 0^ = а^, (3.199)
from eq. (3.194) we obtain
Л2 . Г_________
U" 4 L - •
Л2
^oo 4 ^ml^Pn Ртт)'
. Ami Pmm) ^m ’
j&n
(3.200)
(3.201)
When comparing eqs. (3.201) and (3.196) (we must take g,, = 1, i = m,l,
as the model of non-degenerate states is used for eq. (3.201)), we may
conclude that the integral absorption coefficient of the probe radiation in
the presence of a strong field is given by the same Einstein coefficients Bmi
and Btm and such values of the populations pu and pmm which exist in the
gas with allowance being made for stationary excitation, relaxation
processes and induced transitions. As for the spectral densities of the
Einstein coefficients, they turn out to be different for absorption and
induced emission because of the non-linear interference terms (compare
the expressions (3.190) and (3.193) for emitted and absorbed energies !%'
and !%’).
Earlier we discussed the case when the state I unperturbed by the
external field possessed a lower energy than the state m. Because of this
there were no NIEFs in absorption. If Et > Em the NIEFs will be absent in
emission (/—>m) but be present in absorption.
When the probe field method is applied to the case of a transition both
of whose levels are perturbed by the strong field the role of NIEFs
becomes greater since they exist in absorption as well as in emission under
any initial conditions. Substituting the expressions (3.173) and (3.174) in
the formula (3.177) for we may obtain a relation
= 2ftwM |GM|2 Re Г dt [ dft x
Ло Ло
Р„[<Ж(Г1)>(Г, fl) - *(r, Г1)о(/1)]К,е8*<'-'*>; (3.202)
®(f) = a(r>X0, Ф(', h) = S\f, MPXS(f, fi).
176
Resonance radiation processes
[Ch. 3
By means of the unitary transformation
exp(iS2r), («)/> = QAi (3.203)
the integrand in the relation (3.202) may be written as a product of
multipliers depending only on z = t — tx and rt = —10; consequently, the
expression for is reduced to
= 2nftwM |G„|2 Re (<r„„ - <rmm)J„m(QM) +
S <T„;Jym(QM) - 4(Я)°^ '>
j^n
JA(fiM)=-f 0A(T)eiO"TdT,
Л Jo
<ЫТ) = Sinj(r)Simk(T) exp[i(Qm - £2„)t];
a = f e-in'o(t)ein' dt, St(t - tt) = e-in'S(t, r1)e“fl.
Ло
(3.204)
(3.205)
(3.206)
The terms of the expression (3.204) proportional to onj and ojm correspond
to the absorption and emission of the probe field; sums over j contain
non-linear interference terms. Therefore, according to the general ideas,
NIEFs in this case exist in the probe field absorption as well as the
emission. The integrated absorption coefficient is, at the same time,
determined as before, by the population difference of combining states.
As emphasized in sections 2.1 and 2.2 the probability amplitude
formalism employed until now in the theory of the probe field method is
quite limited from the viewpoint of taking the relaxation processes into
account. At the same time the relaxation properties evidently must
manifest themselves in explicit expressions for the correlation functions
ф,*(т) and, consequently, also in the absorption and emission spectra of
the probe field. In order to take into consideration more general patterns
of relaxation we must turn to the density matrix formalism.
Consider the application of the probe field method for the cases already
discussed assuming the atomic states to be non-degenerate and describing
the relaxation within the limits of the model of relaxation constants (see
section 2.4). In addition, the atoms are taken to be stationary (the effects
of motion are treated in ch. 5).
In the kinetic equation for the density matrix we shall write down in the
explicit form the Hamiltonian of interaction VM of the atom with the probe
§3.3]
The method of a probe field
177
field:
p + Гр = R(2) - i(Vp - pV) - i(VMp - pVJ 4- Q. (3.207)
Here Г is the matrix of relaxation constants; the diagonal matrix Q
describes the excitation of atomic states as a result of collisions, and R(2) is
the in-term of the spontaneous transition matrix. Represent p as
p = p° + pM, (3.208)
assuming pM to be a small value due to the interaction with the probe field.
In a first approximation with respect to VM the equations for p° and pM are
p° 4- Гр° 4- i(Vp° - p°V) - R(2) = Q, (3.209)
pM 4- Грм 4- i(VpM - pMV) - R(2) = —i(VMp° - P°VM). (3.210)
It is to be noted that the left-hand sides of eqs (3.209) and (3.210) are the
same, i.e. they possess the same evolution operators. The differences are
concentrated in the right-hand sides: p° is excited by collisional processes
(Q) resulting in the population of atomic levels; pM is excited by the probe
field according to the matrix p°.
For the probe field resonant with the adjacent transition m-l (see fig.
3.14), from eq. (3.210) we obtain (the suffix p is dropped)
Pmi + (Tml 4- i Aml)pml 4- i VmiPi, = iG^-^ipu ~ pOmm),
j
Рц 4- (rjt 4- 1Дц)рц 4- i Vjk(t)pkl = -iG^-'a^p^m, j^m. (3.211)
к
A monochromatic field is assumed to be resonant with each transition
i-k\
vjk{t) = -Gjke-ia*, p°m = р/те-^‘, (3.212)
where Qjk is the difference between the frequency of the corresponding
spectral component of the strong external field and the transition
frequency a>jk. The new variables
p;7 = r;7e“i(4‘+o"")', pml = rm/e“iO'‘f (3.213)
obey a system of linear equations with constant coefficients and constant
right-hand side:
[C,; - i(QM - Aml)]rml - i Gm/r;7 = iGM(p?z - p°mm),
j
[i;z - i(QM 4- Qjm - zA;Z)]r;Z - i 2 Gjkrkl = -iG^. (3.214)
к
178
Resonance radiation processes
[Ch. 3
Hence the rjt do not depend on time and are linear combinations of the
right-hand sides of the equations (3.214). In particular, the element rml
which is necessary for calculating the work done by the probe field may be
written as follows:
fml iG^lt Pmm) Jtj(^p.)Pjm ,
(3.215)
where the coefficients JZm(i2M) and Jij(Q) may be calculated by well-
known rules. Substituting the expression (3.215) in the formula for the
probe field work we find
= —2ftwM Re(iG*rmZ) = 2лйюм |GM|2 x
Re Лт(я)(р» - P°mm) - S JtifMPjm
L / J
Re J Jim(^n) d£M = 1; J МД.) = 0,
j^m.
The relation (3.216) is analogous to eq. (3.200) which had been obtained
earlier within the scope of the probability amplitude formalism. The
differences between eqs (3.216) and (3.200) are due to the excitation
processes (relation (3.216) assumes the excitation of any level j and eq.
(3.200) takes into account only the level m excitation) and to the
relaxation constants’ values. As in eq. (3.200) the integrals of NIEF terms
with respect to are equal to zero.
Let us discuss the resonance scattering in a three-level system (see fig.
3.11) in more detail. In this case the calculation yields the following result:
= 2лйюм |GM|2x
1 R [Гл< - i(QM - Q - 4,z)](p?z - pL.) ~ iGp„m
л [Г„,-ЦЯ-О- ^)][rmZ-i(QM-AmZ)] + |G|2' '
The expression (3.217) reflects all the main effects, which have already
been generally discussed above. The denominator is a square polynomial
in the frequency Q^, i.e. it contains two resonances at frequencies
^2/41 = ^1’ Q12 = °^2t
<*1,2 = 2{Гт1 + r„t + i(^2 4- Amt + Ant) ±
V[T„Z - Гт1 4- i(Q 4- Anl - Aml)]2 - 4 |GI2}. (3.218)
§3.3]
The method of a probe field
179
The expressions for the parameters at and a2 whose real parts specify the
resonance widths are different from the aforegoing values owing to
different types of relaxation processes (compare with eq. (3.147)). The
existence of two resonances in eq. (3.218) must be interpreted, evidently,
as a result of the splitting of level m into sublevels ml and m2 owing to
the interaction with the strong external field.
The numerator of eq. (3.217) also contains the saturation effects (the
population pZim of the level m depends on the field intensity; see eq.
(3.132)); the term iGpnm describes non-linear interference effects.
By varying the conditions we may separate particular effects. If
P^nm = P°nn, then p°m = 0 (see eq. (3.134)), NIEFs are absent and is
determined by populations only. Under not very strong intensities of the
external field, i.e.
|G|2« |ГЛ, -Гт1 + i(fi + Anl- zV)|2,
and provided that Гт1 » Гп, the splitting of the scattering line shows up in
the following way (fig. 3.16): in the exact resonance (Q = 0) in the centre
Fig. 3.16. The line contour of the resonance scattering in the absence of NIEFs:' (а) Я = 0; (b)
Я = » Qi', (c) Q » Г^.
180
Resonance radiation processes
[Ch. 3
of the line there is a “negative” structure transforming as |D| increases
into an asymmetric contribution and under still greater values of |D| into
the second component of the resonance scattering doublet (combination
scattering). With large enough values of |D| and |G| the graphic
representation of versus is similar to that in fig. 3.13.
If p°u = pJw., then is due to non-linear interference phenomena and is
of necessity alternating (fig. 3.17). In the general case when p°mm ¥= p°u and
pnm ¥= 0 the sign of the work done by the probe field can also change with
the change in i.e. over some regions of the spectrum the probe field
may be absorbed and over the others it may be amplified by the medium.
As has already been observed, the notions of stepwise and two-
quantum transitions are applicable only under small enough field inten-
sities and sufficiently large values of |Я]. Indeed, if the term |G|2 in the
denominator of the expression (3.217) cannot be omitted or expanded in a
power series, then stepwise and two-quantum transitions cannot be
treated separately, i.e. it is impossible to interpret them as independent
radiation processes. Under such conditions adequate interpretation is
based on the saturation effect, NIEFs and level splitting. Now consider
the relation between the latter effect and traditional notions.
Two-quantum and stepwise transitions in their conventional sense are
connected with the first intensity approximation of the strong field, i.e.
with the expansion of the expression (3.217) in a power series of |G|2
accurate to the terms of order |G|2. Using eqs (3.180)-(3.182) we obtain
= 2ha>lt |GJ2Re{
N,~Nm
Гпа — i(£2M — Ami)
[G|2 Г2Гтя Nn — Nm
Гт1 - i(- Дт1) L Гт + (Q- Дтп)2
1___________Nn~Nm !
Д, - i(DM - Q - Дп1) Гтп + i(X2 - Дтп)
________1_____________Nt~Nm
Гп1 — i(£2M — Q — A„i) Гт1 ~ i(^2M — Дт/)-
(3.217a)
The non-linear part of the expression (3.217a) proportional to |G|2
consists of three components. The first allows for the field-induced
population change of the level m (saturation effect). The second com-
ponent is connected with NIEFs, and the third is due to the level splitting.
The traditional understanding of combination scattering (photon ha> is
absorbed, photon й<ом is emitted) obviously implies that the initial state of
§3.3]
The method of a probe field
181
(a) (b)
Fig. 3.17. The line contour of resonance scattering under equal populations of combining levels m
and I.
the atom is the level n, i.e. that the level n is populated. Consequently,
ordinary combination scattering is described by the part of the NIEF term
proportional to Nn. Under reverse combination scattering the photon й<ом
of the weak field is absorbed and the photon ha) is emitted. This process
requires the population of the level /, i.e. it is due to the level splitting and
corresponds to the part of the third non-linear component in eq. (3.217a),
proportional to Nt. The stepwise transition n-m-l implies the population
of the level n. Therefore this process is described by the part of the
saturation term proportional to Nn.
Thus radiation processes understood traditionally are in unambiguous
agreement with the three principal effects of non-linear spectroscopy, i.e.
saturation effect (stepwise transition), NIEF (combination scattering) and
level splitting (reverse combination scattering).
It follows from the expression (3.217a) that the excitation of the
intermediate level m leads to processes of all three types as Nm appears in
all the non-linear components. Physically this is quite clear since the level
m interacts with the two fields G and GM. The actual population of level m
effectively decreases each of the terms and can even lead to a sign change
of the effect. In classical papers devoted to quasi-resonant and non-
resonant processes intermediate level populations have not been treated
and in this respect the present results will add to the physical picture of
the phenomena.
Until now in all the examples the level I has been assumed to be lower
than the level tn (Et<Em, the conditions corresponding to the observa-
tion of combination scattering). If Et> Em we deal with two-photon
absorption or induced two-photon fluorescence (fig. 3.18a). The expres-
sion for is obtained here from eq. (3.217) by the change of the sign and
182
Resonance radiation processes
[Ch. 3
Fig. 3.1S. Three-level systems and radiation transitions: (a, c) two-quantum absorption and two-
photon fluorescence; (b) resonance scattering “through the lower state".
the replacement Q^—> — — <om/ and the change of sign of oj mt
means that in the matrix element of the interaction Hamiltonian VM
different terms must be retained in the relation (3.141) when combination
scattering and two-photon absorption are analysed):
=2лй<ом |GM|2x
1 [Д; + i(flM + D+ Ля/)](р^ш, - p(/,) + \Gpnm
л e[rn/+i(O„+ Я+ Дй/)][Гт,-Н(Я+ 4»,)] + IGI2’
Changing the indices, we may also find from the relation (3.217) the work
of the field when the adjacent transition includes the level n but not m,
i.e. under combination scattering “through the lower state” (see fig.
3.18b)
m++n,
Let now the probe field be resonant with the transition m-n interacting
also with a strong external field. Then eq. (3.210) proves to be more
complicated compared with the case of an adjacent transition (the suffix p
of pM is dropped)
pmn + (Гтп + i4™)pm„ + iGe '^(pmm - рии) = -iGMe ^'(pL. - P°n);
p„ + rtp„ - A^p^ ± 2 Re(iG*ei“pm„) = T2 ReliGje^'0^];
j = m, n.
(3.220)
We may conclude from the structure of the right-hand side of the equation
for Ру that the solution of eqs (3.220) must be written in the following
§3.3]
The method of a probe field
183
form:
= ,E' + r*e,f'; pmn = rmne iOlJ + rmne(E (3.221)
e = Я,, - Q,
where the amplitudes r;, rm„ and rmn are independent of time. It follows
from eqs (3.221) that the dipole moment induced by the probe field
possesses two spectral components: one with the oscillation frequency coM,
and the second with the oscillation frequency 2<o — cuM (to be more
precise, the latter is induced both by the probe and the strong fields):
d(t) = 2 Re(d„me-i“"^pmn)
= 2 Re{d„m[rm„e~in"' + rm„e-i(2a,"<“'‘)']}
(3.222)
The atom’s polarization at the frequency oi)^ = 2a> — co^, symmetrical to
the frequency cuM about co, may be accounted for studying the right-hand
side of eq. (3.22Q) for p/7. The probe field performs the work on the
polarization induced by the strong field; because of the difference between
the frequencies <oM and a> this work oscillates with a differential frequency
(Од — co to which correspond the population beats of p;;. The latter
interacting with the strong field bring about polarization at the combina-
tion frequency (the term pmn — pnn in the equation for pm„). This is the
interpretation of NIEFs in this particular case.
The system of equations for r;, rm„, fmn (of the fourth order) following
from eqs (3.220) and (3.221) may be solved for any values of the
parameters characterizing the atom and the external field (see problem
(11)). However, the obtained expressions are too cumbersome. Much
simpler are the relations for a particular case:
<2 = 0; Гт + Г„ = 2Гт„ = 2Г; Атп«Гт + Гп, (3.223)
when the absorption coefficient of the probe field is given by
= _ 2 iG|2 Re l + r/(r + iQM)
а + Ъ 1 1 (Гт-НЯд)(Гй-НЯд) + 4|О|2’
= £ Nn~Nm
“ 4л mnl + 4|G|2/rmrn’
(3.224)
where a is the absorption coefficient of the strong field. With >0 it
follows from the expression (3.224) that
* t , 4|G|2
ГтГп ’
(3.225)
184
Resonance radiation processes
[Ch. 3
i.e. as |G|2 increases the decrease in the value is greater still than that
of a; the difference between and a under cuM—><o also holds in the
most general case (see problem (11)):
ft 2|G|2F/1 1 Am„\
Г + ^\Гт Гп ГтГ„Г
The seeming discontinuity of the absorption coefficient as a function of
frequency is due to the fact that in eq. (3.224) the dipole moment
components oscillating with frequencies 2cu — cuM, a>, 2cuM - co are not
taken into account. If cuM #= a> the contribution of these components to the
work of the probe field oscillates with the frequencies 2 |cuM — cu| and
|<oM — cu| and on the average is equal to zero. With cuM—> co the beats are
rather slow and the above-mentioned contribution to must be allowed
for. As a result, we obtain the notion of an instantaneous value of the
absorption coefficient in the field with slowly changing amplitude and such
a quantity is continuous under small <oM — <o.
In fig. 3.19 the dependence of on is shown, employing the
formula (3.224). If Гт = Гп, under comparatively small intensities of the
external field (4 |G|2 < ГтГп) the absorption coefficient afl proves to be a
monotonic function of the frequency (see fig. 3.19a). With the further
increase in |G|2 side maxima located near |ЯМ| = Г +2 |G| appear.
Between these maxima and the point = 0 there is a region of negative
values of (it is assumed that Nn — Nm > 0 and a > 0; in the opposite
case we must use the word “emission” instead of “absorption” and vice
versa). Thus within a certain range of frequencies the probe field is not
absorbed but amplified by the medium (see fig. 3.19a). The amplitude of
the side maximum on the plot of increases with the increase in |G|:
under |G|2» it follows from the relation (3.224) that
1Ч.1-Г + 2Ю1
so we have > a when |G| > 4Г.
If Гт Ф Г„ for small |G| the behaviour of as a function of is a little
different: in the absorption line centre there appears a minimum which
deepens and widens as |G| increases. This resembles the spectral line
contour at an adjacent transition (compare fig. 3.19b with fig. 3.16a).
However, when |G| is sufficiently large plots have a form similar to that
in the case Гт = Гп.
The sign of the integrated (over DM) absorption coefficient is given by
the sign of the population difference of the levels m and n. Consequently
§3.3]
The method of a probe field
185
the passage from absorption to amplification under the frequency change
is due to NIEFs and to the fact that the external field perturbs both of the
combining states m and n. The latter’s role can to a certain extent be
illustrated by comparison of fig. 3.19a,b with the plots in fig. 3.20 (see
section 3.4) showing the frequency dependence for the “net” emission of
the probe field.
The probe field method is one of the basic methods of non-linear
spectroscopy. Its theoretical foundations were developed in the 1960s.
Since then it has been employed in various applications in many ways and
the experimental investigation of the main phenomena—level splitting and
NIEFs—have been more or less exhaustive. The principal applications of
the probe field method are connected with the study of the processes
taking place during collisions, under superfine level structure as well as
the generation of coherent radiation harmonics.
186
Resonance radiation processes
[Ch. 3
A classification of the phenomena presented in this section (saturation
effect, field splitting of levels, non-linear interference phenomena) was
suggested and developed in refs [28-30]. In later research these phenom-
ena were studied anew independently and named differently [31].
3.4. Spontaneous emission of atoms interacting
with the external electromagnetic field
Spontaneous emission is a result of the interaction of excited atoms with
zero oscillations of the electromagnetic field. Therefore to treat it
theoretically a quantum description of atoms as well as radiation is
necessary. At the same time, the classical approximation is quite sufficient
for adequate interpretation of an atom’s interaction with a strong external
field. Therefore it seems useful to take a twofold approach, where the
internal motions of an atom and zero oscillations of the field are given by
a quantum treatment and the strong external field is treated classically.
The Hamiltonian corresponding to this approach is
Й = Йа +hVft) + Й{+Й'; = (3.226)
where Йв and are the Hamiltonians of an isolated atom and its
interaction with the classical strong external held, and Й and Й' are the
Hamiltonians of the remaining part of the field and its interaction with an
atom. The wavefunction of the “atom + field” system will be represented
in terms of the expansion
= 4»a, (3.227)
where a is a column of probability amplitudes and V is a row of the
eigenfunctions of the Hamiltonian Йл + Д, i.e. the elements of 4* contain
the wavefunctions of an atom as the multipliers
Wi = 1руе-‘£"/л; ЙД,, = (3.228)
and the wavefunctions of the quantized field
й^ = йюА(иА + i)^A. (3.229)
The elements of the column a correspond to different states of the atom
and held. For example, a(m, n^; f) denotes the probability amplitude of a
state in which the atom is in the state m and there are nA photons in the
held mode with suffix A.
§3.4]
Spontaneous emission of atoms
187
The equation for amplitudes is of a standard form, i.e.
a= —iV(t)a —^H'a, (3.230)
ft
and the elements of the matrix H' are given by the relation [2]
H'(j, nA I к, «л + 1) = —i<o;*d?*V2jift/<oAVV«A + 1- (3.231)
Here cuA and nA are the frequency and the number of photons of the A
mode, and V is the system volume.
We consider the problem of the spontaneous relaxation of the excited
states of an atom. In order to simplify the calculations we assume that the
excited state m of the atom optically combines only with one state /, which
possesses less energy (Et<Em) and is metastable. Let the atom at the
initial time be excited to the state m, and let there be no photons in the
field (nA = 0). Then from eq. (3.230) there follows the system of equations
a(m, 0; t) = 0 | /, lA)e-iOi'a(/, 1A; t),
n л
a(l, 1A; t) = -7Н4/, 1AI m, 0)eiO*'a(m, 0; t);
ft
DA = <oA - O)mr, a(m, 0; t0) = 1; a(l, 1A; t0) = 0 (3.232)
Further, we substitute the formal solution
a(l, 1A; t) = ~H'(l, 1A I m, 0) [ 0; it) dt, (3.233)
" Ло
in eq. (3.232) for a(m,0;t), dropping for simplicity the arguments of
1A):
d(m,0;t) = — ^7- [ 2 0; t,) dt,. (3.234)
* Ло л
Summation over the modes resonant with the transitions m-l may be
replaced by integration over their frequencies and propagation directions:
Г 2
E e~io‘T-> I е“,ПлТрл dcuA dO; pA = V
A J (2jtc)
where pAdcoAdO is the number of the modes within the range cuA,
cuA + dcuA and the solid angle dO. Integration over <oA in eq. (3.234) yields
188
Resonance radiation processes
[Ch. 3
2л <5 (t — g) after which the equation for a(m, 0; r) takes the form
a(m, 0; f) = — ут;а{т, 0; f), (3.235)
where the following notation is introduced:
IX/'I2 /уЗ 1
ym, = 8*2 44 Pa = 2 |dm/|2 = - A^. (3.236)
n nc 2
Thus, owing to the spontaneous emission of photons ha>k the excited state
of the atom m exponentially quenches with a rate ym/ equal to one-half of
the first Einstein coefficient Amt.
The calculations which yielded eqs (3.235) and (3.236) were carried out
within the scope of the model of non-degenerate states. When the
degeneration of atomic levels and spontaneous emission polarization are
allowed for, the quantity |dmZ|2 in eq. (3.236) must be substituted by
|(m|| d ||/)|2/3gm where gm is the statistical weight of the state m and
{tn || d ||n) is the reduced matrix element of the dipole moment:
4 to3
2ym/ = - | {m || d 11« ) |2 = A^. (3.237)
3 nc gm
The derivation of eq. (3.235) can easily be generalized to the cases of /
level instability, of the interaction of an atom with the external field (see
problem (12)), and of spontaneous transitions from the state m to several
states with energies less than Em. In the latter case, for example, eq.
(3.232) for a(m,0;t) contains not one but several sums over modes
resonant with the transitions m—*l, m—*n, m—*J etc. This results in the
equation
a(m,Q;f) + yma(m, 0; 0 = 0, ym = 2 ymj. (3.238)
Thus the introduction of relaxation terms into the eqs (2.5) and (2.11)
which are due to spontaneous transitions may be considered justified. The
justification of the in-term R(2) of the spontaneous transition matrix in the
kinetic equation for the density matrix may be also given by the method
applied above (see problem (14)).
Now we turn to the problem of the spontaneous emission spectrum, i.e.
of the probability for spontaneous emission of a photon with a certain
frequency <oM. In the Hamiltonian H' we must separate the part H,',
describing the interaction of a certain field mode with an atom, study
transition induced by it and find the corresponding work of the field. The
separation of one mode from the large number of modes cannot alter the
§3.4]
Spontaneous emission of atoms
189
process of spontaneous relaxation. Thus the equation for a has the form
a + ya = —iV(t)a - 7 Hj.(t)a. (3.239)
n
The initial conditions corresponding to the problem of spontaneous
emission are those under which at an initial time t0 the atom is in the
excited state and the field is characterized by the absence of photons. Let
m and / denote the indices of the upper and lower atomic states between
which the spontaneous transition occurs. Then the initial conditions and
the expression for H^(t) are written as
a(m, 0M; t0) = 1, a(l, 1M; t0) = 0; (3.240)
HXt) = P;pmH'*(/n, 0 11, lM)eia< (3.241)
The matrix contains* a single element (H^),m, which describes the
transition m-l because it appears in the right-hand side of the equation for
a lower state amplitude a (1,1M; t) (the column fV) in the product with the
upper state amplitude a(m, 0; t) (row 0m).
Assuming as before in the probe field method
a = b + c+ (3.242)
where c+ is a small correction due to H^(t), we obtain the equations
b + yb = -iV(t)b,
c+ + yc+ + iV(t)c+ = - г H^,(t)b (3.243)
n
for b and c+. The amplitude b describes the evolution of the atom under
the action of the external field and spontaneous relaxation in the absence
of /л mode photons. The correction c+ to the amplitude describes the
transitions accompanied by the emission of a photon The quantity
2y, |c+(/, 1M; t)l2 characterizes the rate of atomic transitions from the lower
level / and simultaneously the rate of energy increase of the pi mode.
Therefore the total energy transferred to the pi mode over the total
evolution time is given by
f lc+(l, 1M; 0I2 dt. (3.244)
Ло
* 11,1M) is introduced into the relation (3.241) for completeness of the analogy with the
term in the expression (3.167) containing G* and describing the probe held emission.
190
Resonance radiation processes
[Ch. 3
It can readily be shown if we make use of eq. (3.243) and allow for the
zeroth initial conditions for c+, that the expression (3.244) is equivalent to
= —2ftwM Ref^H'* [ eia^'a(tn,0; t)a*(l, 1/, f) dtl. (3.245)
l й Ло J
Comparison of eqs (3.243) with (3.170) and (3.171) and eqs (3.245) with
(3.143) enables one to conclude that the calculation of the spontaneous
emission spectrum in quantum theory is formally analogous to the
estimation of the work of the classical probe field if in the latter problem
we separate the emission and adequately define the value of the
parameter G*. For the case of interaction with a single mode /л we must
take
G* = -1H'*(m, 0 11,1M) = (3.246)
ft
Instrumentation has a finite spectral and angular resolution and the
emission energy is measured experimentally for a number of modes with
closely spaced frequencies and propagation directions. Therefore it is
advantageous to normalize I GJ 2 in the following way:
|GM|2 = |Н;/Й|2рм dtoM dO = 4^<4 dO. (3.247)
1ОЛ
Thus all the results obtained in section 3.3 for the emission spectrum of
the probe field are extended to the spectrum of spontaneous emission of
atoms interacting with the external field if by |GJ2 in the relations of
section 3.3 we shall mean that defined in the relation (3.246) or (3.247).
The absorption and emission processes can be separated in the
following way. Using eq. (3.239) we set up an equation for the quantity
aaf and perform a statistical averaging procedure, which implies that aaf
passes to the density matrix. Such averaging leads to the emergence of the
statistical terms S and R considered in sections 2.3 and 2.4 but does not
alter the form of the dynamic terms. The above-mentioned procedure
yields the equation
p = -i[V(r)p - pV(t)] -1 (н> - pHj + S + R, (3.248)
ft
where is defined fy the relations (3.241) and constitutes the formal
distinction between eqs (3.248) and (3.207) (compare eqs (3.241) and
(3.167)). The physical meaning of this distinction is the separation of the
net emission.
§3.4]
Spontaneous emission of atoms
191
In a first approximation with respect to H' from eq. (3.248) we obtain
P = P° + Pm> (3.249)
p° + i[V(t)p° - p°V(t)] - S - R = Q, (3.250)
Pm + i[V(t)pM - Pm V(t)] - S„ - RM
= -|(Н;р0-р°Н;), (3.251)
fl
where Q is the inelastic part of the in-term of the collision integral. Let us
specify the equation (3.251) for a spontaneous transition m-l from the
level m perturbed by the external field to the unperturbed level I (see fig.
3.14; and the index p in pM is dropped)
Pa + i 2 У)к(1)рк1 - Sjt - Rji = j*m;
к
Pmi + i^ Ут^)Рц -S^- Rml = -iG^e-'^'p0^. (3.252)
j
Unlike eq. (3.211) the right-hand side of eq. (3.252) for pmt contains the
population of the upper level m only. At the same time, NIEFs for
spontaneous emission and a probe field are identical (the terms p"m in eqs
(3.252) and (3.211)). If E,> Em (spontaneous transition l-+m, Н^«РЩ,)
then instead of the equations (3.252) it follows from eq. (3.251) that
Ph + i 2 Vjk(t)pu - Sjt -Rjt = O, j*m\
к
Рпа + Ъ Vmi(t)Pil - Sml - Rml = iG^pk (3.253)
j
In this case the population of the level I (the upper level) appears;
furthermore, the non-linear interference terms are absent, in accordance
with the assumption that the level I, the initial level for the radiation
process under consideration, is not perturbed by the external field.
Spontaneous emission for the transition m-n (H^«p*pm), both of
whose levels are perturbed by the external field, is described by a more
complicated system of equations which follows from eq. (3.251):
Pmm "b 2 Ref i Vm/Pfm) Smm Rmm =
' i '
pnn + 2 Re(i 2 Vnjpjn^ - S„n - Rnn = -2 Re(iG^-iOlJp0nm),
Pnm+i'E (VnjPjm - PnjVjm) ~ Snm ~ Rnm = iG^C^P^ (3.254)
192
Resonance radiation processes
[Ch. 3
Comparison of the systems of equations (3.254) and (3.220) shows that for
spontaneous emission only the population of the upper level m and the
non-linear interference terms created by the right-hand side of the
equation for the diagonal element pnn of the lower level n remain.
Consequently, the lower level population in the right-hand side in eq.
(3.220) and NIEFs due to the right-hand side in eq. (3.220) for pmm
describe the probe field absorption.
Thus we may employ the results obtained in section 3.3 if we carry out
the aforementioned modifications of the expressions for the work of the
probe field. By way of example, for spontaneous emission accompanied
by the transition m^>l (see fig. 3.11) the formula (3.217) may be used if
one omits the term p°u. For the transition l^>m (see fig. 3.18a) in the
expression (3.219) only the term p°u must be retained, whereas p°mm and
the non-linear interference term proportional to pm„ must be dropped.
The plots of the spectral density of spontaneous intensity for the
adjacent transition m^l (fig. 3.11) versus frequency are qualitatively
analogous to those discussed in section 3.3 (figs 3.13 and 3.16). The
obvious distinction between the spontaneous intensity and the work of the
probe field consists in the fact that the latter can have different signs in
close spectral ranges (fig. 3.17), while the spectral density of spontaneous
emission is essentially positive.
The latter refers also to spontaneous transitions between the states m
and n perturbed by the external field, and in this case the line contours of
amplification or absorption and spontaneous emission may differ drasti-
cally. By way of illustration we consider spontaneous emission in the level
scheme of fig. 3.15 under additional simplifying assumptions: only the
level m is excited; w = wm„; Гт = Гп= Г. Then it follows from the general
expressions of problem (13) [11] that
_ = —— л/ Г11----------1 x
" 16л2 m l\ P + 4|G|2/
Г 1 1/2
1/2
Г2 + (Ц, - 2 |G|)2 Г2 + (Ц, + 2 |G|)2J
|G| Г -2|G| QM+2|G| 11
Г2 + 4 |G|2 Ir2 + (Ц, - 2 |G |)2 + Г2 + (Ц, + 2 |G|)2 J J'
(3.255)
The spectral density of spontaneous emission in this case is a triplet whose
components are located near the frequencies = Q = 0 and = ±
2 |G|. Such a picture of the spectrum is due to the splitting of the levels m
References
193
Fig. 3.20. The line contour of spontaneous emission
rm = r„/io.
for the transition m-n: (а) Гт = Гп\ (b)
and n into the pairs of sublevels ml, m2 and nl, n2. Figure 3.20 shows
the plots of - versus , which differ drastically from the plots of fig.
3.19 depicting the line contour of the probe field amplification.
References
[1] W. Heitler, Quantum Theory of Radiation (Oxford University Press, London, 1960).
[2] 1.1. Sobel’man, Vvedenie v Teoriyu Atomnykh Spektrov (Fizmatgiz, Moscow, 1963) [An
Introduction to the Theory of Atomic Spectra (Pergamon, Oxford, 1972)].
[3] L.A. Vainshtein, 1.1. Sobel’man and E.A. Yukov, Vozbuzhdenie Atomov i Ushirenie Spektral-
nykh Linii (Nauka, Moscow, 1979) [Excitation of Atoms and Broadening of Spectral Lines
(Springer, Berlin, 1981)].
[4] H.R. Griem, Plasma Spectroscopy (McGraw-Hill, New York, London, 1964).
[5] H. Rabitz, Annu. Rev. Phys. Chem. 25 (1974) 155.
[6] S.G. Rautian and LI. Sobel’man, Usp. Fiz. Nauk 90 (1966) 209 [Sov. Phys. Usp. 9 (1967) 701].
[7] V.N. Faddeeva and N.M. Terent’ev, Tables of the Values of the Function w(Z) = e~'1 2 3 4 5 6 7^l +
I e~'2dt) (Pergamon, London, New York, 1961).
Vn Jo /
194
Resonance radiation processes
[Ch. 3
[8] B.D. Fried and S.D. Conde, The Plasma Dispersion Function: the Hilbert Transform of the
Gaussian (Academic Press, New York, 1961).
[9] T.A. Andreeva, V.A. Aleksejev and 1.1. Sobel’man, Zh. Eksp. Teor. Fiz. 64 (1973) 813 [Sov.
Phys. JETP 37 (1973) 411].
[10] R. Dicke, Phys. Rev. 89 (1953) 472.
[11] S.G. Rautian, Tr. Fiz. Inst., Akad. Nauk SSSR 43 (1968) 3.
[12] S.M. Rytov, Vvedenie v Statisticheskuju Radiofiziku, 2nd edn (Nauka, Moscow, 1976).
[13] A.I. Burshtein and Yu.I. Naberukhin, Zh. Eksp. Teor. Fiz. 52 (1967) 1202 [Sov. Phys. JETP 25
(1967) 800].
[14] V.A. Aleksejev and 1.1. Sobel’man, Zh. Eksp. Teor. Fiz. 55 (1968) 1974 [Sov. Phys. JETP 28
(1969) 1044].
[15] A.I. Burshtein and S.I. Temkin, Spektroskopija Molekuljamogo Vraschenija v Gasakh i
Zhidkostjakh (Nauka, Novosibirsk, 1982).
[16] A.P. Kazantzev, Zh. Eksp. Teor. Fiz. 51 (1966) 1751 [Sov. Phys. JETP 24 (1967) 1183].
[17] V.N. Rebane, Opt. Spektrosk. 26 (1969) 643.
[18] G. Nienhuis, Physica 74 (1974) 157.
[19] B.K. Matzkevich, I.E. Evsejev and V.M. Ermachenko, Opt. Spektrosk. 45 (1978) 17.
[20] S.G. Rautian, A.G. Rudavetz and A.M. Shalagin, Zh. Eksp. Teor. Fiz. 78 (1980) 545 [Sov.
Phys. JETP 51 (1980) 274].
[21] S.G. Rautian, in: Proc 9th Int. Conf, on Spectral Line Shapes, Vol. 5 (Ossolineum, Wroclaw,
1989), p. 503.
[22] V.A. Aleksejev and A.V. Malyugin, Zh. Eksp. Teor. Fiz. 80 (1981) 897 [Sov. Phys. JETP 53
(1981) 478].
[23] V.S. Butylkin, A.E. Kaplan, Yu.G. Khronopulo and E.I. Jakubovich, Rezonansnoje Vzaimode-
istvije Sveta s Veshchestvom (Nauka, Moscow, 1977).
[24] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill,
New York, 1955).
[25] S.H. Autler and C.H. Towns, Phys. Rev. 100 (1955) 73.
[26] R. Karplus and J. Schwinger, Phys. Rev. 73 (1948) 1020.
[27] A.K. Popov, Vvedenije v Nelineinuju Spektroskopiju (Nauka, Novosibirsk, 1983).
[28] S.G. Rautian and A.A. Feoktistov, Zh. Eksp. Teor. Fiz. 56 (1969) 227 [Sov. Phys. JETP 29
(1969) 126].
[29] T.Ya. Popova, A.K. Popov, S.G. Rautian and A.A. Feoktistov, Zh. Eksp. Teor. Fiz. 57 (1969)
444 [Sov. Phys. JETP 30 (1970) 243].
[30] T.Ya. Popova, A.K. Popov, S.G. Rautian and R.I. Sokolovsky, Zh. Eksp. Teor. Fiz. 57 (1969)
850 [Sov. Phys. JETP 30 (1970) 466].
[31] Y.R. Shen, Spettroscopia non Lineare (North-Holland, Amsterdam, 1977), ch. 6.
Bennett structure for systems with large
Doppler broadening
4.1. Velocity distribution of atoms under the
interaction with a plane monochromatic wave
At comparatively low pressures (of the order of ITorr and less) gas
particles over characteristic relaxation times l/Г travel a distance sig-
nificantly exceeding the radiation wavelength Л. Thus for atoms charac-
teristic values of the relaxation times are approximately 10-8s and at a
mean thermal velocity u~105cms-1 the mean free path l=v/r~
10-3cm, i.e.
I = v/Г » Л ~ 10-4-10'5 cm. (4.1)
The relaxation times of the vibrational levels of the electronic ground
states of molecules may be still larger and the inequality (4.1) holds
although the wavelength of the vibrational-rotational transitions increases
compared with wavelengths of the electronic transitions.
The inequality (4.1) may be rewritten in the following form:
kv/T»l, к = 2л/к, (4.2)
which implies that over the relaxation time l/Г the phase of the field
owing to the Doppler effect changes by a value of well over unity or the
Doppler broadening essentially exceeds the homogeneous impact
broadening.
Under the above-mentioned conditions the motion of atoms and
molecules substantially affects their interaction with the radiation. Within
the scope of linear theory the thermal motion brings about the Doppler
line broadening discussed in section 3.1. Still more drastic changes
characterize non-linear phenomena. Let us consider the simplest case of a
plane monochromatic travelling wave with a frequency a>. In the atomic
coordinate system the field acting on the atom is of frequency oj — к • v
with a shift of к • v due to the Doppler effect. By virtue of the inequality
(4.2) the resonance conditions are satisfied not for all the atoms but for
195
196
Bennett structure for systems with large Doppler broadening
[Ch. 4
those for which the Doppler shift к • v is compensated (accurate to Г) by
the difference Q = to — a>m„ of the field frequency and the frequency of the
transition wmn. In other words, the field action on the atoms proves to be
selective in velocities and as a result the v distribution for the populations
of combining levels is no longer at equilibrium and acquires a sharp
structure.
The field-induced change in the velocity distribution of atoms must of
itself lead to certain phenomena, e.g. to the appearance of narrow
non-linear resonances on the lines with Doppler broadening, charac-
terized by an unusual dependence of the output power of single-frequency
gas lasers on the radiation frequency etc. The selectivity of the field action
can, however, be manifested in a different way. It has been stated in ch. 3
that, for probe field spectroscopy, together with the saturation effect the
coherence of combining levels is of great importance as well as their field
splitting. It will be recalled that the role of these factors critically depends
on the differences between Bohr frequencies and the field frequencies, i.e.
for the moving atoms on co — wm„ -k-v and coM — comZ — кц • v (&M is the
wavevector of the probe field). Therefore the motion of atoms is
important from this point of view as well.
In the non-linear spectroscopy of moving atoms the field geometry is of
special importance. Indeed, let us expand a field with a certain geometric
configuration into plane waves and take into account Doppler frequency
changes when going over to the atomic coordinate system. To each spatial
harmonic there then corresponds a frequency ы - к • v. Consequently,
the interaction of a moving atom with a field of complex configuration is
equivalent to the interaction of a stationary atom with a non-
monochroma tic field. In particular a standing plane wave in the atomic
coordinate system is bichromatic and its spectral components possess the
frequencies a> — к • v and a> 4- к • v. However, it was shown in section 3.2
that the evolution of an atom depends most strongly on the field spectrum
if the field is sufficiently intense. For example, in a bichromatic field the
splitting of atomic levels into an infinite set of sublevels takes place (see
eq. (3.122)). We may come to the same conclusions using the “time
language”: the moving atom passing in succession different points in space
“feels” the inhomogeneous field geometry which, consequently, manifests
itself also in the spectroscopy of non-linear resonances.
If the motion of atoms is important their velocity changes under
collisions must be important too. In linear spectroscopy elastic
scattering may manifest itself only as the Dicke effect (see section 3.1)
since the velocity distribution which is as a rule at equilibrium is not
§4.1]
Velocity distribution of atoms
197
influenced by collisions. The sharp structure in the v distribution due to
selective interaction of the coherent field with atoms is very sensitive not
only to the quenching processes and disorientation but also to the smallest
(diffraction) velocity changes. In this respect non-linearly spectroscopic
experiments are analogous to experiments with atomic beams. It is not,
therefore, surprising that the problems associated with spectroscopic
manifestations of elastic scattering have been well developed in connec-
tion with non-linear resonances with gas spectra.
Thus it must be clear that the non-linear spectroscopy of moving atoms
(molecules) is characterized by complexity and diversity of phenomena.
This accounts for the rather extensive discussion of the theory of
non-linear phenomena under substantial Doppler broadening.
Consider first the simplest problem of moving atoms’ interaction with a
plane travelling monochromatic wave:
E(r, t) = + ^*е'(“'-* г)]. (4.3)
Let the wave frequency oj be sufficiently close to the frequency ajmn of the
transition m-n, so that the conditions (3.93) of the applicability of the
resonant approximation are fulfilled. Then
Vmn = - = -Ge~i(a,~kry-, (4.4)
G = dmn%!2h, Q=oj-ajmn,
where dmn is the matrix element of the dipole moment operator. In the
model of non-degenerate states and in the absence of velocity changes
under collisions (model of relaxation constants) the equations for the
density matrix have the form (see eqs (2.60) and (3.125))
+ v • V + гЛРтт = qm-2 Relieve*1»-* ')];
d + v V + r")p'” = (ln+ Amnpmm + 2Re[iG*pm„ei(a“‘r)];
\at /
+ V * V + Гтп + 1 = ~ pnn)^a'k r)- (4.5)
The expression (4.4) for Vm„ and the eqs (4.5) differ from those
employed in section 3.2 in the explicit introduction of characteristics of
atomic motion (the phase of the field contains the term к • r and the
operator v • V is introduced; compare with eq. (3.125)).
198
Bennett structure for systems with large Doppler broadening
[Ch. 4
On the assumption that the excitation processes are spatially homoge-
neous and stationary, eqs (4.5) may be reduced to the algebraic equations
pmn = pe"i(O'"*'’); (4.6)
\r-i(Q-A-k-v)\p = -iG(pmm-pnny, Гт„ = Г; Amn = A;
ГтРтт = -2 Re(iG*p) + QmW(v);
r„Pnn=Am„pmm + 2Re(iG*p) + Q„W(v). (4.7)
Further, the atoms will be taken to be excited with a Maxwellian velocity
distribution, i.e.
W(v) = (Vn; v)-3exp(-v2/v2), v2 = 2T/m. (4.8)
The form of eqs. (4.7) is the same as that of eqs. (3.131) for atoms at
rest. Their motion was reflected in the substitution Q—> Q - к • v and the
density matrix elements in eqs (4.7) depend on v as on a parameter. The
term к • v in the combination Q — к • v is connected with the fact that in
the rest system of the atom the frequency of the radiation acting on it is
shifted by к • v because of the Doppler effect.
Owing to this correspondence the solution of eq. (4.7) is obtained from
the solution of eqs (3.131) expressed in terms of the formulae (3.132)-
(3.135) via the substitutions Q—> Q — к • v and (2,—> C?;W(v):
r 1 2Г \G\2 (Nm — Nn) n *
(4-9)
Pmm
Pnn
Pmm Pnn
L Гп \ Гт / П + (й- A — k- v)2J
Г Гк
1-
r.+(a-s-k.^N~-N^
(4.Ю)
p — —iG
_____Pmm Pnn________
Г — i(£2 — A — к • v) ’
2 |G|2 / 1 ! 1 Л \
Г \Гт Г„ rmr„)
(4.И)
(4-12)
Д = П/ТТк;
The quantities Nm and Nn are the populations of the levels m and n in the
absence of the field (integrated over velocities). Their relation to Qm and
Qn is given as before by the formula (3.135).
Changes in populations caused by the electromagnetic field are de-
scribed by the second terms in square brackets of the expressions
(4.9)-(4.11). These terms contain the resonance denominator 7^(1 4- k) 4-
§4.1]
Velocity distribution of atoms
199
(Q — A — k'v)2 which is dependent, in particular, on the value of the
velocity v projection on the wavevector к. Thus the electromagnetic field
alters the velocity distribution of the populations. The resonance de-
nominator is minimal at
k-v = Q-A. (4.13)
The equality (4.13) implies that the field frequency in the atomic
coordinate system is in exact resonance with the transition m-n and the
interaction of an atom with a field is the most intense. The range of the
atomic velocities for which the interaction with a field is effective is
determined, as is seen from the resonance denominator, by the quantity
Ц=П/1 + ~к.
If the condition
rs = r\/l + K«kv (4.14)
is fulfilled, which under not very strong fields is equivalent to the
condition (4.2), the changes of population distributions in velocities due to
the field have the form of sharp structures against the background of the
wide Maxwellian distribution. These structures were called Bennett holes
(or dips) and peaks [1]. With Nm >N„ (population inversion), according to
the expressions (4.9) and (4.10) there is a hole in the velocity distribution
at level m and a peak at level n (fig. 4.1). With Nm <N„ the reverse is
true, i.e. there is a hole at level n and peak at level m.
The Bennett hole and peak are centred on resonance velocities defined
by the relation (4.13) and have a Lorentz form with halfwidth rs =
rVl + K- With variation of the radiation frequency their extrema are
displaced in proportion to Q - A.
The relative amplitudes hm and hn of holes and peaks are given by the
relations
, Nm-Nn_____________________2|G|2_______________
m Nmrm r + 2\G\2(lirm + l/rn-Amjrmry }
= Nm — Nn / _Л^\________________2 |G|2_____________
" Nnrn \ rmJ r + 2\G\2(l/rm + l/rn-Amn/rmrn)' ’
The quantities hm and hn are proportional to the lifetimes 1/Гт and 1/Г„ of
levels m and n respectively. Depending on the excitation conditions the
factors (Nm - Nn)/Nm and (Nm — Nn)/N„ can take arbitrary values and,
accordingly, hm and hn can also take any values. The field intensity
dependence for hm and has the form of a curve with saturation. When
200
Bennett structure for systems with large Doppler broadening
[Ch. 4
Fig. 4.1. Population distribution in velocities of (a) the upper and (b) the lower levels at Nm > N„.
fields are small (к «1) the values of hm and hn are proportional to the
field intensity; when the field is large (k»1) they asymptotically
approach constant values.
Asymptotic values of hm and hn are in agreement with the equalization
of the populations at levels m and n for the group of atoms with resonance
velocities (4.13). This fact can easily be traced in the expression for the
population difference (4.11). For resonance velocities the population
difference under к »1 tends to zero as 1/к (saturation effect).
Thus in a system with substantial Doppler broadening the saturation
effect is characterized by the following. As the field intensity increases,
population equalization takes place to the greatest extent for the atoms
with resonance velocities. Furthermore, with the intensity increase, new
groups of atoms whose velocities are becoming more and more different
from the resonance velocity are becoming involved in interactions with the
field. The latter fact is reflected in the increasing width of the Bennett
hole and peak:
rs~ Г+ |G|2 4-i-4^-1 k«1;
\Fm Гп rmrj
rs-\2r\G\2(^ + ^-^=}, k»1.
V \ГИ Гп ГтГпГ
(4-17)
(4-18)
§4.1]
Velocity distribution of atoms
201
When the field is relatively weak the broadening is proportional to the
field intensity and when it is strong the broadening is proportional to its
amplitude. If the field’s intensity is such that
|G| \/2Г(Г~1 + Г'1 — Ат„/ГтГ„) ~ kv (4.19)
then the populations of atoms with practically any velocity are equalized.
The parameter к which specifies the field intensity dependence of the
half-width and amplitude of the Bennett structure will be called the
saturation parameter. It is proportional to the relaxation time of transition
coherence (value 1/Г) and to the effective total lifetime at levels m and n
(factor 1/Гт + 1/Гп -Ат„/ГтГп).
Consider the expression for the work done by the field
P =-2й<о Re(iG*p)
--^\Gfr(r + ^_-AP:-kvr), (4.20)
where the angular brackets denote velocity averaging. The integral over
velocities is reduced to known functions at arbitrary values of rjkv (see
problem (15)); nevertheless, here we shall confine ourselves to the
approximation (4.14) and using the expression (4.11) we obtain
P = 2h(oNn~Nm |G|2exp[-• (4.21)
Vl + к- kv KL X kv / J 7
The diagram of the work performed by the field as a function of Q has the
form of a Gaussian curve with the maximum at Q = A, i.e. the maximum
is shifted from the transition frequency by a value Л determined by
collisions. The curve width is specified by the parameter kv which
depends on the radiation wavelength, temperature and mass of the
particles interacting with the field.
As can be seen from eq. (4.21) the quantity P as a function of Q
remains unaltered and the field intensity affects only the coefficient of the
spectrum factor exp[—(£2 — A)2/(kv)2]. As the field increases, the value of
P first rises proportionally to the intensity (k«1); then it becomes
proportional to the field amplitude (к » 1).
The results obtained for the model of non-degenerate states can easily
be extended to systems with degenerate states in the following cases. Let
the external field be polarized either linearly or circularly. Transitions
between magnetic sublevels of states m and n induced by such a field are
shown in figs 4.2a and 4.2b respectively. In the first case the quantization
axis is selected along the electric field, and in the second case it is selected
202
Bennett structure for systems with large Doppler broadening
[Ch. 4
(a)
Fig. 4.2. Scheme of transitions caused by waves with (a) linear and (b) circular polarization.
along the wavevector k. If the spontaneous transition m—>n and the
collisional exchange between sublevels can be neglected then the interac-
tion with each transition (see fig. 4.2a) and - 1 (see
fig. 4.2b) takes place independently*. Therefore the equations (4.7) and
the results following from them might be used, pu denoting the magnetic
sublevel population which will subsequently be designated by Рц(М).
The quantity |G| in this case stands for the amplitude of the matrix
element of the interaction for the transition mM-nM’ and will be
designated by G(M). Assuming magnetic sublevels M in the absence of a
field to be uniformly populated (N^M) = NJgj, gy = 2J; + l), proceeding
from the formulae (2.76) and (4.9)-(4.12) we have
4. _ AU_____________2Г |<j(Af)|2__________‘
-gm Гт Г2]! + k(M)] + (Q - Л - к v)2\ ’
’A6. AU_______2Г |G(AT - g)|2____-
-g„ Гп ^[1 +k(M'— a)] +(Q — Л —к • v)2\ '
(4.22)
(4-23)
* Spontaneous transitions m-»n connect the sublevels M' = M, M ± 1.
§4.1]
Velocity distribution of atoms
203
P = -2tiwNm„ |С|2^?ехр
kv
Q — A\2~
kv / .
x
Nmn=Nm/gm-Nn/gn- G = dmn%l2tf, dmn = {m\\d\\n)l\/3-,
G(M) = - M' I la);
k(M) = 11G(M)|2 (^ + 7) = ~ M' I la)2. (4.25)
* '*»! *n'
Here {m ||</|| n) is the reduced matrix element of the dipole moment. The
quantity a is the index of the circular components of the electric wave
field: for a linearly polarized wave a = 0; for a wave with circular
polarization a = 1 of a = — 1.
By virtue of the relations (4.22) and (4.23) the radiation field brings
about a non-equilibrium distribution in the magnetic sublevels just as it
leads to a non-equilibrium velocity distribution. Non-uniformity of sub-
level distribution is due to the M dependence of the matrix elements of
the interaction Hamiltonian for transitions (Clebsch-Gordan
coefficient in the expression (4.25) for G(M)). Therefore some sublevels
can remain unperturbed by the field and their populations are not at all
altered*.
It is seen from eqs (4.22) and (4.23) that the velocity distribution for
field-perturbed sublevels differs for different M: the amplitudes and
widths of the corresponding Bennett holes depend on M. The difference
in width increases with the increasing intensity of the field. Only under
weak fields (ic(Af)«1) do the holes have the same halfwidth Г.
Each of the transitions contributes to the work of the field
(4.24). The frequency dependence of P in the used approximation
r\/l + k(M) «kv coincides with that in the model of non-degenerate
states (compare with formula (4.21)). The degeneracy was reflected by the
fact that instead of one saturation parameter к there appears a set of the
quantities whose number equals min{Jm, Jn}.
Let us consider how the disorienting processes in systems with
substantial Doppler broadening influence the velocity distribution and
work of the field. We shall proceed from the equations for the density
matrix in the polarization moment representation. In the model of
relaxation constants and that of isotropic collisions when atoms interact
* If collisional exchanges exist between the M sublevels this conclusion obviously does not hold.
204
Bennett structure for systems with large Doppler broadening
[Ch. 4
with a plane monochromatic travelling wave the equations (2.67) may be
transformed to
Г pm„ = -i(G™ pm„ - G” p„m) + —,
mrttim л\^птгтп тпгпт/ | ’
[Г + iA - i(£2 - к • v)l]pm„ = -i(G^„pmm - GX„p„„). (4.26)
The relaxation matrices and excitation vectors may be written as follows:
f/(^9 I ^191) ^KKi^qqi^jK J Qj(Kq) Qj^xO^qO,
Amn(icq | k^i) = Атп(к) <5KK1<5W1; I(jKq | к^) = drridWl; (4.27)
Г(кд | к1Я1) + iA(xq | k^i) = (ГК + iAr)drridw;
riK = 2yj + v(/7) - v(jjK \Цк);
Гк + iAK = ym + У„ + v(mn) — v(mnK | тпк). (4.28)
The quantity Атп(к) is given by the formula (2.95). By the resonance
approximation, the interaction matrix U£ in eqs (2.67) is represented as
= —G,7 exp{i[(w„ T a))t ± к • г]}, (4.29)
where the sign is chosen so that the quantity |<о/7Т<и| is at a minimum
(resonance approximation). The matrices G* according to the relation
(2.69) are given by the formulae
G^Kq | Ki^i) = (-l)K_/'_y‘V2K + 1 V2ki + 1 x
{ ! Kj у 1 2 (-1)k,_<?,<k^i - 11^)GI>CT;
Ujt Jj Ji J a
Gija = difa/2* = (-
Gfrxq | Kiqi) = (-ly-^'G^K -q\K1- qt). (4.30)
Here are the circular components of the electric field vector.
Finally, instead of the non-diagonal elements pi7(r, t) appearing in the
equations (2.67), the p,7 entering in eq. (4.26) are independent of r and t
and are connected with p,7(r, t) by the relation
Pij(r, I) = Po exp{i[(<O;7 T (O)t ± к • г]}.
§4.1]
Velocity distribution of atoms
205
In the absence of the electromagnetic field, from eq. (4.26) we have
, , .. N,W(v)
4,-£=, (4.31)
At an arbitrary field intensity it is rather difficult to solve the system of
linear algebraic equations (4.26). The rank of the system of equations in
the general case is (2J + I)2 where J — max{Jm,Jn} and obtaining recipro-
cal matrices is rather cumbersome. Therefore we shall solve eq. (4.26) by
iteration with respect to the field amplitude. Taking eqs (4.31) as a zero
approximation and making use of the explicit form in formula (4.30) of
the matrices G* we may obtain the expressions for the polarization
moments:
2Nm„r amK
+(O- a-t igM ’
= ' *}; (4.32)
'"''m Jm *'n*
Г TV
pnn(Kq) = W(v)|j^=== (5K0 +
2Nmnr
/ &ПК к "I
«яг = (-1)1+у-+Аз[1 1 *}; (4.33)
'-*'n •'n
I(xq) = 2 (-ly-^lal - O! | icq)GaG*-
Г=Ц-, 4=4,; Ga = Gmna, (4.34)
where I(xq) is the so-called field polarization tensor in the icq
representation (see, for example, ref. [2]).
The field-induced changes in the polarization moments of levels m and
n, as can be seen from the expressions (4.32) and (4.33), are proportional
to the field polarization tensor. Owing to the dipole character of the
interaction к can take only values 0, 1, 2*. Therefore the field affects the
value p„(00) (alters the total populations of the levels) and, in addition,
* In higher orders of perturbation theory the values of к will be greater.
206
Bennett structure for systems with large Doppler broadening
[Ch. 4
establishes the orientation p/t(lq) and alignment puQq) at the levels
j — m,n. The dipole nature of the interactions is responsible also for the
fact that only Гк = Ц, i.e. the relaxation constant of the dipole moment
described by the density matrix element pm„(Kq) with the value к — 1,
enters in the formulae (4.32) and (4.33).
The field-induced part of the velocity distribution of polarization
moments, as in the model of non-degenerate states, is a sharp structure in
the form of a Bennett hole (peak) whose halfwidth equals Г, i.e. it is
characterized by the relaxation constant of the dipole moment. The
amplitudes of holes and peaks are proportional to the relaxation times
l/fj*. of the polarization moment of the Kth rank.
The term proportional to Атл(к) in eq. (4.33) and due to spontaneous
transitions m—causes an effective change in polarization moment
relaxation time of the level n. For p„„(00) this time is effectively
decreased, for р„„(кд) with к 1 both a decrease and an increase in the
relaxation time are possible (see section 2.3).
The expression for the work done by the field defined by the formula
(2.63) and in the approximation Г « kv has the form
P(Q) = 2ftw Re 2 iGa(pU^))
a
Vn Г /Й — A
= -2ha)Nmn — exp - ——-
kv L \ kv
Г|Л'’|2 1 V4 I Г/ \|2 Amn (к)а
тк^пк "11
|G| -7Z 7^ + 7---------7-7-------
* Kq *-*тк * пк *тк*пк
(4.35)
|G|2 = S |G„|2.
a
Changes in P{Q) due to the degeneration of states mostly refer to the
non-linear part of P(Q). Its amplitude proves to be dependent on a
greater number of relaxation constants than in the model of non-
degenerate states (see the relation (4.21)). There are eight independent
constants in the relation (4.35): Г, A^,, Гтк, Гпк, к = 0, 1, 2.
Another significant property of the system with degenerate levels is the
fact that the non-linear part of P(Q) depends on the field polarization.
For linear field polarization the following components I(icq) are non-zero
(the quantization axis is selected along the electric wavevector):
л/з /2
Z(00) = -y|G|2; Z(20) = —-y - |G|2. (4.36)
§ 4.2] Non-linear resonances due to Bennett holes 207
In this case according to the expressions (4.32) and (4.33) the field affects
the population of levels (руУ(00)) and produces alignment (p/;(2^)). In
accordance with the above the relaxation constants rj0 and rj2 enter in the
expression for the field work. The constant is absent because a linearly
polarized field produces no state orientation.
For circular polarization of the radiation we have (the quantization axis
is directed along the wavevector)
\/3 v/5 1
Z(00)=-^-|G|2; Z(10)=-^-|G|2; Z(20) =-/= |G|2. (4.37)
j 2 Vo
The circular polarization field alters the population of levels and produces
orientation and alignment. The non-linear part of the field work contains
the complete set of relaxation constants.
Thus the change of field polarization state alters the relative weight of
the terms with different values of к in the expression (4.35).
4.2. Non-linear resonances due to Bennett holes
The simplest non-linear resonance is a dip on a plot of the work done by
the field as a function of its frequency when moving atoms interact with a
standing monochromatic wave.
Let us represent an external field as a sum of countertravelling plane
waves with frequency cd:
E = (4.38)
Under such an external field, processes occurring in an atom are more
complicated than those in a travelling monochromatic wave. Indeed, in
the atomic coordinate system the atom is affected by a bichromatic field
whose spectral components have frequencies cd ± k • v. According to the
results of section 3.2, atomic energy levels in such a field split into a
complicated system of sublevels. In terms of a density matrix this fact
implies that in the laboratory coordinate system the matrix elements are a
set of space harmonics with periods k/s (s = 1,2,3,...) determined by
the wavelength Л. In particular, populations of levels prove to be
modulated in space with a period Л/2.
Let us consider this problem in the model of relaxation constants and
non-degenerate states in more detail. Let the field (4.38) be resonant with
208
Bennett structure for systems with large Doppler broadening
[Ch. 4
the transition m-rv, then
-Vmn = G^a~^ + G^a+k'\ G1>2 = (4.39)
We then expand the density matrix elements into a Fourier series:
Pa(r) = 2 Pmnir) = е'й 2 pse“* r. (4.40)
The equations for the expansion coefficients are
(Tm + 2\sk ’V)pm2s = qm8s0 -
p2s+i ~ GiP-2s-i 4- G2P2S-1 ~ G2P-2S+1)}
(T„ + 2isk • v)p„ 2. = q„8s0 + Amnpm +
l(G* p2s + l ~ + G*p2s-1 ~ G2p-2s + l)'i
[Г — i(£2 — A) + i(2s + l)Jt • v]p2j+1
~ —i[Gi(pm2j — Pn2s) + G2(pm2j+2 — Ригя-г)]- (4.41)
For populations even space harmonics differ from zero; for non-diagonal
elements the odd harmonics are non-zero. It can be easily seen that eqs
(4.41) are a chain where the harmonics of different orders are coupled.
This coupling disappears only under Gi = 0 (or G2 = 0), i.e. when only one
of the travelling waves is present.
Equations (4.41) may be solved analytically only in some specific cases.
One such case, corresponding to exact resonance (Q — A — 0) and to
equal relaxation constants (Гт = Гп — Г), is reduced to the problem of the
bichromatic field (3.100) studied in section 3.2 (see problem (16)). Others
are characterized by the fact that under certain conditions harmonic ampli-
tudes fall off with the increase in their number and the chain of equations
(4.41) can be broken at some step. This is, for example, the case when the
amplitudes Gi and G2 of the countertravelling waves differ greatly. For
significantly different relaxation constants and when the condition
IG^+IG^CCir-^lir-rj . (4.42)
is fulfilled the solution can be represented as a series of successive
approximations [3,4]. At relaxation due to spontaneous decay such
conditions correspond to those discussed in section 3.2 (see eq. (3.115)).
The effects due to higher space harmonics in pv(r) turn out to be very
important in some physical problems such as the problems of the spectral
density of spontaneous emission (ch. 5), of probe field absorption and of
power resonances in ring lasers [5,6]. There are, nevertheless, a range
of problems which can be solved accurately enough by making use of
§4.2]
Non-linear resonances due to Bennett holes
209
simplified version of eqs (4.41). The simplification is based on the
following physical representations. When the inequality (4.2) is fulfilled
the atom during its lifetime at levels m and n passes a considerable
number of wavelengths, which is accompanied by monotonic relaxation.
For level populations in this case the coordinate-averaged field intensity
plays a significant role, i.e. the part of pu(r) that is variable in space is
small in comparison with the constant part. These considerations enable
one to assume that pa(r) does not depend on coordinates and pmn(r)
should be sought as
pmn(r) = e-i“(p+ei*'’ + p_e-i*r), (4.43)
which means that eqs (4.41) retain only lower space harmonics:
= qm(y) - 2 Re[i(Gfp+ + G2p_)];
^ряя = q„(v) + Amnpmm + 2 Re[i(Gi*p+ + G?p_)];
[Г - i(£ - А T к • v)]p± = ~iG12(pmm - p„„). (4.44)
The equations (4.44) adequately describe, in particular, such a charac-
teristic as the work done by the field, i.e.
P =-2ftw Re(i(Grp++ G2*p_)),
the expression for which can be readily obtained from eqs (4.44):
P =—2ha>(Nm — N„) x
___________F[|G1|2x(v)-b|G2|2x(-v)]W(u)_____________
1 + 2Г(1/ГМ + 1/Д - A^/FUZMIGdMv) + |G2|2x(—v)]
х(и) = 1/[Г + (Й-Д-Л-v)2]. (4.45)
Under the conditions Г = Гт — Г„, Q — A — A^ = 0 the exact expression
for P obtained using the results of section 3.2 (see problem (16)) differs
from the approximate expression (4.45) by 10-15% [3]. Moreover,
rigorous numerical solution of eqs (4.41) when the relation between the
relaxation constants is arbitrary and Q — A #= 0 [7,8] shows that here, too,
the discordance with relation (4.45) does not exceed 15%. Thus the
physical considerations which led to eqs (4.44) proved to be justified.
The range of applicability of eqs (4.44) is limited, of course, and we
must bear this in mind when particular phenomena are being analysed.
In the approximation of first non-linear corrections from the formula
210
Bennett structure for systems with large Doppler broadening
[Ch. 4
(4.45) it follows that
P = 2Ha(N, - N„)r(w(v)l\G,\2x(v) + |G2|2x(-v)] X
[ 1 - 2f(i +1 - ^y)[|G,|2x(v) + |G2|2 x(-v)]1) . (4.46)
*m*n'
The same expression for the field work is obtained from the exact
equations (4.41) by the method of successive approximation and averaging
over coordinates. We shall assume that the condition Г «kv is satisfied;
then
P = 2ft<o(N„ - Nm) — exp
Q-A\2'
kv / .
[|G1|2 + |G2|2 ——
^M[|G1|4+|G2|4 +
-
2|G1G2|2F2 1
r + (&-4)2J
x
(4-47)
The terms proportional to |Gi|2 and |G2|2 give the linear part of the work
done by the field. The result of interaction with two travelling waves
enters additively into this part. As a function of Q it is an ordinary
Doppler contour as in the case of interaction with one travelling wave.
The remaining terms in the expression (4.47) are due to non-linear effects
and cause P to decrease. The terms proportional to |GJ4 and |G2|4 are
evidently responsible for interaction with each of the waves individually
and lead to the proportional decrease of P. A similar term will appear in
the expression (4.21) if we expand 1/V1 + к into a series and limit
ourselves to the first correction. The last term in formula (4.47) is
non-zero in a comparatively narrow range of values of Q — A, i.e. it
Fig. 4.3. Work done by the standing wave field versus frequency.
§4.2]
Non-linear resonances due to Bennett holes
211
appears as a dip on the plot of P(Q) located near Q = A and possessing a
halfwidth Г (fig. 4.3). The fact that this term is proportional to the
product | GJ21 G2\2 indicates that it is due to the influence of both of the
countertravelling waves.
In order to account for the nature of the dip on the plot of P(Q) let us
turn to the expression for the population difference. Allowing for the first
non-linear corrections we may obtain from eqs (4.44)*
P™ - pm = ~ 4) W(v){ 1 - 2г(± +1 ) x
*m*n'
IGil2 |G2|2
Г2 + (Й - A - к • v)2 Г2 + (Й - A + к • v)2.
(4.48)
The velocity distribution contains two Bennett holes symmetric about the
point Q = A. Each of the holes is caused by one of the travelling waves. If
|й- А\»Г there are practically no overlaps of the holes. This implies
that waves interact with different groups of atoms. At |й—A|«=rthe
Bennett holes begih to overlap and at Q — A = 0 their overlap is the most
complete. Under such conditions the two waves interact with one and the
same group of atoms, which leads to an increase in the non-linear effect.
It is this very circumstance (Bennett holes overlapping) that is responsible
for the dip on the plot of P(Q).
Note that the shape of the dip on the plot of P(Q) is the same as the
shape of one of the Bennett holes. Such a correlation between the spectral
line shape and the velocity distribution of the populations is typical not
only of the case discussed here. As will be shown subsequently, it is
characteristic also of more complicated relaxation mechanisms and
different radiation processes. This fact is important because non-linear
resonance contours provide information on such physically evident but not
easily measured characteristics as velocity distribution.
The width of the dip in P(Q) is significantly smaller than the Doppler
widths therefore it can be separated from the background of the Doppler
line even at a comparatively small amplitude. The parameters of the dip
(halfwidth Г and maximum shift A) characterize an individual atom.
With the field intensity increasing in accordance with the broadening of
Bennett holes (see formula (4.11)) the dip on the plot of P(Q) broadens
too. Assuming IGJ2 = |G2|2 = |G|2 (standing wave), we obtain from the
* Formula (4.48) may be derived from eqs (4.41) by the method of successive approximations with
respect to |G|2 and averaging over coordinates.
212
Bennett structure for systems with large Doppler broadening
[Ch. 4
relation (4.45) as a result of averaging over velocities
Vn Г /£-Д\21 2 IGI2
Р(О) = 2Н^-Ю^ехр[-(—
/(Й-Л) =
2|G|2 / 1 t 1 Am„\
Г \Гт Гп ГтГпГ
Q-A
у~ гУГьк’
(4.49)
This formula holds true under the condition TV1 + K«kv, i.e. when the
halfwidth of Bennett holes Д is small in comparison with kv. Formula
(4.49) differs from the result for independent interaction with each of the
travelling waves in the factor/(£2 — A), which is selectively dependent on
frequency with effective interval у «1, and describes the dip in Р(й);
at the same time
/(O) = VB+1; Z(”) = 1- (4-50)
Thus, the dip in P(Q) broadens, obeying the same law as a Bennett hole
in a travelling wave.
Since the width of the dip depends on the field intensity the formula
(4.49) is not immediately applicable for describing field broadening in
systems with degenerate states. However, under conditions which lead to
the formula (4.24) (absence of disorienting collisions, neglected spon-
taneous transitions m —>n), the correct result can be obtained after the
following modification of the expression (4.49). |G|2 must be replaced by
|G(Af)|2 = |G|2 —M'\la)2,
where cr=0, ±1 for linear and circular field polarization respectively,
after which the expression (4.49) must be summed over M. In the
resulting profile of P(Q), which is the set of dips with halfwidths
rVl + k(M) characterizing individual transitions , the shape of
the combined dip evidently depends on the statistical weights of the levels,
the value of |Jm — J„| and the wave polarization.
The line shape of P(Q) is also strongly affected by disorienting
collisions. Let us assume the model of isotropic collisions and estimate the
first non-linear corrections as in deriving the relation (4.35) for the case of
a travelling wave. In eqs (2.67) for the density matrix a field (4.38) is
§4.2]
Non-linear resonances due to Bennett holes
213
introduced and the lowest space harmonics are retained, i.e. the p}j are
assumed to be independent of the coordinates and the quantities pm„(r, t)
and {/‘„„(г, f) are taken to have the following forms:
1Цг, t) = G/„+„e-i(“-‘^ + G'm~
0 = pX.e_i("‘*’r) + P^.e_i(“+*r),
P^q) = (-1)л"-л+,7р*:(к - q). (4.51)
The explicit form of the matrices G%„ is given by the formula (4.30) where
we must substitute G™„ for Gmna; by convention we write
Gtma = <%iadmJ2h = Gla‘, G^a = %2admn/2h = G2„.
Equations (2.67) for the density matrix take the form
rmPmm = QmW(v)/V^4 + l-
Pmn ’ ^nm Pmn ^mn rnm ^mn rnmj 5
r„p„„ = Q„W(v)/V2J„ + 1 + A^pmm -
KG^p+jn + GX«P^m — G”Xp^m — G^p™);
[Г + iA - i(£2 T к • v)l]p*„ = -i(G^p,„m - G„*p„„). (4.52)
The elements of the matrices Г, and Г + iA are given by the formulae
(4.27) and (4.28). The solution of eqs (4.52) is sought quite similarly to
the solution of eqs (4.26). In particular, for the work done by the field we
can obtain the formula (compare with the relation (4.35))
Р(Й) = -2ftw Re 2 i<GfopX.(l<T) + G^p-^a))
= -IhwN^—yexp[-(^—1[X (|G1CT|2+ |G2ct|2-
kv L \ kv / JI a
1Л(*?)|2 + IA(k?)|2 + 2Ix(Kq)I2(Kq) X
* Kq L 1 "i Л) J
^тк &пк у4т„(к) 1
— + ~----^~^аткапк J. (4.53)
-*тк tnK tmKlnK -1J
Terms containing |Gi„|2, |G2ct|2 and |/12(jc<7)|2 correspond to the separate
interaction with each of the travelling waves and they have evident
analogues in the formula (4.35). The term containing the product
Ix(Kq)I2(Kq) is due to the overlapping of Bennett holes in the velocity
distribution of the polarization moments pu(Kq) and describes the dip on
the plot of P{Q). The dip is of Lorentzian shape with a halfwidth Г
214
Bennett structure for systems with large Doppler broadening
[Ch. 4
determined by the relaxation of the induced dipole moment (to be more
precise, of the density matrix element ртя(к’^) with the value к — 1). It
can be seen from the relation (4.53) that the shape of the dip in the weak
field approximation does not depend on the field polarization. Neverthe-
less, a change in polarization leads to a change in the dip amplitude, the
amplitude at different polarizations being determined by different sets of
relaxation constants rjK and Ат„(к). Thus, for linear field polarization,
according to the formula (4.36) the dip amplitude contains relaxation
constants of populations of levels (к = 0) and alignment (к = 2). For
circular polarization there is an additional orientation relaxation constant
(see the formula (4.37)).
Note that the relation (4.53) is applicable to arbitrary polarization states
of countertravelling waves (elliptic and natural polarizations). If the
countertravelling waves are similarly polarized, then Iifjcq) = Z2(k<?) and
in the relation (4.53) the quantity 1 + Г^Г2 + (Q — Л)2] is taken outside
the symbol of the sum over Kq. Consequently, in the present case the
dependence of the non-linear part of the field work on frequency is similar
to that in the model of non-degenerate states.
Non-linear resonance in the gain or absorption spectrum of a standing
wave brings about the characteristic property of resonance on a plot of the
output power of a gas laser as a function of frequency. We are not going
to give a detailed treatment of the various numerous processes occurring
in lasers but consider the simplest case of a one-mode laser. Let all the
radiation losses be caused by its output through one of the mirrors of the
resonator. Then under stationary conditions the energy emitted by gas per
unit time from a unit cross-section of active medium is equal to the density
of energy flux
-p(e)/-£W(i-r),
oil
where r is the reflection coefficient of the mirror, I is the length of the
active medium of the laser, and P(Q) is the work done by the field per
unit time in a unit volume. For Р(й) we shall use the simplest expression
(4.47), putting for it |Gi|2 = |G2|2 — |G|2. The relation (4.54) can then be
reduced to the form
(4-54)
222.1 ir?|2
к —— —
г\гт
_ q — 1 — [(Й — A)/kv]2
= 1 + ^/[Г2 + (Й - Л)2] ’
(4.55)
§4.2]
Non-linear resonances due to Bennett holes
215
8x\/jtNm-N
rl=~^------
The equality (4.55) defines the stationary value of the intensity of the
generated radiation. The quantity tj has the meaning of the maximum
excess of the gain in the active medium over the losses. Oscillation is
possible at T] > 1 and takes place under variation of radiation frequency
within the range [(£2 — A)/kv]2< rj — 1. The relation (4.55) holds true
provided that T] — 1«1; this condition follows from the fact that the
expression (4.47) for P is valid when the saturation parameter is small
(к «1). The power of oscillation as a function of frequency is shown in
fig. 4.4. On a comparatively wide profile a dip appears, related to the
resonance term in the denominator of expression (4.55). This dip is called
a Lamb dip*. Its emergence was caused by the dip in the plot of the work
done by the field P(Q) (eq. (4.47)) as a function of frequency, i.e. is due
to the overlapping of Bennett holes in the velocity distribution of
populations. In the approximation Г«kvS/ri — 1 the Lamb dip has a
halfwidth V2 Г. In the model under consideration its contrast (the relation
of the amplitude of the dip to the maximum profile amplitude k(Q)) is
(4.55a)
and reaches the value 1/2 at x »1.
In real lasers the contrast of the Lamb dip is always less than that given
by the formula (4.55a). This is due to the velocity change of atoms caused
by trapping of radiation or collisions (see section 4.3).
If an absorbing cell with the same gas as the active component of the
Fig. 4.4. Power of oscillation as a function of frequency.
The Lamb dip was first predicted in ref. [9].
216
Bennett structure for systems with large Doppler broadening
[Ch. 4
laser medium or another gas whose transition has a similar Bohr
frequency is placed inside the resonator of a laser, the plotted oscillation
power will be characterized as well as the Lamb dip by an additional
resonance whose parameters are determined by the characteristics of the
absorbing gas transition. The work done by the field in the formula (4.54)
in the present case is made up of the negative part due to the amplifying
medium and the positive part due to absorption in the absorbing cell:
P = - a, exp[-(-1, _ |£|2 [1 - 0,(0,) |£|2] +
L x fCU \ / J
Г {Qi- Л,\21
a2exp -mi-02(a2)m
\ kv2 ! -
(4.56)
where the suffixes 1 and 2 refer to the amplifying and absorbing media
respectively.
In the model corresponding to the formula (4.47) we have
0,.2(0,.2) = 0,,2{1 + ^/[Л,2 + (£1.2 - Л.2)2]}•
Frequency-independent values a12 and 0,i2 can be obtained from the
above-mentioned relations for the work done by the field. It must be
taken into account that the cross-sectional areas of the beams in the
absorbing and amplifying laser elements may be different and then a and
P must include the relation between those areas.
The expression
„ n — 1 — a2/a, — e? + EiOtilai
’ <4-57)
e, = (Qt — A^/kty; г/ — 1 — a2/a, « 1; rj = 8n,atl/c(l — r) .
Fig. 4.5. Characteristic dependence of the output power of a laser with a non-linear absorbing cell on
the frequency.
§4.2]
Non-linear resonances due to Bennett holes
217
is analogous to the relation (4.55). As well as the ordinary Lamb dip on
the plot of the radiation power versus frequency there is an additional
non-linear resonance described by the term /32(£?) and shaped as a peak
with a halfwidth Г2. The maximum of this peak is placed on the Bohr
frequency of atoms in the absorbing cell. It does not, generally speaking,
coincide with the transition frequency of the amplifying medium. A
characteristic frequency dependence of |$|2 at Vi« v2 is shown in fig. 4.5.
The halfwidth of the peak Г2 may differ significantly from that of the
Lamb dip. If the amplifying and absorbing media are different, the
difference between the halfwidths of the peak and the dip can be large.
Lasers with non-linear absorbing cells* are widely used in laser physics
owing to the fact that as distinct from the active medium of a laser the
conditions in the absorbing cell may vary in a much wider range. In
particular, in such lasers very narrow non-linear resonances have been
attained which enabled one to carry out highly accurate investigations of
fine physical effects and to achieve the stable frequency of laser radiation.
Information on specific applications of lasers with non-linear absorbing
cells and the physical results obtained by means of these may be found in
ref. [6].
From the equations for the density matrix there follow the relations
(Pmm(v)) = Nm + Р1ЬыГт-,
(pm(v))=N„-P(l -Amnirm)lb(»rn, (4.58)
where the (p77(v)) are the velocity-integrated populations of levels
j = m,n. In the system with degenerate levels by (p77(v)) the total
population of the level j (E« v))) is meant and by 1} the
constant of population relaxation is implied, i.e. rjK at к = 2.
Thus all the characteristic features of the field work P are fully extended
to integrated populations of combining levels. However, the level
population determines the frequency-integrated intensity of spontaneous
emission from this level. For example, for the transition m-l (I is any of
the lower levels) the number of photons emitted per unit time is given by
the relation
P
The plotted frequency dependence of wml contains a constant component
AmlNm and a profile proportional to the frequency dependence of the field
* The inverse Lamb dip caused by saturation absorption was first observed in 1967 [10, 11].
(4-59)
Wml Aml
218
Bennett structure for systems with large Doppler broadening
[Ch. 4
work. Consequently, all the results obtained above for the work done by
the field P are extended to the intensity of spontaneous emission from
combining levels m and n. In particular, under interaction with a standing
wave there is a non-linear resonance on the plot of wml(Q) due to the
overlap of Bennett holes [12].
Note also that spontaneous emission is polarized if atomic states
perturbed by the field are degenerate.
Measurement of the integrated intensity of the spontaneous emission of
atoms in a strong external field has been suggested in ref. [13] as a method
of investigation. This method was one of the first to be applied to the
solution of spectroscopic problems: for measuring Einstein coefficients
[13,14], for obtaining cross-sections of inelastic processes under atom-
atom [15] and atom-electron [16] collisions, in order to study disorienting
collisions (see ref. [2]), rotational relaxation and hyperfine splitting, and
for other purposes.
4.3. The effect of collisions on Bennett holes and
non-linear resonances
In the previous sections non-linear phenomena were considered on the
assumption that collisions do not bring about velocity changes and
therefore their effect can be described by means of relaxation constants.
At the same time, it is quite clear from general considerations that elastic
scattering must manifest itself most strongly in the structure of non-linear
resonances. Indeed, owing to the selectivity of atomic interactions with a
coherent external field a non-equilibrium part of the velocity distribution
of atoms appears (Bennett structure; section 4.1) and collisions must
remove this non-equilibrium component.
Because of velocity change under collisions atoms “leave” the velocity
interval corresponding to effective interaction with the field. On the
contrary, the part of the atoms which practically did not experience
the action of the field as a result of the collision can acquire the resonance
value of the velocity and owing to that take part in induced transitions.
Therefore the velocity change of radiating atoms during collisions with
perturbing particles with an equilibrium distribution (thermostat) more or
less alters the Bennett structure which manifests itself, for example, in the
§4.3]
The effect of collisions on Bennett holes
219
shape of the Lamb dip which essentially copies the velocity distribution
(see section 4.2).
Apart from the direct effect on the velocity distribution, elastic
collisions result in frequency modulation of the dipole moment of an atom
which influences the shape of the spectral lines as shown in section 3.1.
The latter evidently must reveal itself in the shapes of non-linear
resonances.
The variety of manifestations of elastic scattering in non-linear spec-
troscopy is partially due to the complexity of the differential cross-section.
According to section 2.5, under typical conditions of atomic collisions the
differential cross-section consists of three parts, i.e. practically isotropic
scattering, classical scattering at comparatively small angles (of the order
of the ratio between the interaction energy and the energy of thermal
motion) and sharply selective diffraction scattering. The latter part is
characterized by the angle (see eq. (2.274) and below)
Od = K/pw, Х = П/ци,
where Л is the de Broglie wavelength, pw is the Weisskopf radius, and p
and и are the reduced mass and relative velocity of the colliding particles.
The corresponding component of the collision integral kernel has a width
u
$«= —u0d = ft/mpw
m
or in terms of the Doppler frequency shift
2лй 103
ks=------= 0.6 Xy-—-MHz.
In the latter equality A is the wavelength expressed in microns, pw is the
Weisskopf radius expressed in Angstroms and M is the atomic weight.
If M = 20, A = 1 and pw = 10, then ks = 3 MHz, i.e. it turns out to be
comparable with the width of the resonances that are far from narrow.
Thus, as distinct from the spectral line broadening under conditions of
an equilibrium velocity distribution (section 3.1), the shape of non-linear
resonances must be sensitive to the finest effects of elastic scattering, let
alone the isotropic scattering*.
* The possibility of diffraction scattering being manifested in non-linear resonances was first
mentioned in a short note in ref. [17]. After detailed theoretical analysis (3,18-25] and development
of experimental techniques numerous varying phenomena due to velocity changes at collisions have
been observed (see refs [6,26-29]).
220
Bennett structure for systems with large Doppler broadening
[Ch. 4
Mathematically, it is comparatively difficult to analyse the role of elastic
scattering, since a system of integrodifferential or integral equations must
be solved. The appropriate formalism has not been applied to the
problems of linear spectroscopy and therefore is not adequately de-
veloped. This accounts for the particular emphasis placed on methodology
in subsequent chapters.
We shall begin the analysis of the effect of elastic collisions with velocity
changes with the simplest model of non-degenerate states. First let us
consider the resonance interaction of atoms with a plane monochromatic
travelling wave. Abandoning the assumption of a small velocity change
during the collisions, which leads to the model of relaxation constants,
instead of eqs (4.7) we have
(2yy + Vj)p^v) = ?z(v) =F 2 Re[iG*p(v)] +
^mnPmm (v)<5;„ + J A„(v I v^p^Vi) dv,,/ = m, tv,
[y + v - i(Q - к • v)]p(v) = -iG[p„,„,(v) - p„„(v)] +
J A(y | v,)p(v,) dv,; (4.60)
Vj, = J Ац(ух | v) dv,; v = J A(v, | v) dv,; (4.61)
y = ym + y„; vy = vyy; v = vmn; f} = 2yy + vy-vy
T + iA = y + v-v, (4.62)
where the out-frequencies vy and v and the kernels of the collision
integrals and A are determined by the general formulae of sections 2.4
and 2.5. The other designations in eqs (4.60) are the same as in eqs (4.7).
Elastic collisions with velocity changes are described in eqs (4.60) by the
in-terms integrated over velocities.
Employing the method of iteration with respect to the amplitude of
the electromagnetic field solutions of eqs (4.60), it is convenient to express
them in terms of the Green’s functions satisfying the equations
(2y, + v^Fjjtv | v') = J A#(v | vJFjfa | v') dv, + <5(v - v'); (4.63)
[y + v - i(£? - к • v)]F(v | v') = J A(v | v,)p(v,) dvi + d(v — v ')•
(4.64)
§4.3]
The effect of collisions on Bennett holes
221
The terms of the iteration series are expressed in terms of the Green’s
function in the following way:
p2s+1(v) =-icj F(y | v')[p£m(v')-p*(v')] dv';
pL.(v) = -2ReiG*J Fmm(v | v^p21"1^') dv';
p^n(y) = Amn IF„„(v | v')p^,(v') dv' +
2ReiG*JFnn(v | v')pz’-1(v')dv'. (4.65)
As a zeroth approximation the solutions of the equations (4.60) are used
for populations in the absence of the held. If atoms are excited with
a Maxwellian velocity distribution and the difference v7 — v7 of out- and
in-frequencies does not depend on v then in consequence of the property
(2.247) of the collision integral kernels we have
PW») = = NmW(v), I? = 2yy + - v-,
*-m
P°nn{v)=Y qn{v) + ^qm{v) =NnW(y), (4.66)
so that the collisions do not violate the equilibrium distribution. The
quantities 1} include the rates of spontaneous decay and the quenching
processes v7 — v,.
We write out, proceeding from eqs (4.65) and (4.66), the expressions
for the populations of levels and work done by the field in terms of
Green’s functions, allowing for the first non-linear corrections, i.e. in the
approximation which will be frequently used:
P^(v) = N„,W(y) - 2 |G|2 Nmn x
J Fmm(v | vOF'^ | v2)IV(v2) dvi dv2;
PM = NnW(y) + 2 |G|2 Nmn f F„„(v | v,)^ - v2) -
Am„Fmm(v, | V2)]F'(v2 I Vj)W(v3) dvi dv2 dv3;
222
Bennett structure for systems with large Doppler broadening
[Ch. 4
P{Q) = |G|2 [J F'(y | vOWCv,) dv dv, -
2 |G|2J {Fmm(v1|v2)<5(v2-v3) +
F„„(v, I v2)[<5(v2 - v3) - A^F^Cvz I v3)]} x
F'(v | Vi)F'(v, | v4)W(v4) dv dv, dv2 dv3 dv4
(4-67)
F'(v | v,) = Re F(v | v,); Nmn = Nm — Nn.
In order to understand the role of elastic scattering one must obtain
Green’s functions. Equation (4.64) for a “non-diagonal” Green’s function
coincides with eq. (3.18) for F(vkQ\ v'). The solution of eq. (3.18) and
the physical meaning of the Green’s function was discussed in con-
siderable detail in section 3.1. With the help of this function the
expression for a non-diagonal element p of the density matrix is obtained
not only in the linear approximation with respect to the field p1 but also,
in conformity with the relation (4.65), for any order 2s +1 of perturbation
theory.
The diagonal Green’s functions essential for the analysis of non-linear
phenomena are quite clear physically. The function T’/v |v') describes
the stationary velocity distribution of the population of the level j
providing that atoms are excited to the level / with a specific value of the
velocity v'.
It immediately follows from eqs 4.63 and 4.64 that the Green’s
functions Fjjfv | v') and F(v | v') may be written as
I •’’) = 6(v - •’’) + I (468)
2У/ + V;
F(„ + !„), (4.69)
where ^(v |v') and F(v |v') are regular functions. Thus the Green’s
functions consist of two parts: the initial distribution <5(v — v') unper-
turbed by collisions and regular parts Fi: and F characterizing the action of
collisions. Some idea of the relative significance of these parts can be
gained from considering the integral over v of /^(v |v'). If the fre-
quencies are independent of v the ratio between integrals of regular and
singular parts of the Green’s function FtJ(v | v') equals
Ъ1Г^. (4.70)
§4.3]
The effect of collisions on Bennett holes
223
The quantity яу is the mean number of collisions over the interval 1/1].
The formulae (4.68)-(4.70) and the subsequent results can be easily
interpreted “in terms of time”. For this purpose the notions of mean
lifetimes are introduced:
1 1 ____________________11
T/ 2yy + vy-v; T*' Ff + Vi 2yy + v/
r2y = ry - rv = яут17 (4.71)
The time t, is the total lifetime at the level j limited by inelastic processes
(quenching) and radiation damping; rv is the mean lifetime at this level
between the instant of excitation and the first velocity-changing collision
or the mean lifetime between two successive collisions. Finally, t2/ is the
time complementing tv up to the total lifetime r7, i.e. t2/ is the time
interval after the instant when the initial velocity distribution is disturbed
by a collision (the first after the moment of excitation).
The introduction of the notions of the times ту, т1у and т2у enables one to
understand the meaning of relations (4.68)-(4.70). The singular part
enters into the Green’s function Fn(v | v') with the weight rv because r17 is
the mean free time over which the atomic velocity does not change. The
integral of the regular part differs from rv by a factor of яу and enters into
the Green’s function with a weight т2у equal to the “remaining” time when
collision-induced migration in velocity space occurs.
The obtained relation between the parts of the Green’s function is
reflected directly in the velocity distribution of the atoms. To formulate
the corresponding conclusion the solution of eq. (4.60) must be written
making use of the Green’s function (4.68):
pmm(v) = NmW(v)~ pmI(v) - pm2(v);
Pmi(v) = Tlm x 2 Re[iG*p(v)];
pm2(v) = J F^v | v') x 2 Re[iG*p(v')] dv'. (4.72)
According to eq. (4'72), the field-induced population change can be
broken into two parts: pmJ(v) and pm2(v). The first is due to the term
<5(v — v') in Fmm(y | v') and replicates the form of a selective “source”,
Re[iG*p(v)]. This part of the distribution will be referred to as a Bennett
hole to maintain the relation with the model of relaxation constants. The
second part pm2(v) which is called a collisional part is due to collision-
induced migration in velocity space and is wider than the Bennett hole.
224
Bennett structure for systems with large Doppler broadening
[Ch. 4
The particular forms of the Bennett and collisional holes depend, of
course, on the explicit forms of the kernels. However, there is a universal
relation between the integrated characteristics of the parts pml(v) and
pm2(v) identical to the relation (4.70). Indeed, if the collision frequencies
are independent of v then
<Pm2(v))/(pml(v)) = т2т1т1т = пт (4.73)
(the angular brackets denote integration over v). Thus, regardless of the
particular cause of elastic scattering, the ratio between integrated values
of the collisional part and the Bennett hole equals The relation
(4.73) is the subject matter of the areas theorem*.
If spontaneous transitions m -»n may be neglected (A^, = 0) then for
the velocity distribution of the population of the level n the relation (4.73)
is fulfilled where m is substituted for n. Otherwise, Amn=£0, cascade
transfer of a non-equilibrium component from the level m to n occurs, i.e.
apart from the fundamental collisional structure in p„„(v) there will be a
structure due to the collisions at the level m (see eq. (4.67)). pmn(v) as
before can be represented as a sum of the part p„i(v) unperturbed by
collisions (Bennett distribution) and a collisional part p„2(v) without
particularizing what collisions (at the m or n level) cause the latter part.
For the ratio of the values p„i(v) and p„2(v) integrated over velocities we
can readily obtain from eqs (4.60)
<pn2(v))I<p„i(v)) = ti„ = r,„(l - Am„rlm);
in = t2»(l - Am„r„,) - ti„ = T2„(l - Am„r„,) - A^t^t^. (4.74)
The quantities fln and give the weights with which p„i(v) and pn2(v)
enter into the velocity distribution.
Note that may be negative, i.e. the collisional part of the
distribution may have a sign opposite to that of the Bennett distribution.
In particular, if the state m decays only to produce the state n, then
AmnTm = 1, and the integral over velocities of p„„(v) proportional to
itn + *2п becomes zero, which means that there is a change of sign of the
distribution part p„„(v) due to the interaction with the field.
The explicit expression for the regular part of the Green’s functions
may be obtained by iteration from eqs (4.63) and (4.64), taking the
* The distribution pmm(v) is essentially non-equilibriutn only for the projection of v on the
wavevector. On this one-dimensional distribution is the notion of area based as well as the name of
the theorem.
§4.3]
The effect of collisions on Bennett holes
225
singular parts to be a first approximation:
/=1
A*jP(v | v') = J Afj(y | vJAj!-1^ | v') dvu
A)y°(v | v') = A„(y | v'); (4.75)
__________1________v ( A(v । V1) dV1 V
V V у + v - i(42- к • v) z?i J у + v - i(42- к • vj
4(vi | V2) dr2^(vz-! I v')dvz-i
у + v — i(I2 — к • v2) y + v — i(I2 — k- v')
The function AjP(v | v') is called the kernel of the /th order. It is easy
to verify the relation
I v) dVi = v).
The quantity Ayy(u | v')/vy is the probability (in the usual sense) of the
velocity change v'-*v as a result of I collisions or, in other words, the
velocity distribution (normalized to a unity) arising from the distribution
<5(v — v') due to I collisions. Evidently, collisions broaden this distribu-
tion and consequently its maximum value falls off with increasing I. Each
of these distributions enters into the Green’s function PJf(y | v ’) with a
weight of
1 / vy у _ 1 / vt у
4 + V 2yy + vy \2yy + vy/ ’
which is given by the product of the time of the atom’s being at the level j
between two successive collisions by the probability [vy/(2yy + vy)]z of the
atom’s remaining at the level I up to the (/ + l)th collision.
The convergence of the series for Fu follows from the fact that terms of
the series decrease no more slowly than the terms of a geometric
progression with the denominator VjKTj + vy) = ny/(l + ny). Therefore the
effective number of terms of the series is equal to лу.
By analogy with nt the quantity n = v/(y + v' — v') for a non-diagonal
Green’s function may be introduced. It is difficult to estimate the effective
number of terms of the series (4.76) when the specific type of the kernel
A(v I v') is not known. Nevertheless, it can be shown that this number
does not exceed ny and actually may prove to be significantly smaller as a
226
Bennett structure for systems with large Doppler broadening
[Ch. 4
result of a decrease in the factors [y + v — i(I2 — к • v)]-1 owing to their
dependence on the velocity and Q.
If n]«1 then on the strength of the relation (2.230) the inequality n «1
is satisfied and so in the formulae (4.75) and (4.76) we may confine
ourselves only to the first terms of the series (Z = 1), i.e.
F(v | v') =-----— ---------- d(v - v') +----\-----------7 ,
у + v —1(12 — к • v) L у + v — i(I2 — к • v )J
1 ” ’> - 2^ К - V ’’+ 2^ 1 ” '>
(4-77)
The approximation (4.77) means allowing for the first corrections for
velocity-changing collisions.
The case of large n, and n is of particular interest because velocity
changes become significant. To obtain Green’s functions one must specify
the forms of the kernels А„(у | v') and 4(v | v'). It stands to reason that
to calculate the values of sums in the relations (4.75) and (4.76) is
impossible for arbitrary kernels, and therefore a question of kernel
modelling arises.
Consider one of the simplest models, the model of strong collisions,
according to which after collision the atomic distribution becomes
Maxwellian, independent of the velocities before collisions:
| vj = V/W(v), 4(v | vj = vW(v). (4.78)
The summation in the relations (4.75) and (4.76) for the kernels (4.78) is
readily carried out:
F"^v lv')=z-7r-R(v-v') + n>W(v)],
2yy + Vf
(4.79)
F(y | v') =----------—7-— --------- (<5(v - v') +
у + v — i(I2 — к • и) I
vW(v)
у + v — i(I2 — к • v')
vW(v!) dvi
у + v — i(I2 — к • Vj).
(4.80)
Formulae (4.79) and (4.80) can also be obtained by direct solution of eqs
(4.63) and (4.64).
Making use of the expressions obtained, we calculate the velocity
distribution of the populations. If we are going to employ the method of
field amplitude iteration it will be necessary first to find the non-diagonal
§4.3]
The effect of collisions on Bennett holes
227
element pl(v). From eq. (4.65), using the zeroth approximation and the
Green’s function (4.80), we obtain
p\v) = —iG(Nm -N„)j F(v | v')W(v')dv'
= —iG(Nm - N„)y(v)[l - v(y(v))]-*, (4.81)
y^ = Y + v^-k-vy <Hv)>=fy(»)dv. (4.82)
The similarity of this expression to eq. (3.49), should be noted. But for
the replacement v—>v', both expressions equally depend on 12.
However, unlike eq. (3.49), p'(v) determines the shape of the radiation
(absorption) line with a given velocity v provided that the atoms have
been excited with a Maxwellian velocity distribution. The close connection
between pJ(v) an<^ the expression (3.49) is a consequence of a symmetry
relationship for the Green’s function
F(v |v')W(v') = F(v'|v)W(v), (4.83)
which holds true for any real kernel and follows from the relation (2.246)
for kernels.
On substituting the expression (4.81) into eq. (4.65) for p,y(v) using the
Green’s function (4.79) we obtain
pmm(v) = Nm W(v) - 2 |G|2 (Nm - Nn) x
Re{[l - ^(v)»-1^^) + T2mW(v)(y(v)>]}; (4.84)
p„„(v) = NnW(v) + 2 |G|2 (Nm - Nn) x
Re{[l - vCy^))]"1!^^^) + T2„W(v)(y(v))]}. (4.85)
Consider the parts of expressions (4.84) and (4.85) which are due to the
influence of the electromagnetic field (field correction to the velocity
distribution). By virtue of general principles (see relations (4.72)) the field
correction is a sum of a Bennett distribution (the term proportional to
y(v)) and a collisional term (proportional to W(u)). The in-term in eq.
(4.60) for a non-diagonal density matrix element caused the appearance in
the expressions (4.84) and (4.85) of the factor [1 — v(y(v))]-1, reflecting
the effect of frequency modulation. This is a complex factor and therefore
the Bennett distribution in eqs (4.84) and (4.85) takes on additional
asymmetry compared with the model of relaxation constants where it is
proportional to Re[y(v)].
228
Bennett structure for systems with large Doppler broadening
[Ch. 4
The collisional part of the distribution is characterized by a velocity
dependence according to the Maxwellian distribution W(v). To take
account of this circumstance we shall call this part the homogeneous
saturation band: velocity homogeneity is due to the employed model of
strong collisions, in accordance with which within the Maxwellian dis-
tribution an atom with any velocity may acquire the resonance velocity
and interact with the field.
The ratio between the areas of the band and of the Bennett distribution
equal and т2т1т:Хп according to the areas theorem (see eqs (4.73)
and (4.74)).
In the most interesting case of dominant Doppler broadening (y + v'«
kv), when the Bennett distribution looks like a sharp structure, the effect
of the factor [1 — v(y(v))]-1 in the relation (4.84) becomes negligibly
small:
+ + 1Q), (4.86)
where w(z) is given by the formula (3.32). Under these conditions the
effect of frequency modulation is so great that the integral term in eq.
(4.60) for p(v) can be neglected, i.e. it may be assumed that 4(v | Vi) = 0.
By way of example, the expression (4.84) after this takes the form
Pmm(v) = NmW(v) - 2 |G|2 (Nm - N„)W(v) X
у + v'
lTlm (y + V')2 + (^ - v" - к • v)2
Vir
r2m”exp -
kv L
Q - v"\2'
kv / .
(4-87)
The Bennett distribution as in the model of relaxation constants has a
Lorentz form. Its halfwidth is, however, equal to ym + y„ + v'. The
dependence of ym + y„ + v' on the concentration of perturbing particles is
determined by the out-frequency v' and not by the difference v' — v' as in
the model of relaxation constants.
Let us write down the expression for the population difference of the
levels, which is necessary for evaluating the work done by the field:
pmm(v) - p„„(v) = (Nm - Nn)W(v) x
-2 IP|4 Ti(r + v') .
V 1 1 l(y + v')2 + (i?-v"-jt-u)2
7^ exp
kv
(4.88)
^2 ^2m(l ^2л(1 AmnTm).
§4.3]
The effect of collisions on Bennett holes
229
The quantity can be considered as the effective time of interaction with
the field from the time of excitation to the first velocity-changing collision
(independent of the level an atom occupies). The quantity t2 has the
meaning of the time of interaction with the field after the first collision up
to the time of quenching of the two levels.
The ratio of amplitudes of the homogeneous saturation band and of the
Bennett hole is
/-Y + v't2 Г /42-v"\2'
Уя——------exp —1,-1
kv Ti L \ kv / J
(4.89)
Despite the small value of (y + v’)/kv the ratio (4.89) may be far from
small owing to the factor t2/t*. A large value of t2/t! is typical of
systems where the cross-sections of inelastic processes are significantly
smaller than those of elastic processes.
The expression for the work done by the field P to the approximation of
the first non-linear corrections is obtained by substituting the values of the
populations (4.84), (4.85) and the Green’s function (4.80) into the
expression (4.65) for p3 whereupon in accordance with the definition of P
we obtain
P = -2ft<u Re(iG*p) = -2ft<u |G|2 (Nm - Nn) x
{Re 1 -«(>>(«)> “ 2 |G|2 [Re ~
2|о|Мкег?ШЫ1' (4-90)
In limiting cases there are the following expressions for P:
7 + v'-v'»kv: P=-2ti(o\G\2(Nm-Nn)x
{r+(a-^~2|G|2(ri + T2)x
Г Г I2)
_Г + (12- 4)2J F
r + i4 = ym + y„ + v- v;
7 + v'«kv: P = — 2ha) |G|2(Nm — N„)~^ex.p
kv
2т2Ул Г (Q — v"\2])\
. - exp -I I H.
kv L \ kv / JJz
(4-92)
This was first noticed by Kazantzev and Surdutovich [20].
230
Bennett structure for systems with large Doppler broadening
[Ch. 4
In the first case the velocity changes under collisions do not show up at all,
i.e. the formula (4.91) can be derived by assuming in eqs (4.fj0) that
A„(y | Vj) = vy<5(v - Vi); A(v | v,) = vd(v - vx).
This is quite natural since when the condition r»kv is satisfied the
interaction with the field is not sensitive to atomic motion.
In the opposite limiting case (relation (4.92)) the times r, and r2 show
themselves differently in P as a function of frequency. The term
proportional to r2 and describing the band of homogeneous saturation is
characterized by somewhat different dependence on £2 from those of the
other terms. Thus the strong collisions bring about deformation of the
non-linear part of the work P(£2). In this respect we must draw your
attention to the different results obtained within the model of relaxation
constants, where the line retains the Doppler form whatever the field
intensity is.
If v"#:0 then the linear (in |G|2) part of P is an asymmetric function of
frequency (see the formula (3.60) and its discussion). In particular, the
function
Re{<y(*>)>/[l-(v' +iv")<y(v)>]}
attains its maximum value under (see problem (17)) £2 = v" — 2v",
although its first moment is equal to 12 = v".
Let us now study the effect of strong collisions on the simplest
non-linear resonances due to the interaction of atoms with a standing
wave. As in deriving eqs (4.44), the effects of space modulation of the
populations will be neglected. According to the problem under study we
proceed from equations containing terms integrated over velocities:
(2yy + Vj)p„(v) = qf(v) =F 2 Re[iGrp+(v) + iG2*p_(v)] +
AmnPmm (v)djn + f A„(v I v^p^Vi) dvB
[y + v - i(I2 T к • v)]p±(v) = -iGlj2[pmm(v) - p„„(v)] +
j A(v | Vi)p±(vi) dvt. (4.93)
The notation used in eqs (4.93) is analogous to that of eqs (4.44).
An analytic treatment of the interaction with a travelling wave showed
that if the condition y + v'«fcv is fulfilled the in-term of the non-
diagonal collision integral allows insignificant corrections for non-linear
resonances (in the model of strong collisions). Therefore, seeking the
§4.3]
The effect of collisions on Bennett holes
231
solutions of eqs (4.93) in the model of strong collisions, we shall take
4(v | Ui) = 0. Then* in the relations (4.65) which define the populations,
iG*p2l-I(v') w*ll be replaced by the quantity
iG*p+-1(v') + iGZp^v') = [p^2(v') - p*~2(v')] x
+
Ifrl2
_y + v — i(I2 — к • v
|62|2
у + v — i(I2 + к • v)J
The diagonal Green’s functions Ft/(v | v') in eq. (4.65) are given by the
expression (4.79). On repeating the standard calculations of the work
done by the field to the approximation of the first non-linear corrections
we obtain
vit
P(Q) = -2ha)(Nm - Nn) — exp
kv
Q - v"\2~
kv / .
x
(|G1|2 + |G2|2-{[|G1|4+|G2|4 +
2 |GiG2|2(y + v')2 I r,
(y + v')2 + (Q - v")2J у + v'
(IGJ2 4- |G2|2)2^^T2exp
kv
Q - v"\2
kv /
(4-95)
Velocity-changing collisions lead to the following modification of the line
contour P(£2) compared with the model of relaxation constants (compare
eq. (4.47) with eq. (4.95)). The dip on the plot of P(£2) still retains the
Lorentzian shape, but its parameters (halfwidth and amplitude) are not
determined by the same characteristics as in the expression (4.47). The
halfwidth у + v' is equal to the sum of the radiation halfwidth and the real
part of the non-diagonal out-frequency v whereas the halfwidth of the dip
in the model of relaxation constants comprises v' — v'. The amplitude of
the dip in relation (4.95) is proportional to the time of the interaction with
the field Ti between two successive collisions and not to the total
interaction time
1 1 / 4m„\
t=t1 + t2 = — + —(1 —— ,
Г Г X Г /
* The imaginary part of the in-term at the same time brings about interesting effects even at |V"| «kv
(see problem (17)).
232
Bennett structure for systems with large Doppler broadening
(Ch. 4
as in relation (4.47). Velocity-changing collisions give rise to the ap-
pearance of a spectral structure proportional to t2 which in the model of
strong collisions has a Gaussian shape exp[—(Q — v")2/(fcv)2] and is due
to the band of homogeneous saturation in the velocity distribution of
populations (the name “band” will be retained for the corresponding term
in P(I2)).
All the parts of expression (4.95) that are functions of £2 attain their
maximum (or minimum) values at the point Q = v". However, when the
relation (4.95) was derived the in-term of the non-diagonal collisional
integral was not allowed for. If the imaginary part of the in-term is distinct
from zero (v"#:0) the extremum of non-linear corrections is still under
I?! = v". The linear (in |G|2) part of P(I2) is at a maximum at the point
£22 = v" — 2v" (see the discussion of the formula (4.90)) and problem
(17)). It must be recalled that the maximum of the line profile in the
model of relaxation constants corresponds to the frequency Q=v" — v".
The ratio of the area of the band to that of the dip is
r2 /|G1|2 + |G2|2\2
Tj \ |GiG2| /
(4.96)
i.e. it is proportional to the ratio of effective times t2 and The ratio of
the amplitude of the band to that of the dip is
^y + v'r2/|G,|2+|G2|2\2
_ I , . I
kv \ |G1<jr2| /
(4-97)
The model of strong collisions is distinguished from other models since
it admits an exact analytical solution for the density matrix under an
arbitrary intensity of a monochromatic travelling wave. Assuming the
kernels of the collision integrals to be given by the formula (4.78) we write
eqs (4.60) in a matrix form:
K(v)p - vW(v)(p) = W(v)Q.
(4.98)
The matrices appearing here are given by the relations
/2ym + vm
ВД =
iCr
\ —iG*
0
2y„ + v„
-iG
iG*
iG*
-iG*
g-ik-v
0
-iG
iG
0
g* + ik • v
§4.3]
The effect of collisions on Bennett holes
233
g = у + v — iI2;
/Qm\ /Vm 0 0 ° \
Pnn 1 л I Qn ]. v = 0 v„ 0 ° 1
P = ; Q = p I 1 0 1 ° 0 V ° /
\p«/ \ 0 / \ 0 0 0 V*/ (4.99)
Premultiplying eq. (4.98) by К ‘(v) and integrating over velocities we
obtain
(p) = [l-(K-1(v)W(v))v]-1(K-I(v)W(v))Q, (4.100)
where 1 is a unit matrix. On substituting expression (4.100) into eq.
(4.98), by simple manipulations we obtain
p = K-1(v)[l - v(K-1(v)W(v))]-1W(v)Q. (4.101)
The matrices K(v), 1—(K(v)W(v))v and 1 — v(K-1(v)W(v)) are not
singular and therefore the formulae (4.100) and (4.101) are correct.
The expression for K-I(v) *s rather clumsy (see problem (18)). Note
that K-1(v) determines the solution of eq. (4.98) in the absence of the
in-terms of the collision integrals (v = 0), i.e.
p = K"1(v)W(v)Q,
(4.102)
which formally corresponds to the case already studied (model of
relaxation constants) (see section 4.1).
The solution (4.101) has a relatively simple form when collisions are
accompanied by large phase shifts and v = 0 may be assumed. For the
difference of populations and the work done by the field it follows from
the relation (4.101) that
Pmm(v)-Pnn(v)
= (Nm — Nn)W(y) Г___________к(у + v')2_____’
1 + 2t2 |G|2 (K'(v)) L (y +v')2(1 + k) + (I2- v"-k‘v)2.
= (N W )[»'(;) 2|Gf»,r(») + 2|GfT2<r(v)>H'(v)1
”'L ' ’ 1 + 2 |G|2 r2(r(»)> J’
K = 2|G|2T,/(y + v);
v + v' — i( 12 — v" — k'v)
y(„)=ух») - ,п.)- (y/vT(1+;)+(Q_^_;.pr
(4.103)
P(fi)=-2»<a(N.-N.)|G|2T7?f^L (4.104)
234
Bennett structure for systems with large Doppler broadening
[Ch. 4
The values erf r, and t2 are given by the formula (4.88). At т2 = 0 (the
absence of in-terms) the expressions (4.103) and (4.104) for the difference
between populations and field work formally coincide in shape with the
corresponding expressions (4.11) and (4.20) of the model of relaxation
constants. Changes due to the fact that t2 is non-zero amount to
multiplying the population difference and P(I2) by a factor [1 +
2 |G|2 т2(У'(г))]-1 independent of velocity and increasing saturation. In
the difference of populations it leads to the emergence of a band of
homogeneous (non-selective)-saturation. By the general area theorem, the
ratio of the band area to that of the Bennett hole (see relation (4.103))
does not depend on |G|2 and is equal to t2/ta. The halfwidth of the hole is
(y + v')Vl + *-
In the case v^O, on combining the equation for ptj and p the
expression for the population difference may be reduced to the form
Pmm(v) ~ Pnnfv) = (Nm - N„)l l + 2T2Re(^ x
к L \ 1 Ут 1Уп / -
[W(u) — к(у + v')y'(v)] —
<P> 1]
iG(Nm-N„)JJ
к(у + v') Re vY(v)
(4.105)
The first term in relation (4.105) depends on v with v = 0, being an even
function of Q — v" — к' v. It follows from the definition of Y(v) and the
fact that v and (p) are complex that the second term of the expression
(4.105) contains both even and odd functions of Q — v" — k'V. In
general, the Bennett hole proves to be asymmetric as was already noted
when analysing the formula (4.84) obtained by the iteration method. It is,
however, characteristic that the second term in the relation (4.105) is
comparatively small, being a function of the order of |v|/fcv of the first
term also when the fields are comparatively large but under the condition
that (y + v')Vl + к« kv. This also implies that the asymmetry of the
hole is not very large and the expression (4.105) under (y + v')vl + k«
kv passes into eq. (4.103).
The expression for the work done by the field under v #= 0 is clumsy so
we shall reduce it for a particular case allowing for the first corrections for
the parameter v/kv:
IGI2 Vir Г (Q-v" + 2v"V
P(Q) = 2ka,(Nn-Nm)-}~^^ 1-*----------- +
Vl + к kv L (kv)2
У/nV _ 1 (1 + K)(y + *9 + л IGI2 (*i + *2)]
fcfiVl + * kvy/1 + к J ’
|(y + v')Vl + к — i(£2 — v")| « kv, 2 IG|2 t2 « fcvVl + k.
§4.3]
The effect of collisions on Bennett holes
235
Formula (4.106) describes the behaviour of P(Q) in the vicinity of the line
maximum, i.e. near the point Q = v” — 2v". The position of the maximum
of the profile is the same as in the linear theory and does not depend on
the field intensity within a fairly wide range of the latter’s change.
The term proportional to v' gives rise to an increase in the intensity at
the maximum which in its turn reflects the effect of the line narrowing (the
Dicke effect). The increase in field intensity (the increase in к) reduces
the effect of this term and, on the contrary, increases the role of the last
(negative) term in the expression (4.106). Thus non-linear effects result
only in broadening of the line*.
The model of strong collisions may be extended by allowing for the
degeneration of states and, for the case of molecules, rotational level
splitting as well. As to the M projections of the angular momentum, the
model of strong collisions corresponds to the conditions under which the
collisions result in an equilibrium (homogeneous) distribution in M
sublevels irrespective of its type before the collision. We may treat strong
collisions resulting in a Boltzmann distribution in rotational levels J
similarly. The model of strong collisions may be formulated for other
degrees of freedom as well.
Within such a generalized model of strong collisions the in-term of the
collision integral may be represented as
Sm(nMn'M', v) = v^ Wiv)^ p(nMn'M', vj dvj +
v) +
Д/ “t" 1 Mi
^(«ОтгтЕ [ p(nA/1n'Mi,v1)dv1 +
1 Mi J
v^W(v)WB(J)^^x
I p{aJ}MAa'JAMA, vj dvn
Mih J
n = aJ. (4.107)
Here WB(J) is the Boltzmann distribution in rotational levels which is the
same for the states n and n'. Each of the terms in the expression (4.107)
describes the result of a collision during which certain changes of a state
take place. The first term corresponds to the velocity-changing collisions
* More detailed analysis of solution of the equation (4.98) is given in ref. [21].
236 Bennett structure for systems with large Doppler broadening [Ch. 4
when the internal quantum numbers do not change. The second term is
connected with elastic collisions during which the angular momentum is
completely disoriented but the velocity remains constant. Collisions with
simultaneous disorientation and velocity changes are described by the
third term in formula (4.107). Finally, the fourth term, which is physically
meaningful only for molecular gases, characterizes collisions resulting in a
Boltzmann distribution in the rotational levels. Such collisions can be
expected to result in an equilibrium distribution in velocities as well as in
magnetic sublevels.
The relation (4.107) establishes practically the most general form of the
in-term of the collision integral within the model of strong collisions.
Obviously, under different conditions not all the terms of the expression
(4.107) are equally important. For example, to introduce the fourth term
for atoms does not make sense. On the contrary, for molecules it may be
the main term.
The model of strong collisions is the only known model to allow for
velocity changes and internal quantum numbers and at the same time it
makes the exact solution for the density matrix possible under any
electromagnetic field intensity. As far as the velocity changes are
concerned the possibilities of this model have already been investigated.
As to disorientation and rotational relaxation, the reader must turn to
problem (19).
Now let us proceed to the analysis of another limiting case opposite to
the model of strong collisions, assuming that during the collision the
velocity is changed by a value much less than a mean thermal value v
(model of selective scattering). It should be recalled (see section 2.5) that
diffraction effects correspond to small scattering angles (0 ~ 10-2) and the
corresponding part of the cross-section at the same time can substantially
exceed the cross-section of isotropic scattering. Therefore the model of
selective scattering undoubtedly reflects real processes taking place under
collisions and the problem consists only in determining the significance of
the difference from the model of relaxation constants where a velocity
change is not allowed for at all.
In the model of selective scattering the kernels depend on the difference
of velocities before and after the collision (difference kernels; see relation
(2.259) and subsequent discussion). With the interaction with a plane
wave in mind we shall use one-dimensional difference kernels (see relation
(2.276))
A„(y - и,), A(v - Uj),
(4.108)
§4.3]
The effect of collisions on Bennett holes
237
where v and u, denote the velocity projections on the wavevector k. The
equations for one-dimensional Green’s functions follow from eqs (4.63)
and (4.64):
(2y, + v^F^v -v')=lAu{v - - u'jduj + d(y - v'), (4.109)
[y+ v—i(Q—kv)]F(y | v') = JA(v- Vt)F(vj | u'jduj + <5(u — v'),
(4.110)
where v, and v = v' + iv" are “one-dimensional” frequencies. The diago-
nal Green’s functions F^v — v') as well as the kernels depend on the
difference v-v'. Since the factor y + v — i(£2-kv) contains v, the
non-diagonal Green’s function does not depend on v — v', although the
kernel A(v — Uj) is a difference kernel.
Equations (4.109) and (4.110) will be solved by means of a Fourier
transform*. We introduce the Fourier transforms of Green’s functions:
^(r) = J Ftl(v — v') exp[ifc(v - u')r] d(u - v');
F(t I v') = J F(y | v’) exp(ifcur) du. (4.111)
An algebraic equation for follows from eq. (4.109) and its solution is
given by the formula
^T)_2y7 +^-АДт)
_ 1 Гц 1
2y, + v,. L 2г/ + ^-Ай(т)Г
Л(т) = / An(v ~ ui) exP[i^(u “ Ui)t] d(u - vt). (4.112)
The function F(r | v') satisfies the differential equation
/ d \
(y + v - ifi + —IF(r I v') = A(t)F(t I u') + exp(ifcu'r),
\ dr/
A(r) = J A(v — Vj) exp[ifc(u — ujr] d(u — uj, (4.113)
* The iteration series (4.75) and (4.76) may be used with the same result. It can easily be seen that a
Fourier transform with respect to v results in a mixed representation corresponding to the Fourier
transform of the Wigner representation with respect to both variables.
238
Bennett structure for systems with large Doppler broadening
[Ch. 4
which is readily solved:
F(r|v') =
dtj exp —(y + v — i£2)(r — 14) +
(4.114)
The inverse Fourier transform of the functions (4.112) and (4.114) yields
+ к f" Луу(т) exp[-ifc(u — и')т] 1
2л: J-о 2yy + у, - Д/У(т) J
(4.115)
F(u|u') = ^-
dtj exp[i(Q — kv)r
[(И-ку'Ут^ф^ф *(Ti); (4.116)
ф(т) = exp] —(Г + izl)r — I [v — Л(т')] dr' [;
I Jo j
T + izl = y4-v — v. (4.117)
The function ф(т) defined by the relation (4.117) has the meaning of a
correction function for an atomic oscillator which appears in the linear
theory of line broadening (see section 3.1). In particular, the spectral line
contour for atoms excited with a given velocity v' is defined by the
integral of F(y | v') over v and, according to eq. (4.116), it is equal to
fF(u|u')du=f 0(r)exp[i(Q — fcv')T]dr. (4.118)
J Jo
The term —(Г + iA)t in the correlation function reflects, as usual, its
damping due to spontaneous decay, inelastic processes and phase modula-
tion. The integral term in ф(т) describes the effect of frequency
modulation owing to velocity changes during the collisions (see expres-
sions (3.40) and (3.41)).
The solution of non-linear problems with the help of Green’s functions
is not difficult but the appearance of the results is rather clumsy.
Therefore it is good practice to treat different effects individually and only
then to turn to the general case. We have already discussed in section 3.1
the effect of frequency modulation on the spectral line shape, i.e. the role
of the non-diagonal collision integral. Now it seems logical to analyse the
distinctions between phenomena in the absence of “phase memory” and
to explain the role of diagonal collision integrals. To treat this simple case
§4.3]
The effect of collisions on Bennett holes
239
will be of independent interest, in particular with respect to the electronic
transitions of atoms and molecules. Electronic states during collisions are
perturbed, as a rule, in quite a different way, so the in-term in the
non-diagonal collision integral can be neglected (A(u | v') = 0). The form
of a non-diagonal Green’s function under such conditions is simple, i.e.
6(v — v')
(4.119)
and velocity-changing collisions influence directly only the populations of
levels.
Let us consider diagonal Green’s functions (4.115). As an approxima-
tion of the function A^v — uj a model exponential kernel (2.282) will be
used:
A/y(u - Ui) = exp(—|v - vd/^); A„(x) = . (4.120)
Then Fjj(r) and 7^(v — v') are given by
1 Г n- 1 'P-
---------- 1+;----7—’~77,—7 > = (4.121)
" 2y, + v,L 1 + (1 + п,)(Цт)2] 'Г/
/^(u-u') = —— <5(u-u')+-—expf - —Ц .
,л 2y, + vyl 2s, VI + и, s,Vl + n,/J
(4.122)
The regular part of the Green’s function, which is a velocity distribution
diffused by collisions, is a symmetric function of the difference v — v’,
which is due to the use of a difference kernel. The parameters of the
given distribution are the halfwidth s, of the kernel (at the level j) and the
effective number of collisions n,. The halfwidth of the regular part of
Green’s function is equal to sj\Sl+nif i.e. for large n, it is proportional to
y/nr This law (the law of large numbers) is characteristic of the
distribution variance of a random quantity which undergoes nf successive
sudden (discontinuous) changes, in each of them the variance being equal
tO Sj.
Making use of the Green’s function (4.122), we may find the limits of
applicability of the difference kernel: the velocity variance resulting from
n.j collisions must be much less than the mean thermal velocity
SyVl + rij « V.
(4.123)
240
Bennett structure for systems with large Doppler broadening
[Ch. 4
Otherwise, the friction effects discussed in section 2.5 will necessarily
show up, leading to an asymmetry of the distribution (pulling in the
direction of v = 0).
If the number of collisions n, is large (ny»1) we can conclude from the
relation (4.121) that the integral in the expression (4.115) and the
properties of the regular part of the Green’s function are determined by
the behaviour of Лу(т) in the range of small values of r. For an arbitrary
kernel we can use the expansion
A/(T) = ^[1 - a^k | r| - bfakt)2 + ...], (4.124)
The numerical factors a, and bf depend on the particular form of the
kernel and on the behaviour of А/У(и - и,) as |u - i/J—>o°. According to
the general theorem of Fourier transform theory we can draw the
following conclusions. Let Atj(y — Uj) at large values of |u — uj fall off as
(y — Vi)-2 or slower; then in the expansion (4.124) the term linear in r
differs from zero, and a, > 0. If, on the contrary, the wings of the kernel
fall off faster than (v — th)-2 the expansion begins with the quadratic term
(a, = 0, bj >0). In the latter case, limiting ourselves to the quadratic term
in the expansion (4.124) we obtain an expression for the Green’s function
Fjjfy — v') coinciding with eq. (4.122) with the exception of the substitu-
tion Sj^rfbjSj.
Thus, to the accuracy of the above-mentioned replacement, the Green’s
function (4.122) under ziy»l is the universal Green’s function for all
kernels in which the wings fall sufficiently quickly.
When aj 0 the width of the regular part of the Green’s function under
large rij is proportional to i.e. this case is outside the limits of
applicability of the law of large numbers. Generally speaking, kernels with
a slow decrease at |u — i/J—>o° in principle may exist. However, in what
follows the Green’s functions will be considered with regard to the model
of the exponential difference kernel. In addition to the agreement with the
law of large numbers such a kernel contains the discontinuity of the
derivative at v = inherent in the real one-dimensional kernels (see
section 2.5).
Making use of Green’s functions (2.119) and (2.122) on the basis of the
recurrence relations (4.65) we obtain the velocity distribution of popula-
tions to the approximation of the first corrections for the intensity of the
electromagnetic travelling wave:
f N -N 1 Г fv + v')2 T)
Pii(y) = W(v) N/T2|G|2-^-------------; • — —, — + n/Zi(x) ,
"v ' v 'I ’ 2yy +v, y + v'L(y + v')2 + x2 ’ ’ 'JJ
(4.125)
y + v
/cS.Vl + rij
§ 4.3] The effect of collisions on Bennett holes 241
7 z x (y + v')2 Г ехр(-|т)|/Цл/1 + ny)
Л) 2ЦУГ+^1„ (y + v')2 + (x-r/)2 4
Re[ci £, sin %,- - si %, cos , (4.126)
x = Q - v" - kv, §,= [y + v' - i(Q — v" — kv)]/кз,\/ТТп;.
Here ci § and si § are the integral cosine and integral sine of a complex
argument:
f"sint . _ f" cos? ZJ
si § = — I ---dr; ci § = — I -------dr. (4.127)
J? r r
The velocity distribution of the population at the levels j = m,n, apart
from the Bennett hole of the usual Lorentzian shape with a halfwidth of
(y + v')/k, contains according to the formula (4.125) a collisional
structure described by the function Z,(x).
The velocity distribution for the level n has, generally speaking, a more
complicated form. The point is that the spontaneous transition m^n
results in the transfer of a non-equilibrium structure created at the level m
to the level n. In order to simplify the formula (4.125) this factor is not
reflected in it, i.e. the Einstein coefficient Amn is assumed to be small
enough. The expression free of this restriction is obtained in problem
(20).
The collisional distributions Z7(x) are described by the convolution of
the exponential parts of the Green’s functions (4.122) with the Bennett
distribution, and therefore the width of Z7(x) is always greater than that of
the Bennett distribution. On the contrary, the characteristic width of Zy(x)
must be considerably smaller than v since only in this case is the use of
difference kernels justified. Therefore the Z7(x) have the fdrm of sharp
structures against the background of a Maxwellian distribution (collisional
peaks and dips) symmetric like the Bennett distribution about the velocity
v = (£2 — у")/к.
The distributions Z7(x) are expressed in terms of tabulated functions
(4.127). Simple relations for limiting cases may, however, prove to be
useful:
у + v' » fcS' Vl + nf.
________(y + v')2________
(У + v')2 + (£2 — v" — kv)2
2 Rc + +
[y +v' + i(Q-v"-£v)]3
(4.128)
242
Bennett structure for systems with large Doppler broadening
[Ch. 4
у + v'« ks/Vl + nf.
„, . л y + v
Z(x) = - -—x
2 + n,
exp(—| £2 — v" — kul/ksyVl + и,) .
(4.129)
collisional dip is the Bennett hole slightly
In the first relation the
broadened by collisions; its shape is similar to the Lorentzian shape with a
halfwidth equal to
|(y + v')[l + 3(1 + иу)(кзу)7(у + v')2].
/0
If y + v'»ks7Vl + л, it is not justifiable physically to break the non-
equilibrium structure into a Bennett hole and a collisional dip. It is
advisable to treat them as a whole:
(y + v')2
/ - ,42- ~2 + W>Z(X)
=(1+„д (^7/ 2_2Re.
L(y + v)2 + x2 (y + v-ix)3J
The halfwidth d, of such a “compound” dip is given by
(4.130)
<5, = (y + v
(4.131)
The relative correction to the halfwidth due to the velocity change is
quadratic with respect to the small parameter ksj\/nil('Y + v').
In the other limiting case у + v'«ksf\jl + n, the collisional dip is
represented as a separate structure with an exponential shape wider than
the Bennett hole; its halfwidth “at the level 1/e” is /c5yVl + Character-
istically, the halfwidth of the function Z7(u) is proportional to the
halfwidth of the kernel of the collision integral which, in its turn, is
directly related to the halfwidth of the differential cross-section of elastic
scattering (see formulae (2.280) and (2.281) and their discussion). Thus
the measurement of the collision integral may prove the basis for studying
scattering amplitudes.
The ratio of the amplitudes of the collision dip and of the Bennett hole
in the approximation (4.129) is
л y + v'
2 kstVl + и,П'
and for rij»1 may be fairly large.
(4.132)
§4.3]
The effect of collisions on Bennett holes
243
The aforegoing discussion of the limiting cases may give us an idea of
the behaviour of the non-equilibrium part of the velocity distribution with
the change in density Nb of the perturbing particles. Let radiation
relaxation constants 2y; and ym + y„ be small in comparison with ks,. The
out-frequency v' (in the same way as other collision rates) is proportional
to Nb. In the range of very low densities the factor (4.132) becomes much
less than unity, the collisional dip has a small amplitude and the
non-equilibrium structure in the velocity distribution is determined by the
V
Fig. 4.6. Components of the non-equilibrium parts of the velocity distribution:---, Bennett hole;
-----, collisional dip. For explanations see the text.
244
Bennett structure for systems with large Doppler broadening
[Ch. 4
Bennett hole of Lorentzian shape and has a halfwidth у + v' linearly
dependent on the density (fig. 4.6a). As the density rises, nt = v7/IJ
increases, reaching the constant value v7/(v/ - vz). If vy/(v, - v7)»1 then
there exists a range of values of JVb such that for it the factor (4.132) is not
small but the condition у 4- v'« foyVl + is still fulfilled. In this range
the non-equilibrium part of the velocity distribution has a marked double
structure (see fig. 4.6b) including a Bennett hole and the collisional dip
whose amplitudes are comparable and widths differ greatly. The col-
lisional dip shape in this range is given by the formula (4.129). The
broadening of the collisional dip is slower than that of the Bennett hole
(«Vn, + 1) and at v, — vz» 2y, its width no longer depends on density. As
Nb increases further there comes a point when the non-equilibrium part
again appears as a single shape, but now because the widths of the two
dips become comparable, i.e. + v' ~ fcsyVl + n, (see fig. 4.6c). Subse-
quent changes of the non-equilibrium part are given by the formula
(4.130). As the density increases, the distribution (4.130) approaches a
Lorentzian distribution with a halfwidth of у 4- v'. The difference between
the halfwidth <5, of the distribution (4.130) and the halfwidth у 4- v' falls
off in inverse proportion to Nb.
A useful characteristic of the non-equilibrium part of the velocity
distribution is its halfwidth as a whole. A graphical representation of the
qualitative dependence of <5, on the density Nb is shown in fig. 4.7. Unlike
Fig. 4.7. Qualitative relationship between the halfwidth 6, of the non-equilibrium part of the velocity
distribution and the density.
§4.3]
The effect of collisions on Bennett holes
245
the model of relaxation constants, the <5, dependence on density has a
non-linear character. First <5, increases in comparison with y + v'; then the
plot rather slowly comes closer to the straight line y + v'. At large and
small densities the plot of <5, asymptotically approaches y + v' (see fig.
4.7). The intermediate region where the distance between the plot and the
straight line is at a maximum coincides with the condition y + v'~
ksf\/l + tij and the order of magnitude of this distance may be evaluated
using the formula (4.131)
[<3, - (y + v')]max ~ ksf ~, П‘ - ~ (y + v') .
V1 + fij 1 + tij
If tij > 1 the broadening due to the velocity change is of the order of the
Bennett hole width.
Let us consider the structure of a non-linear resonance in the work done
by the standing wave field. Proceeding from eqs (4.93), at A(v | vj = 0 we
take for simplicity G} = G2 (standing wave). In the recurrence relations
(4.65) the quantity p+-1 + p_-1 appears instead of p25-1 and the diagonal
Green’s functions are given by the formula (4.122). For simplicity let
Am„ = 0. Generalization to the case Am„ #= 0 presents no difficulties but
leads to more cumbersome expressions. Averaging over velocities will be
performed on the assumption that ks^l + л,«kv and y + v'«kv,
which implies the separation of non-equilibrium parts in the distribution
Paly) in the form of sharp structures. Simple calculations lead to the
following expression for the work done by the field:
v/л Г (Q — v"\2
P(Q) = 4hoj(Nm-Nn) |G|2—exp -I———) x
kv L \ kv / J
Л 2|G|2 y 1 Г (y + v')2
\ y + v',X,2y,. + v,.l (у + v')2 + (£2 — v")2
n,[Z,(0) + Z,(i2-v")]}).
„ч , (У + v')2 Г exp(-|i?|/s,Vl + л,)
} 2s,Vl + л, (у + v')2 + (X2 - v" - kij/2)2 1
(4.133)
(4.134)
On the plot of P(Q), apart from the dip with a Lorentzian shape due to
overlapping Bennett holes, there are also two resonances described by the
function Z,(i2 — v") and caused by overlapping collisional dips in p,,(v).
The function Z,(£2— v") as can be seen by comparing formulae (4.134)
and (4.126) coincides with the function Z,(£2 — v" — kv) if in the expres-
sion (4.126) a substitution k—>k/2 is performed for Z,(x). Consequently,
246
Bennett structure for systems with large Doppler broadening
[Ch. 4
with the exception of this substitution, each term of the sum over j = m,n
in the expression for the field work (4.133) selectively dependent on £2
repeats the non-equilibrium part of the velocity distribution at the
appropriate level.
Thus all the results referring to non-equilibrium parts of the velocity
distribution can be easily extended to the non-linear resonance in the plot
of P(Q) which is a superposition of the Lorentzian contour [1 + (C -
v")2/(y + V')2]-1 and collisional dips Zm(£2 — v") and Zn(Q — v"). Within
the range of small densities of perturbers the amplitudes of the collisional
dips are negligibly small. Within the range of average densities there
are three spectral components with halfwidths y + v', ksmy/l + nm and
ksny/l + nn. At large densities all three components have similar widths
and the shape of the total non-linear resonance is determined by the
expression
y + v'________k2 + nm) + sX(l + nn)
(y + v')2 + (C - v")2 e 2 (2 + nm + л„)[у + V' - i(X2 - v")]3 ’
(4.135)
its halfwidth being equal to
, , j ,2 + Пт) + ^л^л(1 + «») 1 zj 1
у + v + jk----------------------------------. (4.136)
2 + nm + n„ y + v
Let us sum up the results of our analysis. In the absence of the “phase
memory”, collisions do not affect the shape of the Bennett hole and the
non-linear resonance corresponding to it. The Bennett hole, as in the
model of relaxation constants, has a Lorentzian shape and its halfwidth
y + v' is linearly dependent on the density of perturbing particles. The
velocity change during collisions manifests itself in an additional non-
equilibrium structure which can be interpreted as a result of migration in
velocity space.
Now let us abandon the above-taken restriction A(y | Vj) = 0. Within the
scope of modulation notions this implies that during each collision there is
no complete phase disturbance of the atomic oscillation. Under such
conditions it is possible for frequency modulation of an atomic oscillator
due to its velocity change and the Doppler effect to manifest itself. If the
mean free time is small enough, the line broadening due to frequency
modulation is completely or partially compensated and, therefore, the
width of the profile varies within the range from y + v' to у + v' — v' = Г
on going respectively from small values of Nb to large values (see eqs
(3.43) and (3.44)). Proceeding from these general considerations, it would
§4.3]
The effect of collisions on Bennett holes
247
be expected that the transformation of the velocity distribution and the
Lamb dip that take place with the increase of Nb will qualitatively follow
the same laws as stated above for the case v = 0. However, the quantity
Y + v' = r+v' entering in the formula (4.125) and subsequently should
be represented as varying from Г + v' to Г. This circumstance complicates
the description of the phenomena but does not affect it qualitatively (see
fig. 4.6).
Consider the case of a standing wave, i.e. in eqs (4.93) we put G} = G2,
which formally enables us to use the recurrence relations (4.65), substitut-
ing into them
p*-1^) = рГ *(v) + P-~l(v), (4.137)
F(y | v') = F+(y | v') + F(v | v'), F_(v \v') = F+(-v \-v'),
where F+(v|v') is given by the expression (4.136). Making use of the
property (4.83) for Green’s functions and neglecting the spontaneous
transition m—>n(Am„=0), from eqs (4.65) we obtain the velocity
distribution of the populations:
p„(v) = W(v){ty T 2 |G|2 (Nm - Nn) x
j* dr7 ~kv~ krf) + B(Q + kv- b/)]J , (4.138)
B(Q) = Ref $(r)eiaTdr,
Л)
where the correlation function ф(т) is defined by the formula (4.117).
Let us study the relation between the velocity distribution (4.138) and
the field work expression which is obtained by means of eq. (4.67)
allowing for eq. (4.118):
P(Q) = 2ha)(Nn - Nm) |G|2 [B(Q - kv) + B(Q + Jlv)]W(v) dv -
2 IG|2 S f ЗД)[В(Й - kv) + B(Q + Jlv)]W(v) x
j=m,n *
[B(Q — kv — ki]) + B(Q + kv — kif)] dr/ dv j . (4.139)
Since the widths of the functions B(Q — kv) and F^Tf) are significantly
less than kv and v respectively, the Maxwellian factor can be taken
outside the integral sign with the value v2 = (Q/k)2. Subsequent integra-
248
Bennett structure for systems with large Doppler broadening
[Ch. 4
tion with respect to v is easy:
P(Q) = 4ha> |G|2 (N„ - Nm>7,(i2)[l - 2 |G|2 72(C)];
lx(Q) = (Vjc kv)~' exp[—(Q/kv)2];
I2(Q) = rjB.CO) + B2(^)| +
S fdrj^(rj) + , (4.140)
j=m,n * ' Z / _
where ^,(rj) is the regular part of the Green’s function. The functions
Bi(A:r//2), 62(Q) are given by the relations
B^krf) = Re I
Jo
2
e'^dr,
B2(Q) = Re I </>2l - leiftr dr.
Jo '2/
(4.141)
Making use of the formulae (4.113) and (4.117) it is easy to see that the
substitution k/2—>k is equivalent to
4>(r); B(Q - kif). (4.142)
Consequently, the Q dependence of an individual term of the sum over j
in 72(£2) with the exception of the substitution k/2—>k repeats the Q — kv
dependence of one of the terms of the non-equilibrium part of p/?(v) in
eq. (4.138). The analysis of the non-equilibrium part of the velocity
distribution can thus be combined with that of the non-linear resonance in
the plot of 7>(i2).
Comparison of the formulae (4.133), (4.134) and (4.140), (4.141) shows
that the function B2(Q) plays the role of the Lorentzian in eq. (4.133) and
its difference from the latter is due to a non-zero in-frequency v (see eq.
(4.117)). Thus the structure of the expressions (4.140) confirms the
aforegoing general statements about the role of phase memory.
The most interesting consequences of phase memory are connected with
the structural change of the Bennett hole (the function B(Q — kv) or
B2(Q)). From eqs (4.117) and (4.141) it follows that
B2(Q) = Re ( dr exp| —[y + v' — i(£2 — v")]r + f A(r'/2) dr'
Jo Jo
A(0) = | A(v — Vi) dv = v = v' + iv".
(4.143)
§4.3]
The effect of collisions on Bennett holes
249
If l/(y + v') significantly exceeds the characteristic width 1/ks of the
function A(r/2) then, as can readily be shown, the integral over r' of
A(t'/2) in eq. (4.143) is approximately equal to v/ks and may be
dropped. Therefore
B2(Q) = Re-----—у-—-----—, y + v' «ks. (4.144)
v 7 Y + vi(Q- v") 1 v 7
This approximation corresponds to large velocity changes (in comparison
with (y + v')lk) and the role of the frequency modulation is the greatest.
This situation is typical of low densities of perturbing particles.
In the opposite limiting case of high densities the small values of r are
important in eq. (4.143) and an expansion of Л(т'/2) similar to that
performed in formula (4.124) for a diagonal kernel can be employed:
A(r/2) = v[l — aksr/l — b(kst/2)2],
[ A(r'/2) dr' = vr[l — jaksr — -fabfksr)2], (4.145)
Jo
where s is the halfwidth of the kernel A(y — vt) and the numerical factors
a and b depend on its shape. Substituting expression (4.145) into relation
(4.143) we have
B2(i2) = Rel drexp{—[Г —i(£2 —Л)]т} x
Jo
exp[—va/csr2/4 — v6(£s)2r3/12].
Retaining in eq. (4.146) the leading term only, we obtain
B2(Q) = Re 1——, T + i4 = y + v- v,
Г — i(SJ — A)
and B2(Q) as in the approximation (4.144) is described by the Lorentzian
contour. However, the halfwidth and position of the maximum in eq.
(4.147) have changed in comparison with eq. (4.144). At v'>0 this
change manifests itself as a decrease in the broadening rate of the
non-linear resonance (Dicke effect).
The scope of the approximation (4.147) may be found by evaluating the
quadratic and cubic terms in the expression (4.146). If A(v - uj at
|u —Vi|—>0° is decreasing not faster than (v — vt)~2, then «#=0 and the
applicability criterion of relation (4.147) has the form
v'ksir2 = nksir«l, n = v'/r. (4.148)
(4.146)
250
Bennett structure for systems with large Doppler broadening
[Ch. 4
If the “wings” of the kernel are diminishing more rapidly than (y — uj 2
then a = 0 and in place of the condition (4.148) we find
v' /Лз\2
7\7/
«1.
(4.149)
Therefore, for the limiting cases (4.144) and (4.147), different parameters
are important, the difference being at its greatest for n »1. It has already
been mentioned that the quantity n may be interpreted by analogy with n,
as the number of collisions bringing about a deviation of the atomic
oscillator frequency and occurring over the time l/Г of the relaxation of
the dipole moment.
At a #=0 the cubic term in eq. (4.146) can be neglected and the integral
with respect to r is expressed in terms of a tabulated function:
/ л
(4.150)
/ 2 Г5
w(£) = еП 1 —-г= j e-*2 dt
\ VJt Jo
Г - i(Q - A)
\Javks
At a = 0 the integral in eq. (4.146) cannot be expressed in terms of known
functions.
It is easy to obtain the first corrections for the velocity change by
expanding in eq. (4.146) exponents with r2 and r3 and restricting
ourselves to the first terms of the expansion:
1 a vks b v(ks)2 ]
r-i(Q-A) ~ 2 [Г —i(£2 —A)]3 “ 2 [Г - i(i2 - A)]4 J '
(4.151)
The halfwidth dB and the position of the maximum of this function are as
follows:
B2{O) = Re
дв = Г + lanks + lbn(ks)2/r,
£2max = A + — nks + b ^-n(ks)2/r.
v v
(4.152)
(4.153)
If a#=0 then terms with b in the relations (4.152) and (4.153) must be
dropped and then the correction to the halfwidth and maximum shift is
proportional to the total deviation of frequency nks taking place after n
collisions. If, on the contrary, a=0, i.e. if the “wings” of the kernel
A(y — Vj) are decreasing more rapidly than l/(y — uj2 (see the discussion
§4.3]
The effect of collisions on Bennett holes
251
of the formula (4.124)), then the combination ks^Jn enters 6B and £2max,
specifying the resulting deviation of the frequency and obeying the
diffusion law («Vn) or the law of large numbers.
In the expansion (4.145) of the kernel A(r) in a power series the
frequency v was factored and coefficients a and b were assumed real,
which suggests that the real and imaginary parts of the kernel have the
same shapes. Actually the r dependences of ReA(t) and ImA(t) are,
generally speaking, different. As to the structure of the imaginary part of
non-diagonal kernels, at present this problem is not understood clearly
enough. Therefore we shall not give a more detailed treatment of the way
in which a velocity change affects the shift of non-linear resonances.
If the scattering amplitudes in states m and n are close to each other,
then the kernel A(v | v,) according to general relations of section 2.5 is
“almost real”, i.e. |v"|«|v'|, and frequency modulation manifests itself
mainly as the width of the non-linear resonance. The qualitative be-
haviour of the plotted dependence of the halfwidth 8B of the function
B2(Q) on the density of perturbing particles is shown in fig. 4.8. The
general peculiarity of the curves is their non-linear behaviour. At low
densities, дв = у + v'; at high densities, дв = у + v' — v'. If v'>0 the
Fig. 4.8. The qualitative behaviour of the Bennett hole halfwidth as of the function of perturbing
particles’ density: (1) v'>0, a = 0; (2) v’>0, a^O; (3) v'<0, a=0; (4) v'<0, e^O; (5) 6В = Г;
(6) дв = Г + Ranks', (7) дв = Г— |п| Хв; (8,9) дв = у + v'.
252
Bennett structure for systems with large Doppler broadening
[Ch. 4
slope of the curve decreases as the density increases; if v'<0, then, on
the contrary, the slope of the curve is greater at high densities than at
the low densities. According as «#=0 or a=(), i.e. depending on the
behaviour of A(y — Vi) at |v — vj—>«, either the straight line 8B =
Г + 3anks/4 or <5B = Г is the asymptote. In the latter case the asymptote
has a common point with the plot in the range of low densities.
Consequently, the type of the asymptote of <5B as a function of the density
Nh enables us to predetermine the type of the kernel “wings”.
The total non-linear resonance is a combination of the Bennett hole and
collisional dip and their evolution when the density Nb of perturbing
particles increases is qualitatively illustrated fairly well in fig. 4.6. The
relations (4.140) show that in the range of small values of Nb where
dB«kSj\/l + nf the collisional dip reveals itself as a distinct structure
described by a formula of the type (4.129) (see fig. 4.6b). Both dips have
almost equal widths in the range of Nb where ksf\/l + ~ Г and they are
to be treated as a whole (see fig. 4.6c). The latter is clearly seen from the
asymptotic expression for the halfwidth of the function I2(Q) at high
densitities, i.e. for non-linear resonance as a whole. The first corrections
to the halfwidth Г due to effects of velocity change are additive; therefore,
from expressions (4.131) and (4.152) we obtain
d = Г+ ^[2n(ks)2 + nm(ksm)2 + n„(fcs„)2]. (4.154)
О 1
Thus the contribution of frequency modulation to the broadening of
non-linear resonances is the same as the contribution of the effects due to
changes in the velocity distribution*.
It was noted in section 2.5 that in the general case the differential
cross-section of scattering is a superposition of a narrow diffraction part,
classical scattering through small angles and practically isotropic classical
scattering (see fig. 2.5). Accordingly, the real kernels of collision integrals
are also “multicomponent”. At least two components are to be distin-
guished, one of which specifies scattering through small angles, the other
giving scattering through large angles. Scattering through small angles, as
shown in section 2.5, may be described by a difference kernel model; for
describing scattering through large angles the model of strong collisions
proves to be a good approximation.
Thus the following approximation (the model of non-degenerate states)
* For a more detailed theoretical analysis of the shape of non-linear resonances see refs [23,30].
Experimental results as well as a bibliography can be found in refs [31,32].
§4.3]
The effect of collisions on Bennett holes
253
can be employed for real kernels of collision integrals:
A„„(v | Vi) = A„„.(v - v,) + v^>,W(v),
Ana (y - vj dv = v$,
v , = v<4 + v<2>,
* nn * nn ’ rnn >
(4.155)
where v£>. and are the appropriate “partial” in-frequencies. A similar
presentation is also possible for one-dimension kernels A„„ (v | Vj).
Earlier non-linear phenomena were analysed within the scope of one of
the models of a kernel. The results obtained are generalized to the case of
a two-component kernel (4.155). The calculations are given in problem
(21), where the consequences of such generalization are also discussed.
The velocity distribution of populations and the plot of the work
performed by the field contain a Bennett hole, collisional dip and a
homogeneous saturation band, i.e. they have all the characteristic features
of the model of strong collisions as well as the model of the difference
kernel. The areas of these structural elements at the level / are
proportional to the corresponding effective lifetimes:
1 vO) v<2)T.
Tv + = 1/1], Г] = 2y, + Vy - V, (4.156)
The quantity riy as before denotes the lifetime of state j with a certain
velocity. The time ity specifies the duration of a state in which the atom
has already experienced a collision but still retains a non-equilibrium
velocity distribution. Collisions that significantly change the velocity
reduce the time Finally, r^2) is the time over which an atom at level j
has an equilibrium velocity distribution. The time r£2) complements rv and
t^ to give the total lifetime at the level. Characteristically, rtf is not
influenced by the presence or absence of collisions with small velocity
changes; this is quite clear because is measured from the instant of the
first collision with a strong velocity change, after which velocity distribu-
tion remains at equilibrium, no matter what collisions, strong or weak,
occur afterwards.
Important parameters specifying non-linear resonances are the collision
numbers n, and n. In the model (4.155) the quantities
n(i) = _22----n(i) = y--------------
’ Г, + Ц2>’ r + Rev(2>
(4.157)
act as such parameters.
254
Bennett structure for systems with large Doppler broadening
[Ch. 4
Scattering through large angles (|v — v,| ~ v) affects л}0 and n}2)
similarly to non-elastic processes, i.e. strong collisions restrict the broad-
ening of the collisional dip and its weight in non-linear resonances, and
reduce the difference between the halfwidths of the Bennett hole at small
and large frequencies. In the range of high densities the halfwidth of the
Bennett hole Г + Re v(2) is increased not only by quenching and phase
modulation but also as a result of frequency modulation caused by strong
collisions.
Until now, effects due to velocity-changing collisions have been
analysed within the scope of the model of non-degenerate states because
of the relative simplicity of the finite expressions and their interpretation.
When characteristic effects in a simple model have been revealed, the
results obtained may be generalized to systems with degenerate levels and
the complexity of the expressions will not interfere with the understanding
of the physical aspect.
Consider the interaction of atoms with a standing wave so as to analyse
simultaneously both the velocity distribution and the non-linear resonance
in the work performed by the field. In the representation of polarization
moments we must work from the equations (compare with eqs (4.26))
2ympmm = ^=== + " i(G-P™ - G“„p„m),
Q„W(v) , o
^YnPnn xJ'yj
i(G™pnm - G:mpm„),
[y - i(i2 T к • v)]p*„ = Sm„ - i(G^,pmm - GX„p„„);
Pmn = Pm„ + P™; ptn{Kq) = (-1У”“Л+<?Рт:(К “ q\,
where the matrices GJ are given by the relation (4.29).
Unlike eqs (4.52) which hold true for the model of relaxation constants,
the collisional integrals S,y in eqs (4.158) also describe, in particular, the
processes accompanied by velocity changes.
Let us assume the frequencies and kernels of the collisional integrals to
be diagonal in Kq and independent of q, i.e.
(4.158)
Stj(Kq, v ) = - Vqp^Kq, v) + Aif(KV | KV^p^Kq, v,) dv,. (4.159)
There may be reasons for employing such a model for isotropic collisions
(see formulae (2.176) and (2.189) and the preceding discussion) but,
§4.3]
The effect of collisions on Bennett holes
255
generally speaking, representation of the collision integral in the form
(4.159) is to be considered a postulate.
It is convenient to solve the equations (4.158) with the help of the
Green’s functions F^v | v') and F*„(v | v') which, because the collision
integral is diagonal, have the following structures:
Fj^qv | Kxqxv') = dKK£qqiFjK(y | v');
F*Jjcqv | Kxqxv') = dKKidqqiFi(y | v');
F^(v|v') = ^(v|v')- (4-160)
Also, the functions FjK(y | v') and F*(y | v') satisfy the equations
(2y, + v,}FjK{v | v') = j AjK(v | vx)FjK(yx | v') dvj + <5(v - v'),
[y+ v)]F*(v | v')
= J AK(v | Vi)Fj(vi | v') dVi + <5(v - v'); (4.161)
AiK(v I «О = I kv2), AK(y | Vi) = Атп(ки | KVj).
These equations differ from eqs (4.63) and (4.64) of the model of
non-degenerate states in that in the former equations the kernels and,
consequently, the Green’s functions are parametrically dependent on к.
Mathematically this circumstance presents no additional difficulties and
the results of solving eqs (4.63) and (4.64) may be used.
Similarly to the iteration formulae (4.65), from eqs (4.159) we obtain
Pm»_1(v) = -i| Fmn(v | v')[G™np^2(v') - GX„p™-2(v')] dv',
P™(v) = -ij^(v | v')[G”mp^-1(v') - G^p^rXv')] dv';
P™(v) = -ij FM(v | v')[GX„p™ !(v') - <%nP™ *(»')]dv' +
j Fnn(v | v')Am„p^„(v') dv';
F™,(v | v') = F+„(v | v') + F“„(v | v'); (4.162)
p°(Kq, v ) = dKOdqO . (4.163)
Values of V,- are given by the formula (4.31). Giving no more concrete
definition of the Green’s function than the formula (4.160) we write the
256
Bennett structure for systems with large Doppler broadening
[Ch. 4
components of the first terms of the iteration series allowing for the
explicit form of matrices G* given by the formula (4.30). For simplicity,
the spontaneous transition m —»n will not be taken into account (Лтл =
0). Proceeding from the zeroth approximation (4.163) we have in the first
order
plmn{Kq,v) = -iGqdKlNmn^ F(v | и')W(u') du
(4.164)
Nm„ = Nmlgm - NJgn; g) = 2^ + 1; Gq= %qdmn/2ti.
Since the dipole interaction is considered, only the element p'mn(Kq) with
к = 1 is excited. Therefore the index к = 1 in the non-diagonal Green’s
function may be omitted.
The expression (4.164) is substituted into the second and the third
relations of eqs (4.162) in order to find the field changes in the
polarization moments of the levels. Allowing also for the zeroth ap-
proximation we have
N.W(v)
Pu(*q, и) = <5кАоT2NmJ(Kq)ajK x
FjK{v | v,) Re[F+(ui 1u2) + F~{vx | Vz)]W(iij) duj du2, (4.165)
1
к 1
J’
a™ = (-1)1+к+/т+лз[ 1
a
Here I(icq) is the field polarization tensor defined by the formula (4.34).
The field component of the expression (4.165) as in the model of
relaxation constants (see eq. (4.32)) is proportional to I(Kq) which is
responsible for the dependence of its amplitude on the field polarization.
The model of relaxation constants corresponds to the Green’s functions
W) =
<5(v — v')
F*(v | v') =
<5(v — v')
у + v — Vj — i(£2 T k • v)
(4.166)
The velocity distribution in the expression (4.165) is к dependent, i.e.
§4.3]
The effect of collisions on Bennett holes
257
each polarization moment of the rank к in the general case has its own
form of the non-equilibrium part of the distribution. Under a fixed value
of к, however, this part is expressed in terms of the Green’s function in
the same way as in the model of non-degenerate states (compare with
relations (4.67)).
Now we calculate the work done by the field in the approximation of
first non-linear corrections. The necessary quantity p3mn(lq) in the third
approximation with respect to the field amplitude is obtained by substitut-
ing the expression (4.165) into the relation (4.162). Simple manipulations
yield
P(Q) = 2fttoNnm{|G|2| Re[F+(v | v,) + F~(v | v1)]W(v1) dv dvj -
2 2 I;(k<7)|2 a]K[Re[F+(v | v,) + F~(v | vO^fa 1v2) x
j=m,n J
Kq
Re[F+(vj | v,) + F~(v2 I v3)]W(v3) dv dvj dv2 dv3 j, (4.167)
|G|2 = £|Ga|2.
0
Compare the relation obtained with the corresponding expression (4.67)
of the model of non-degenerate states by assuming in the formulae (4.67)
that Amn = 0 and extending them for the case of a standing wave by a rule
of the form (4.137). The linear part of the field work except for the
evident difference of the factor Nnm = Nnl('2Jn + 1) - + 1) is the
same in expressions (4.67) and (4.167). The non-linear part of the
expression (4.167), in addition to the sum over j = m,n which appears in
eq. (4.67) as well, involves a sum over к = 0,1,2 so that P(Q) consists of
six terms (there are two terms in the model of non-degenerate states).
Apart from the coefficient a]K, determined by the angular momenta of the
states, and the factor \I(Kq)|2, representing polarization effects, the
dependence on к is non-trivially confined to the Green’s functions
| v2). In the simplest model (4.166) the functions FjK are inversely
proportional to relaxation constants. For the general case at the same time
the FjK(vi | v2) with different к are various functions of velocities, i.e. in
velocity distribution and in non-linear resonances the terms with к =
0,1,2 describe non-equilibrium structures with various shapes.
On the contrary, the expressions (4.165) and (4.167) and the cor-
responding expressions (4.67) of the model of non-degenerate states have
something in common which is methodologically very important. In both
258
Bennett structure for systems with large Doppler broadening
[Ch. 4
of the models the velocity distribution and field work are similarly
expressed in terms of Green’s functions which, in their turn, to an
accuracy of the parametric dependence on к, are given by similar
equations (compare eqs (4.161) with eqs (4.63) and (4.64)). This fact,
connected with the assumed model (4.159), enables one to extend the
results obtained in the model of non-degenerate states to the systems with
degenerate levels.
In accordance with the structure of the differential cross-section we
write the collision integral kernel as in formula (4.155):
AiK(y | uj = AjK(v - vO + v;(2)W(u),
AK(v | vj = AK(y - Vi) + vKW(y), (4.168)
where scattering with a small velocity change is described by the
difference kernel AK(y — vj and considerable velocity changes are given
by a model of strong collisions. If
v^=<5Kov}2); vL2)=<5kOv(2) (4.169)
then the term with W(u) represents the model of strong collisions in both
velocities and magnetic sublevels.
Equations (4.161) with kernels (4.168) are solved in quite a similar way
to problem (21) for the model of non-degenerate states. It was shown
above that under the conditions of narrow non-linear resonances the
in-term of the non-diagonal collisional integral corresponding to strong
collisions can be neglected as a result of significant frequency modulation.
In this approximation, proceeding from the results of problem (21) we
obtain (one-dimensional treatment)
1 k
^(vlvd-T^-vJ + j^-x
f-----л«(т) e-K<»-.,).dT+ (4.170)
J 2yy +vy-A/K(T)
Aa(t) = f AjK(v - и1)е*г(и-’”) du;
J— 00
1 1
T“ = 2^’ lS>W = 2r, + r;i;, + rg’
1 v<2)
= j- . y(2) y-’ = 2y, + v, - vjK,
LjK ' V JK ±fK
v)’> = | AjK(v - Ui) du = A,K(0); vjK = + v£>.
§4.3]
The effect of collisions on Bennett holes
259
The expression for the non-diagonal Green’s functions F± remains the
same as in problem (21), although naturally it must be noted that the
frequencies and kernels characterize relaxation of the element pmn(Kq)
with the value к = 1.
From the results of problem (21) and formulae (4.165) and (4.167)
there follow the expressions for the velocity distribution of polarization
moments and for the work done by the field:
NW(v)
Pa(Kq, v) = + 2NnmW(v)I(Kq)a/K x
Jr00
dr ф(т)
0
I I TV-^/k(T)
1 + tvA/k(t).
vlt
+ ei(o+*v)r] + 2t^>(k)
k и
P(Q) = tnhtoN'MQ) |G|2 - 2 |Z(k9)|2/^(Q)
L
A(Q) = (VS kv)-le~iaik^,
4,(0) = 2 £ «J Re Г dr Г1 + ] x
j—m,n Jo L 1 Tiy24yK^T/2j-
/ т\ / т\ П 1
+ф2(^ eiOT +2т^(к)^е-<то)2
\2/ \2/ J kv J
(4.171)
(4.172)
The correlation function ф(т) is as before defined by the relation (4.117).
The field part of the velocity distribution (4.171) as a function of kv and
every term Лк(^) in the expression (4.172) for P(Q) as a function of Q
are similar to the corresponding characteristics in the model of non-
degenerate states. In particular, the areas of the Bennett hole, the
collisional dip and the band of homogeneous saturation in the velocity
distribution Pai^q, u) are related as
т„:т§>(х):Ш (4.173)
Level degeneration and disorienting collisions bring about the increased
number of non-linear resonance characteristics. Non-linear resonance in
the field work involves not three components as in the model of
non-degenerate states but seven (for the general case with different
widths). The weights of the components in the general non-linear
resonance vary according to the field polarization.
260
Bennett structure for systems with large Doppler broadening
[Ch. 4
References
[1] W.R. Bennett, Jr., Appl. Opt. Suppl. 1 (1962) 24.
[2] M.P. Chaika, Interferentziya Vyrozhdennykh Sostoyaniy (State University Press, Leningrad,
1975).
[3] S.G. Rautian, Tr. Fiz. Inst., Akad. Nauk SSSR 43 (1968) 3.
[4] T.A. Germogenova and S.G. Rautian, Zh. Eksp. Teor. Fiz. 46 (1964) 745 [Sov. Phys. JETP 19
(1964) 507].
[5] N.G. Basov, E.M. Belenov, M.V. Danileiko and V.V. Nikitin, Zh. Eksp. Teor. Fiz. 60 (1969)
117 [Sov. Phys. JETP 33 (1969) 35].
[6] V.S. Letokhov and V.P. Chebotajev, Nonlinear Laser Spectroscopy (Springer, Berlin, 1977).
[7] S. Stenholm and W.E. Lamb, Jr., Phys. Rev. 181 (1969) 618.
[8] D.J. Feldman and M.S. Feld, Phys. Rev. A 1 (1970) 1375.
[9] W.E. Lamb, Jr., Phys. Rev. 134 (1964) 1429.
[10] P.H. Lee and M.L. Skolnick, Appl. Phys. Lett. 10 (1967) 303.
[11] V.N. Lisitzin and V.P. Chebotajev, Zh. Eksp. Teor. Fiz. 54 (1968) 419 [Sov. Phys. JETP 27
(1968) 227].
[12] N.G. Basov and V.S. Letokhov, Electron. Technol. 2 (1969) 15.
[13] G.G. Petrash and S.G. Rautian, Opt. Spektrosk. 18 (1965) 336.
[14] T.V. Bychkova, V.G. Kirpilenko, S.G. Rautian and A.S. Khaikin, Opt. Spectrosk. 22 (1967)
678.
[15] J.H. Parks and A. Javan, Phys. Rev. 139 (1965) 1351.
[16] A.S. Khaikin, Zh. Eksp. Teor. Fiz. 54 (1968) 52 [Sov. Phys. JETP 27 (1968) 28].
[17] W.R. Bennett, Jr., Phys. Rev. 126 (1962) 580.
[18] S.G. Rautian and I.I. Sobel’man, Usp. Fiz. Nauk 90 (1966) 209 [Sov. Phys. Usp. 9 (1967) 701].
[19] S.G. Rautian, Zh. Eksp. Teor. Fiz. 51 (1966) 1176 [Sov. Phys. JETP 24 (1967) 788].
[20] A.P. Kazantzev and G.I. Surdutovich, Nelineinaja Optika (Novosibirsk, Nauka, 1966), p. 118.
[21] A.P. Kol’chenko and S.G. Rautian, Zh. Eksp. Teor. Fiz. 54 (1968) 959 [Sov. Phys. JETP 27
(1968) 511].
[22] B.L. Gyorffy, M. Borenstein and W.E. Lamb, Jr., Phys. Rev. 169 (1968) 340.
[23] T.A. Andreeva, V.A. Aleksejev and I.I. Sobel’man, Zh. Eksp. Teor. Fiz. 64 (1973) 813 [Sov.
Phys. JETP 37 (1973) 413].
[24] A.P. Kol’chenko, A.A. Pukhov, S.G. Rautian and A.M. Shalagin, Zh. Eksp. Teor. Fiz. 63
(1972) 1173 [Sov. Phys. JETP 36 (1973) 619].
[25] P.R. Berman, Appl. Phys. 6 (1975) 283.
[26] I.M. Beterov and R.I. Sokolovski, Usp. Fiz. Nauk 110 (1973) 169 [Sov. Phys. Usp. 16 (1973)
339].
[27] R. Vetter and P.R. Berman, Comments At. Mol. Phys. 10 (1981) 69.
[28] S.G. Rautian, in: Proc. 6th Int. Conf, on Atomic Physics (Zinatne, Riga; Plenum, New York,
London, 1979), p. 493.
[29] P.R. Berman, in: Tendences Actuelles en Physique Atomique, eds G. Grynberg and R. Stora
(Elsevier, Amsterdam, 1984), p. 454.
[30] V.P. Kochanov, S.G. Rautian and A.M. Shalagin, Zh. Eksp. Teor. Fiz. 72 (1977) 1358 [Sov.
Phys. JETP 45 (1977) 714].
[31] T. Hansch and P. Toschek, Z. Phys. 236 (1970) 213.
[32] L.S. Vasilenko, V.P. Kochanov and V.P. Chebotaev, Opt. Commun. 20 (1977) 409.
5
Probe field spectroscopy under large Doppler
broadening
5.1. Non-linear resonances in three-level systems
In the previous chapter only one of the main non-linear effects in systems
with large Doppler broadening was considered, i.e. the field-induced
changes of the velocity distribution of atoms. Non-linear resonances due
to this effect have quite varied profiles whose shapes depend on intensity
and polarization of the external field, type of relaxation process and other
circumstances.
Field-induced level splitting and non-linear interference effect (NIEF)
are also of great importance for the spectroscopy of the probe field. Both
of those effects appear mostly as the shapes of the emission and
absorption spectra of the probe field for the transitions involving levels
perturbed by the external field. Therefore in this area one should expect a
still greater variety of non-linear resonances.
In this section we study how atomic motion and collisions affect the
absorption (amplification) of the probe field resonant with one of the
atomic transitions adjacent to the transition m-n which experiences the
action of a strong electromagnetic field.
For a definite example, asshme the probe field to be resonant with the
transition m-l with > 0 (a scheme similar to combination scattering;
see fig. 3.10). The results obtained for this scheme can be extended to any
of the diagrams of figs 3.9 and 3.18.
For the model of non-degenerate states the matrix elements of the
interaction have the following forms
Vmn = ~G exp[—i(I2t - к • r)], Vm, = exp[-i(i2Mt - • r)],
О (О С&тп) (5* f)
where the quantities with the suffix /z characterize the probe field.
The formula (3.217) derived in section 3.3 for the above scheme and
describing the work done by the field in the system where the atom is at
rest may be extended to the gas of moving atoms. To this end the
frequencies Q and Ц, entering in eq. (3.217) must be substituted for
261
262
Probe field spectroscopy under large Doppler broadening
[Ch. 5
Q' = Q — к • v and — кц • v and the entire expression (3.217) must
be averaged over velocities. The substitutions Й—»Q' and
reflect the fact that in the atomic coordinate system the frequencies of the
strong and probe field due to the Doppler effect are equal to respectively
(o — k'V and a)tl — ktl • v.
Thus the work performed by the probe field within a scheme of the
combination scattering type is given by the expression
Рц = 2кш„\С„\2х
— / [C ~ ~ Й' — A„/)](p/z — ftmm) — iGp^nn \
e\[r„, - i(a; - Qf - - i(o; - д,,)] + |G|2/’
Q' = Q — k‘V, Й', = Йм — kp'V, pu = NiW(y). (5.2)
Here Nt is the population of the level I unperturbed by a strong field, and
pmm and рХл are solutions of the equations for the density matrix in the
absence of a probe field which, according to eqs (3.132), (3.134), (4.9)
and (4.11), are given by
p™. = wk - 2|G|2W. - . Tgl;
iGpX. = - |G|2(K - N.)W(y) ;
* s ’ ^mn)
.--- 2 IGI2 / 1 1 А^Д
Ц = ГтпХ/1 + к, к = — + (5.3)
*тп '-*т
The expression (5.2) holds, to be sure, only when atoms move along a
straight path, i.e. in the absence of velocity-changing collisions (model of
relaxation constants). The role of elastic scattering will be studied below.
Averaging over velocities in eq. (5.2) may be performed in the general
form, but the results obtained are rather awkward. In order to reveal the
most interesting effects it is enough to discuss the limiting cases.
First of all, consider the approximation corresponding to a weak enough
field G. Retaining in eq. (5.2) only the first term of the expansion in
power series of |G|2 we obtain (compare with eq. (3.217a))
PM = 2ft<WJGJ2Re^
W(v)
fm/ — — Am/)
pV,-Nm-
|G|2
(2V — N )^=----------------------
Гт Cm + (Й' - Amn)2
§5.1]
Non-linear resonances in three-level systems
263
Nn~Nm_________________1_________+
Pnn + i(£2 — Лтп) r„i — i(£2p — £2 — Anl)
---------------1---------]1\. (5.4)
r_ - i(a; - л_) r„ - i(a; - a’ - d.,)JJ/ ' ’
The non-linear part of the expression (5.4) proportional to |G|2 consists of
three characteristic terms. The first is due to the population change of the
level m under the action of the field (saturation effect). The second term
corresponds to the term iGp*„ in the formula (5.2), i.e. it describes the
NIEF. The last component is the first term of the expansion allowing for
|G|2 in the denominator of expression (5.2), i.e. it is connected with the
field-induced splitting of the level m. Thus, in the approximation of the
first non-linear corrections, three basic non-linear effects can be distin-
guished as additive terms. In section 3.3 it was shown that these terms
should be unambiguously interpreted as the contribution of stepwise
transition (saturation effect), combination scattering (NIEF) and reverse
combination scattering (level splitting).
In this chapter mostly systems with large Doppler broadenings are
analysed, i.e. the condition
P^kv, i,j = l,m,n (5.5)
is assumed fulfilled. The result of the averaging of expression (5.4) as well
as the exact expression (5.2) essentially depends on the mutual orientation
of the wavevectors к and itM of the strong and probe waves. Such
anisotropy is a mathematical consequence of the fact that in Рц there
appears a quantity £2^ — £2' = Ц, — £2 — (kfl — к) • v where the value of the
Doppler term (kfl — it) • v essentially depends on the direction of itM with
respect to it and on the value of the ratio |it|/|itM|. The scattering
anisotropy, therefore, can be said to arise because the Doppler shifts
kfl • v and it • v of the frequencies to,, and <o combine in different ways
depending on the relative direction of wave propagation. For counter-
travelling waves the Doppler shifts are added, for parallel waves they
are subtracted from each other, and when the values |JtM | and |it| are close
they may completely or almost completely cancel one another. The angle
of k,, with respect to it influences separate terms of the non-linear part of
P„ as well as their relationship.
First let us study the case when the probe wave is propagating opposite
to the strong wave (subsequently the notation кц Hit will be employed). If
|Й| e£kv and the condition (5.5) is fulfilled the Maxwellian exponent can
be factored outside the integral sign with a value v corresponding to the
264
Probe field spectroscopy under large Doppler broadening
[Ch. 5
maximum of sharp Lorentzian multipliers, i.e. when v2 = — A^y/k2.
The remaining expressions can easily be integrated using the residue
theorem.
Note that the parts of the integrand of eq. (5.4) which describe the
NIEF and level m splitting have their poles in one half-plane of the
complex variable к • v and consequently the integral of them over v is
zero. As a result, under kfl U& we have
P„ = 2ftco„ |GJ2^exp[ -У - Nm -
2 |r .Л Re_________________(Nn-Nm)/rm__________________1
1 1 к Гт1 + (kjk)!^ - i[O„ - Aml + (k„/k)(Q - Am„)]Г
(5-6)
The non-linear part of the work done by the probe field (5.6) is,
consequently, associated only with the velocity distribution of the popula-
tion. In the plot of Pp as a function of Ц, the non-linear part appears as a
sharp structure of Lorentzian shape with a halfwidth Гт, 4- кцГтп/к and is
positioned in the vicinity of Qft = Amt —k^Q — A^/k (fig. 5.1, solid
line).
Ibe nature of the non-linear resonance in the considered case is
analogous to that in a plot of the work done by the standing wave field
Fig. 5.1. Dependence of the work done by the probe field on the frequency 42M under and
§5.1]
Non-linear resonances in three-level systems
265
(see section 4.2), namely the strong field produces a Bennett hole in the
velocity distribution at the level m which is in the vicinity of к • v =
£2— Amn- The probe field effectively interacts with atoms whose velocities
satisfy the condition k„ • v = — Лт/. If the probe field frequency is such
that (Йм — Ami)lk^ = — (Й — Дтп)!к it interacts with atoms for which the
velocity distribution is changed by a strong field and a non-linear
resonance arises. Its width + к^Гт„1к consists of the width Гт, of the
transition m-l and a Doppler term к^Гтп/к due to the width Гт„!к of the
Bennett structure. These peculiarities are consistent with the interpreta-
tion of non-linear resonances as a direct spectral manifestation of the
Bennett structure in the velocity distribution of atoms.
The vanishing of terms responsible for NIEFs and level splitting may be
explained as follows. Under ТД.Л Doppler shifts for the transitions m-n
and m-l are opposite in sign. The expression (5.4) shows that averaging
over velocities in such a case has a similar effect to integration over
which in its turn makes those terms vanish (see section 3.3).
In the opposite case when the directions of the strong and probe waves
coincide (Jt,x TT&), averaging of the expression (5.4) over velocities gives a
completely different result. The Doppler shifts k^ • v and к • v of the field
frequencies and to are now of the same sign and in the difference
кц • v — к • v they cancel each other. Furthermore^ the NIEF term in
formula (5.4) as a function of velocity has poles in both the upper and the
lower half-planes of the complex variable and consequently does not
become zero when integrated over velocities.
If к„<к then in the case of kfl^k the term responsible for the
field-induced splitting of levels, once averaged over velocities, also differs
from zero (see problem (22)). When kfl> к this term still vanishes as a
result of averaging and the work done by the field is given by the relation
Р„ = 2к(оц | GJ2—? exp
Z^-^X2
\ k„v / .
x
N, - Nm - 2 |G|2(N„ - Nm) Re(—
rC 'I| l&e.
C2
-G ~
к
c2, Д = Pmi "I" ~ Гтп', = Йм2 + Д„1 + Дтп ДтГ,
к
Г2 ~ Г,' + Гпп’, Дп1 ^тп / (^2 Дтп)]
к к
С2 = [Гт, — Г„1 + Гтп + 1(Дп1 ~ &nt ~ 4™)] '.
(5-7)
266
Probe field spectroscopy under large Doppler broadening
[Ch. 5
The non-linear part of the work done by the probe field comprises two
profiles with different widths shifted relative to each other by a distance
д = — Am„ — A„i, i.e. as a result of collisional shifts. Since the
coefficients q and c2 are complex the profiles prove somewhat asymmetric
(to the degree that this difference <5 deviates from zero). If <5 = 0 both
profiles have a Lorentzian shape and are centred at the same frequency
(see fig. 5.1, dotted line). The width Гг of the first profile is the same as in
the case of countertravelling waves. The width of the second profile
Г2 = r„i + (£,, — kyr^Jk consists of the width Гп1 of the forbidden transition
n—l and a Doppler contribution from the Bennett structure with a width
Гтп/к but with the difference wavenumber кц — к, as it should be for a
two-quantum process.
The second profile c2/(T2 — ii3M2) in the expression (5.7) is determined
exclusively by NIEFs; a part of the first profile amplitude (T~l) is due to
the Bennett hole while the second part represented by the term c2 in the
amplitude is also due to NIEFs. When the conditions 6 = 0, Д = Yt + У,
characteristic of spontaneous relaxation are satisfied the interference term
completely suppresses the resonance due to the Bennett hole:
1 к —к
Ci = 0, c2 = ~p-, r2 = Yn + Yi+ \ (Ут+У«),
К
Йм1=Йм2=Йм-^Й. (5.8)
The non-linear resonance in this case turns out to be a single Lorentzian
shape with a halfwidth Г2. If the frequencies of the transitions m-n and
m-l are close enough together (kfl — к « к) then the resonance halfwidth
is equal to the halfwidth of the forbidden transition: Г2 = y„ + y,.
The amplitude of the first resonance c, differs from zero only because of
collisional phase modulation which disturbs the interference [1]. Thus it
may be convenient to single out the first resonance for studying some
peculiarities of collisions.
It follows from expressions (5.6)-(5.8) that is characterized by a
marked anisotropy. In the case of spontaneous relaxation the equality
Г2 — Г, = 2ym holds true, i.e. the width of the resonance for the case of
counterpropagating waves is always greater than when the propagation
occurs in the same direction. For the case of collisional broadening both
inequalities Г2 > Ц and Г2 < Ц can be realized.
Let the wavevectors be not completely collinear. The vectors к and v
will be written as к. = + kL and v = v„ + v± where Ля, v„ and k±, v± are
§5.1]
Non-linear resonances in three-level systems
267
collinear and orthogonal to the vector ^. Therefore ktl'V=ktlvn and
k • v = Иц • Vn + кj. • v±. Integration of the expression (5.4) over v,, yields
again the formulae (5.6) and (5.7) where’ the substitution к—>кп,
Q-+ Q — кj. • Vj. must be performed, a Maxwellian factor added and
averaging over v± carried out. The results (5.6) obviously hold if the angle
& between the wavevectors к and кц satisfies the inequalities
kv |#|« к Ju « rnt + k Гтп = Г2,
kv |n - 0| « rm/ + ^Гтп = Ц. (5.9)
The interference term and the term due to splitting (the second and third
non-linear terms in eq. (5.4)) may be shown to approach zero rapidly
when the angle & exceeds the value &0 = rjkv. In this respect the passage
from parallel to opposing wave propagation actually occurs within the
range of angles between k* and к of the order of $0- As for the first
population term in eq. (5.4), its profile will be described by the
convolution of a Lorentzian and a Gaussian function with respective
widths Ц and kv sin ft. Consequently, the spectral width of this term will
rapidly increase as ft exceeds i?(), attaining a Doppler width kv under
ft = n/2; with a further increase in ft the Doppler contribution will
decrease, practically vanishing when the condition |n — #| < #0 is satisfied.
Therefore, for the isolation of narrow non-linear resonances, it is most
interesting to treat almost collinear wave propagation, and the inequalities
(5.9) specify the necessary accuracy of this collinearity.
Now let us consider the case of large detunings of a strong field
(|Й| »kv), confining ourselves to the approximation (5.4). In this case
{If’p
N,-Nm-(Nm-Nny-~x
I^-i^-A^-k.-v)/ (Nn N,)Q2X
_____________W(v)______________
Гп1 - — Q — Anl) 4- i(£„ - k) • v
(5.10)
and one can see that a spectral separation of stepwise and two-quantum
transitions takes place. The maxima of these lines are at the frequencies
Ц, = Лт, and = Q + Anl respectively. The shape of the stepwise
transition line is the same as that in the absence of non-linear phenomena.
268
Probe field spectroscopy under large Doppler broadening
[Ch. 5
It is isotropic, i.e. independent of the direction кц, and is characterized by
the Doppler halfwidth k^v. On the contrary, the line of combination
scattering possesses a marked anisotropy. Under k^ Цjfc the Doppler shifts
кц • v and к • v are added and the Doppler width of the line is (fcM + k)v.
This may differ notably from the stepwise transition linewidth kflv. When
wave propagation is parallel the Doppler shifts ktl -v and к • v
are subtracted, and the Doppler width is at a miminum and equals
\kf, — fc|D. If — fc|D « r„i the two-quantum transition line has the width
rnl.
A stepwise transition somewhat changes the amplitude of an isotropic
line, and this change may be quite small (|G|2«Q2) although with
N, — Nm = 0 the isotropic line arises only from non-linear effects. The
anisotropic line is due to an external field. The line shapes of combination
and reverse combination scattering under quasi-resonant conditions
(|Й|»kv) are the same, and hence its amplitude is proportional to
N„ — Nt. Note finally that the amplitude of an anisotropic line is unaffected
by the population Nm of the intermediate level.
The aforegoing may be extended to two-photon absorption and
two-photon fluorescence (Et> Em> En; see fig. 3.18a) but in the former
case we must carry out the substitutions —» — Ц, and kfl —» — kfl and in
the latter case Q—» — Q and it—» —it, as shown in section 3.3. For both of
these resonances, consequently, NIEFs are non-negligible in the case of
opposing propagation of waves (itMt|,it). Under such conditions the
two-quantum transition line also has a minimum width (up to Гп1).
The Doppler shift compensation is especially close to being complete
when the frequencies of two waves acting on atoms coincide. These
conditions are satisfied under Rayleigh scattering, two-photon absorption
and two-photon fluorescence.' Experimentally, the method of two-photon
absorption has been developed. In such experiments, absorption of one or
both of the travelling waves which form a standing (or almost standing)
monochromatic wave is measured.
As far as Doppler shift compensation is concerned, absorption and
emission of two identical photons do not differ from the above-discussed
phenomena. However, on actually calculating the line shape one must
allow for field interaction with the two transitions of an atom l-m and
n-m which, nevertheless, does not affect the formal structure of equations
for ртл, pmt and p„i. In addition, the action of both of the opposing waves
must be calculated in the same order of perturbation theory.
Calculations described in problem (23) yield the following expression
for the work done by one of the travelling waves propagating in the
§5.1]
Non-linear resonances in three-level systems
269
opposite direction:
P = h(OinNn
WE -(£Mtv)2
8ft4O? L2kv П, + 1
Ql = o) ~шт„. Q = 2(o- o)i„ + Anl.
The Lorentzian term in this expression describes the contribution to P
from two-quantum absorption with Doppler shift compensation due to
non-linear interaction of countertravelling waves with wavevectors к and
—k, so that the total Doppler shift is к • v — к • v = 0.
Different modifications of the two-photon absorption method (e.g.
measuring the fluorescence from the upper level /) is widely applied to the
investigation of the hyperfine structure of lines, inelastic processes and
other problems [2-4].
It has been found that it is often more convenient to use not a standing
wave but two opposing waves with different frequencies. The point is that
atomic levels are far from being equidistant and an adequate choice of a>
and enables one to attain a greater degree of resonance with an
intermediate level (diminished factor Й2 in the expression (5.10)). As a
result, the coefficient of two-quantum absorption may be considerably
increased, sometimes by several orders of magnitude [3]. Therefore it
proves to be possible to satisfy the condition — k\v«r„i and the
resonance width practically coincides with rnh
Let us return to the discussion of the expression (5.2), abandoning the
approximation of the first non-linear corrections. The analysis is relatively
simple when the field-induced level splitting is small compared with the
Doppler width (|G|«fcv) although with respect to relaxation constants
the value of |G| may be arbitrary. The corresponding calculations are
carried out in problem (24), where the final expression for Рц is given in
which for simplicity it is assumed that A„ = 0:
Vit
>k: P* = 2йю„ |GJ2-^Texp[-(rVfcMv)2] x
{N,-Nm + (Nm-N„)[F±(i2J+/±(i2J]}; (5.11)
k, 2\G\2 (Г±-1£±)/Гот + (1±У1Т^)/2
* Z± e (r± - i£±)(Po- i£±) + |G|2 ’
Г± = Гп1 + Г3(кц *к)/к, e± = Qkjk,
Г^Г^ + Цк./к, г^г^у/i+i.
270
Probe field spectroscopy under large Doppler broadening
[Ch. 5
The “+” and ” signs in eq. (5.11) correspond to unidirectional (£,, ftk)
and opposing wave propagation. The functions f± and F±
respectively represent the interference term (proportional to (1 ± Vl + k)/2
in eq. (5.12)) and the term due to the non-equilibrium velocity
distribution.
Comparing eqs (3.217), (5.2) and (5.12) one can see that the non-linear
part of the work done by the field (5.11) as a function of e± has the same
formal structure as the expression for the work done by the field of
a stationary atom, treated as a function of £2M under Й = 0, Ay = 0,
with relaxation constants Г± and Го instead of Гп1 and Гт!. Thus, the line
shape of eq. (5.12) characterizes a certain “effective atom” whose
relaxation parameters are different for the two orientations of the wave-
vectors.
If |G|2 is small compared with the relaxation constants, the formulae
(5.11) and (5.12) go over into eqs (5.6) and (5.7). The principal
characteristic features of the line shape when the fields are relatively large
are as follows. First of all note the conservation of the main property, i.e.
the anisotropy of the line shape (P* differs from P~). As for a weak field,
the non-linear part is proportional to Nm — Nn, i.e. the “net” splitting
effect which remains under Nm = Nn vanishes as a result of averaging over
velocities*.
The interference term which was absent under in the approxima-
tion of the first non-linear corrections appears in the next order of к (f_ is
proportional to the quantity 1 — Vl + к). Finally, on the strength of the
|G|2 dependence of the effective relaxation constants Го and Г±, non-linear
resonances undergo field-induced broadening. For this reason the field-
induced splitting (the term |G|2 in the denominator of the expression for
F± +f±) is less pronounced than for the case of stationary atoms**.
In the particular cases discussed above the effect of the field-induced
splitting is not explicit. For it to manifest itself effectively such fields are
desirable that the condition |G| »kv is fulfilled. Let us consider this case
in more detail.
Let Nm = N, = 0 and only the state n be excited. If |G| » kv the formula
(5.2) if expanded into simple fractions may be reduced to
fil 2Гт„
“This conclusion holds true if the condition |fcM|>|fc| is fulfilled. In the opposite case (k^<k)
splitting must be taken into account [5,6] (see problem (22)).
** For a more detailed analysis of expressions (5.11) and (5.12) see ref. [7].
Re(W(v)
}); (5-13)
§5.1] Non-linear resonances in three-level systems 271
M2 +(Мг —1/2)Гт/Гт„
(Tmi + Гл,)/2 - i[fi„ — Qj — (к, - М.к) • v]
Aft+ (M2—1/2)1^,/Д»,
(Г^ + Гл,)/2 - i[fiM - Q2 - (*м - М2к) • v]
Q12 = £(Q + Лл/ + А^) ± VIG|2 + ^/4,
м Чм Q } (514)
1,2 2 \ 2ViG7+X?74/
Неге Л/12 is a correlation factor, already discussed in section 3.3 (see eq.
(3.161)). The result of averaging over v can be expressed in terms of
standard functions (see problem (25)).
The work performed by the field Ptl as a function of is represented
by two lines with maxima at = Q, and Q2 so that the distance between
the lines is V4 |G|2 + Q2. Each line is anisotropic, i.e. its shape depends
on the mutual orientation of kfl and к and on the value of the correlation
factor MK2. Under ktl Hit the two lines are narrower than under ktl ft k. In
the particular case Q = 0 the lines have the same shapes and amplitudes as
well as identical angular dependences. If |JtM — M}k | v or — M2k | v is
much smaller than (Гт, + Ц,)/2 the corresponding line under will
have a Lorentzian shape with a halfwidth (Г^ + Гп,)/2.
The above results were obtained for a scheme of the combination
scattering type (see fig. 3.10). Nevertheless, they can be extended to any
of the schemes shown in fig. 3.18.
As already mentioned, the changing over from the process with the
absorption of a photon hio (or й<ом) to the process with its emission must
be accompanied by a sign change, i.e. Q'—>— Q (or >— £?M). In
addition to the sign change of Q' and Q,', the indices must be also
changed. For example, in the scheme of combination scattering through
the lower level (see fig. 3.18b) the substitution I—*g is necessary as well as
m*+n. Finally, under two-photon absorption and induced two-photon
fluorescence corresponding to the three cases of fig. 3.18c the following
substitution is necessary: l—>g,m*+n.
As for non-linear resonances due exclusively to the saturation effect
(see ch. 4), the spectral shape of the work done by the probe field is
determined by the types of relaxation processes and, in particular, its
shape is modified as a consequence of velocity changes at collisions.
Assuming “phase memory” for the transitions m-n, m-l and n-l to be
absent the formula (5.2) holds if the substitution
17->y, + yy + v’r = (5.15)
272
Probe field spectroscopy under large Doppler broadening
[Ch. 5
is performed in it. The entire influence of velocity-changing collisions is
found in pmm. In the approximation of first non-linear corrections in
formulae (5.6) and (5.7) there is an additional collisional structure which
in the model of selective scattering is similar to the collisional dip in
pmm(v) and is described by the function Zm (see eqs (4.125) and (4.126))
on substituting in it Q — к • v—> T Qk^/k, у 4-v'—> ym 4-yz 4-(y 4-
v’ykjk. The signs “4-” and ” correspond to orientations and
k^k.
In the model of strong collisions the expression for the work done by
the probe field is obtained in problem (26). Resonances in the probe field
spectrum are discussed in section 5.4, allowance being made for velocity
changes, phase memory and degeneration of levels.
5.2. The probe field method in two-level systems
In this section the work performed by a probe field which is resonant with
the same transition m-n as a strong field is analysed. Let us compare the
peculiarities of the present case with three-level systems. Induced transi-
tions due to the probe field (<oM) take place between the states perturbed
by a strong field (w) whereas in three-level systems only one state of a
transition resonant with a probe field is perturbed. Owing to that the
qualitative manifestation of the field splitting effect in a two-level system is
different from that of a three-level system: instead of two spectral
components four of them arise (see eqs (3.147) and (3.178)). Further-
more, as already noted in section 3.3 NIEFs in a two-level system lead to
population oscillations of the levels m and n and to polarization at a
combination frequency = 2w — <oM.
It would be natural to expect that the motion of atoms and collisions
would involve qualitatively new changes of probe field spectra. For
example, as a result of population oscillations the widths of some
components of the spectrum are determined by relaxation characteristics
of the levels but not transitions. Velocity-changing collisions involve
conservation of phase memory as a basic characteristic of the oscillating
part and velocity change leads to specific shape changes of the inter-
ference of non-linear resonances.
In the model of relaxation constants, on substituting Q—> Q' = Q —
к • v and Q,, —> • v we proceed from eqs (3.220), which are
§5.2]
The probe field method in two-level systems
273
valid also for moving atoms. When we substitute new variables (3.221) in
eqs (3.220) and neglect spontaneous transitions we arrive at the
following equations for ги, rmn and r*„:
(I] - iE')r„ ± i(G*rmn - Gf^n) = ±iGMp™; j = m, n;
(Г - iQ' - ic')rm„ + iG(rmra - гля) = -iGM(pmm - p„„); (5.16)
(Г + iQ' - iE')r™ - - rm) = 0;
Q' = Q-k'V, Q' = Q-k-v, e'= Q-Q —(ku—k)'v.
The quantities p;; and pmn in the right-hand side of eqs (5.16) satisfy eqs
(4.7) describing the interaction of atoms in the absence of a probe field.
The collisional shift Amn, which is not reflected in eqs (5.16), may if nec-
essary be assumed included in Q and Q^Q—> Q — Amn, Q^-^Q^ — Amn).
The work at the probe field frequency by general rules is given by
Р„(Я) = -2й<°м Re(iG>mn). (5.17)
The work in the case when atoms are at rest is considered in problem
(11). Obviously, in order to extend the result to the system of moving
atoms, it is necessary only to substitute Q—> Q' and —> Q'^ and to take
a velocity average.
In the approximation of first non-linear corrections this result is easily
obtainable directly from eqs (5.16), making use also of the first expansion
terms of Pa and pmn in terms of a strong field amplitude:
W = 2h(o„(Nn - Nm) |GJ2 x
Re^
W(v)
r-io;
I 1 Ч\Гт Г„/Г2 + Q'2
/ 1 1 \ / 1 1 \ 11 \
\Гт-1Ег r„ — iE'/\r + iQ' r-iQ^/Jj/‘
(5.18)
The terms of the expression (5.18) containing 1} — ic' reflect the influence
of NIEFs (interference term). The other components of the non-linear
part of PM(£?M) are due to changes in the populations of levels m and n
(population term).
As in the three-level system, the interference term of the work done by
the field (5.18) is anisotropic with respect to the orientation of the
wavevectors kfi and k relative to each other; when ЛД1Л the quantity e'
274
Probe field spectroscopy under large Doppler broadening [Ch. 5
contains a double Doppler shift 2k-v, when кц ft к the Doppler shifts are
practically cancelled* and e' does not depend on velocity.
Under conditions of large Doppler broadening {kv »Г) and kv »|S2|
the interference term in the formula (5.18) under vanishes as a
result of velocity averaging and the expression for Ptl is transformed to
VJt
РДЯ) = 2Й<оД^ -Nm) |GJ2^exp[-(£Ufcv)2] x
AC I/
(5-19)
The non-linear resonance in the curve is close to = — Q and has
a Lorentz shape with a halfwidth 2Г (fig. 5.2). The resonance amplitude is
proportional to the total lifetime 1/Гт 4- 1/Гп at levels m and n. If we take
Qft = Q and treat expression (5.19) as a function of Q, the non-linear part
of the work (5.19) coincides with the sharp term of eq. (4.47) describing
the dip in the plotted work done by a standing wave field. This
coincidence is predictable as the non-linear resonances in relations (5.19)
and (4.47) arise for the same reason, i.e. the interaction of a field with a
group of atoms whose populations had been changed by a countertravel-
ling wave.
With the interference term is non-zero and integration over
Fig. 5.2. Work done by the probe field versus frequency (two-level system, r«kv, |G| <<kv).
*The difference of the moduli and к of the wavevectors even under |OM-O|~kv is
|kM - k| ~ kv/c, where c is the velocity of light, and it may be taken that = k.
§5.2]
The probe field method in two-level systems
275
velocities in the formula (5.18) yields
v it Г / £2
PM(QJ = 2*(U/i(4-Nm)|GJ2^exp[-^) ]x
1"2|G|2Re^
/11 1 1 \1
I--1---1------1----I
V-C. r„ rm- iE rn - IE/ J
e = - £2. (5.20)
The non-linear resonance is near = Q and is represented by three
spectral components of different widths. Two of them have halfwidths Гт
and Гп determined by level damping (or, to be more exact, by population
relaxation) and are a spectral reflection of time oscillations of the
populations. The population dip is proportional to 1/Гт 4- 1/Г„ as in the
case of countertravelling waves but now it is situated in the vicinity of
e = — Q = 0. The main changes are connected with the interference
term. According to the general results of section 3.3, the integral over e of
this term is zero, i.e. its role consists in changing the shape of the
non-linear resonance, preserving at the same time the “area”. Charac-
teristic shapes of the population dip and interference profile are shown in
fig. 5.3. It follows from relation (5.20) that under e = 0 the values of the
population and interference terms are the same and, consequently, the
interference term contribution is always non-negligible.
By virtue of the inequality Гт,Г„<2Г, which is always satisfied, the
resulting non-linear resonance under is narrower than that for
kfi U k. For the simplest case when the condition 2Г = Гт 4- Г„ is fulfilled it
Fig. 5.3. Typical forms of “population” dip (1) and interference structure (2).
276
Probe field spectroscopy under large Doppler broadening
[Ch. 5
follows from formula (5.20) that
Vit Г /Х2 \2'
РДЯ) = 2ЙЧ1(^-^)|См|2^ехр -M x
KV L \KV /
1- 2|G|2Re
1
.Гт(Гп — ic)
1
Г„(Гт — ic).
(5.21)
As in the three-level system the condition 2ГЬ = Г, + Г; means that collision
phase modulation is absent and the interference of atomic states plays the
predominant role. In the relation (5.21) this manifests itself in the fact
that in non-linear resonance two spectral components with halfwidths Гт
and Г„ and equal amplitudes remained, and the third resonance with the
width 2.Г was completely suppressed. If the two Lorentzians in eq. (5.21)
are treated as a single resonance, its halfwidth is easily found to be у/ГтГ„,
which is less than (Tm 4- Г„)/2 and Г. If Гт~ Г„ the resonance at кц ft к is
about half as wide as that at ft к. If Гт and Г„ differ greatly the two
Lorentzians must naturally be treated separately. In the latter case
resonance narrowing at кц tt & may be very large.
If the condition 2Г = Гт 4- Г„ is not fulfilled, the non-linear part of the
expression (5.20) may be represented as a sum of three Lorentzians:
1 / J_ J_ 1 1 \
2Г-[е \Гт + Г„ + Гт-1Е + ГП-1Е/
= Гтст + Г„с„ + 2Гс
Гт — ie Г„— ic 2Г — ie’
= 1 = 2Г-Гт-Гп Г 1 1
Ci 1Х2Г-1У C 2Г 1гт(2Г-Гл) Гп(2Г — Гт).
(5.21a)
The amplitude c of the Lorentzian, which has a width 2Г and is absent
from the formula (5.21), differs from zero because of collision phase
modulation. At 2Г»Гт + Г„ we have c = cm + c„. Thus the Lorentzians
2Гс/(2Г — ie) may be useful for the study of collisions.
In the case a relatively simple expression for the work done by
the field may be obtained under the condition |G|«kv. On deriving
the corresponding expression it would be more convenient to proceed not
from the general formula of problem (11) but from eqs (5.16) themselves.
The quantities are the amplitudes of the time and space oscillations of
the level populations. Under ЛД1Л these oscillations are similar to the
§5.2]
The probe field method in two-level systems
2П
space oscillations of populations in a standing wave (see section 4.2) and
are to a great extent averaged by the motion of atoms. Thus the term
iG(rmm - r„„) in eq. (5.16) for rmn may be neglected, with results of the
same accuracy as in eqs (4.44) being obtained, whereupon we have
к, ft ft: РДД.) = 2ЙШ, I GJ2
= 2ft<uM(4-AQ|GJ2^exp - (-*) x
kv L \kv/ J
___________1
Vl 4- к Г + Ts — i(£2M 4- X2)_
2 IG I2 /1 1 \
r + rs = r(l + VT+lc)«kv, K^-L±l — + -\ (5.22)
The value of the population difference pnn — pmm was taken from eq.
(4.U).
The halfwidth of the non-linear resonance in eq. (5.22) is a combination
of the halfwidth Г, of the Bennett hole in the velocity distribution of
population and the halfwidth Г. The relative resonance amplitude is
1 — 1/V1 + к and as a function of к it has the form of a curve with
saturation. The value Рц under = — Q decreases with the increase in к
proportionally to Vl 4- к.
To analyse the approximation of very strong fields |G| »kv let us for
the sake of simplicity assume Г = Гт = Гп. Then we obtain an expression
for the work done by the field (see problem (27)):
/ 1 Г г / ОЛ
/>„ = »a>„ |G,I ((₽.. “Л») Rer+jQ, [r_j£, (1 + +
2ioJ----------------------------
lr — i[c 4- Qi — (ftM — 2M2k) • v]
___________M22___________11\
Г — i[c — Qi — (ftM — 2Mik) • v]J. /’
Q1 = Vi224-4|G|2, Mly2 = 1(1 ± QIQ^. (5.23)
Here Mly2 is the correlation factor already used when analysing three-level
systems (see eqs (3.161) and (5.14)). If |G|»kv in the relation (4.48)
the Doppler shift k • v can be neglected for the population difference
p„„ — Pmm', physically, this means that an intensive field interacts with
278
Probe field spectroscopy under large Doppler broadening
[Ch. 5
atoms practically independently of the value of their velocity. Therefore
all the selective dependence on v in eq. (5.23) is shown explicitly.
The result of velocity averaging in the formula (5.23) essentially
depends on the orientation of ktl and k (anisotropy of the line shape) and
on the relation between |Q| and kv. First consider the case of exact
resonance, Q = 0. Here = M2 = 2 and for unidirectional waves velocity
averaging leads to
“\/jr 1
P.W = йюЖ - Nm) |GJ2^ —- x
AC v X “i К
Re
Г
-Г-iQ,,
i|G|i|G|
r-i(QM +2 |G|) Г —i(QM — 2 |G|).
(5-24)
The profile PM(£?M) is symmetric about =0 and formally is a combina-
tion of three spectral components. However, the main role belongs to the
interference terms which have a dispersion shape and are located near
= ±2 [G| (fig. 5.4, curve 1). The amplitude of the central component
of the Lorentzian shape is |G|/r times as small as those of the side
components and on the scale of fig. 5.4 is hardly noticeable. The width of
Fig. 5.4. Line shape of PM(OM) under |G| 0 = 0.
§5.2]
The probe field method in two-level systems
279
the components is given by Г. Note that the profile described by the
formula (5.24) is similar to that for atoms at rest (see section 3.3, fig.
3.19a, curve for |G|/r = 4). The atomic motion was reflected only by a
decreased common multipier у/Hlkv in PM(£?M).
Unlike the unidirectional waves, in the case of their counterpropagation
Doppler shifts are added (see formula (5.23)):
e' = e — (JtM — k) • v = e — 2k,, • v
(Л„ - 2MK2k) • v = (1 + 2Af12)k^ • v,
owing to which there is substantial smoothing of the spectrum (see fig.
5.4, curve 2; for the corresponding calculations refer to problem (28)).
Now let us turn to analysing the expression (5.23) for the limiting case
|;Q|» kv. In the factor 1/[Г 4- i(Q — к • v)] the Doppler shift к • v and Г
can be neglected. The selective dependence on velocity will be retained
only in Lorentzian factors containing e. As distinct from eq. (5.24) the
line shape PM(£?M) is now asymmetric with respect to the point e = 0, and
the side components located in the vicinity of = Q ± Vi22 + 4 |G|2 have
a “bell-like” form (fig. 5.5a,b). Generally speaking, all the components
Fig. 5.5. Line shape of under |G| »kv, |Я| »kv: (а) |Я|«|G|; (b) |Я| > |G|; curves 1 and 2
correspond to the orientations of wavevectors кц ft к and tit.
280
Probe field spectroscopy under large Doppler broadening
[Ch. 5
are characterized by Doppler broadening. The broadening of the central
component is determined by the Doppler shift (kfi — k) v; the Doppler
shifts of the side components are equal to (JtM - 2Mtk) • и and (кц —
2M2k)'V. If |Q|«|G| the side components under and are
similar to each other (see fig. 5.5a); they narrow substantially as the angle
between kfi and к decreases. Doppler broadening under is
completely absent if kv |Я\/Г [G | «1.
Increase of |X2| leaves the shape of the central component unaffected
and the shape of the side components is governed by the expression
±Af22exp[—(e ± £207(^1,2v)2],
kl<2 = - 2M,.2jt| = |Л„ - к T (Я/Я})к\. (5.25)
It is clear from eq. (5.25) that under the side components have the
same widths, i.e.
/t12u = kv |Q|/VI22 + 4|G|2,
increasing with the rise in the ratio |X2|/2 |G|. At the same time the
distance between them, 2VI22 + 4 |G|2, also increases and the “negative”
component becomes relatively less intense.
For countertravelling waves
k12v = (2 ± Q/Vi22 + 4|G|2)kv
and with the increase in |Q|/2 |G| one of the components (that more
remote from wm„ and characterized by a positive amplitude) becomes
narrower and the other broadens. These phenomena are illustrated in fig.
5.5b.
Summing up the analysis of the strong field effects, we note that in the
systems with large Doppler broadening, as for atoms at rest, conditions
are possible under which in some spectral interval the probe field is
amplified and in others it is absorbed. In this respect the case [G| »ki>,
|;Q|»kv, is the most interesting when the amplification and absorption
intervals are separated with a large spacing which can be modified by the
intensity of the strong field.
Now let us turn to the effect of velocity-changing collisions on the
spectrum of work done by the probe field. We confine ourselves to the
simplest conditions. The “phase memory” at the transition m-n will be
assumed absent and the work done by the field will be calculated in the
approximation of the first non-linear corrections. The expression for
§5.2]
The probe field method in two-level systems
281
PM(£2M) will be presented as
Ря(Я) = 2А<»д|Сд12 OV.-AU) yr exp -(-?) +
к ли L '/Ciz/
|G|2 Re/ r” r” + P>” P"
\y + v - i($2M - • v)
r^-rJG'G,- р^-Ари1\С\\ ^NjW^-p,.
(5.26)
(5-27)
In the expression (5.26) the linear part of the work done by the field is
written in the explicit form (the first term in braces). The quantities Др/У
are non-linear parts of the velocity distribution of populations. They obey
equations analogous to eqs (4.60) if the excitation term in the latter
equations is dropped and in the dynamic part we confine ourselves to the
first field intensity approximation.
For the approximation of the first non-linear corrections in eq. (5.16)
for Гц the term containing r*„ may be dropped as this quantity describing
the polarization at a combination frequency appears at higher orders of
magnitude. Adding the collision integral to the equation for and taking
into account the aforegoing with regard to ДруУ, we obtain for py and f)
(2yj + vy)py = J Ajj(v | vjpfa) dv, ±
(Nm-Nn)W(v)(— ~.o, +——(5.28)
\ у + v — 112 у + v + 112 /
(2yy + v, - ie')^ = J Ац(у | t^r/v,) dr, ±
-4W»)(y + v-‘a,_.c; +
(5-29)
Note that eq. (5.28) to a great extent resembles eq. (5.29)*. Moreover,
they become identical under e' = 0. This proves methodologically useful
as one may employ the obtained results without seeking anew the solution
of eq. (5.29).
We are again restricting ourselves to the analysis of the two most
interesting orientations of the wave vectors, i.e. and For
countertravelling waves (ЛМЦ,Л) under large Doppler broadening the
interference term proportional to fm — fn in the expression for the work
* The normalization (5.27) was specially chosen in order to emphasize this likeness.
282
Probe field spectroscopy under large Doppler broadening [Ch. 5
done by the field, may be shown to become zero after velocity averaging
for the same reason as in the model of relaxation constants. Therefore the
non-linear part of the work done by a probe field propagating in the
direction opposite to the strong field is fully due to the effects of level
population changes.
Equation (5.29) is, therefore, necessary only when unidirectional waves
(кц tt Ac) are analysed. Then e' = e - (км - к) • v = e, i.e. it does not
depend on velocity. Furthermore, the condition of a large Doppler
broadening (kv » Г) enables one to ignore the quantity ie in the dynamic
part of eq. (5.29). Thus eq. (5.29) ultimately differs from eq. (5.28) only
by the replacement 2yy-»2yy —ie. Since e is independent of velocity, in
order to obtain solution of eq. (5.29) it is sufficient to substitute in the
solution of eq. (5.28) 2yy-»2yy — ie, which circumstance will be made use
of further on.
We now pass to a discussion of concrete collision models. In the model
of strong collisions the velocity distribution of the population difference
pmm ~ Pnn is given by the formula (4.88). Allowing for the relation
between ри and py expressed in terms of eq. (5.27) and the relation
between py and fj under we come to (the spontaneous transition
m -» n is ignored)
л/л Г /fix2'
PM(fiJ = 2k^(M,-Nm)|GJ2-^exp -(-£) x
/Си L \Ku/
(l-2|G|’Re S {j-^x
' j=mjt '"*4j ~ vj
_________1_________
2(y + v) - i(fiM + fi)
v, Vit Г Zfi\2])\
---------г—exp - I — I f I, (5.30)
7}(2yy + vy) kv rL \kv/ JJ/
л/тт Г / H
PM(fiJ = 2/i^(M,-Nm)|GM|2^exp - M x
kv L \kv/ J
(1 2|G| Ке{(2Ут + Уя1)(2у_+г_)
2y,+ v; Г
j=m,n 2yy + Vj - ie kv L \kv
(5.31)
§5.2]
The probe field method in two-level systems
283
In the case of the counterpropagation of waves, the non-linear part of the
work done by the probe field as a function of contains a resonance (in
the vicinity of = —$2 + 2v") of a Lorentzian shape with a halfwidth
2(y + v') and a band of Doppler width kv. The resonance and band are
caused respectively by the Bennett hole and a homogeneous saturation
band in the velocity distribution of the population difference (formulae
(4.88)). As well as comparing formulae (5.30) with (5.88), we can see that
the dependence of Рц on (see fig. 4.7) is similar to the velocity
distribution.
Owing to the NIEFs reflected by values of in the expression (5.26),
the line shape PM(£2M) under undergoes significant changes as
compared with кц UH. Only the band remains unchanged in the non-linear
part of Рц. The non-linear resonance due to the Bennett hole is, according
to the condition у + v' = ym + y„ + (vm + v„)/2 (see eq. (2.218)), com-
pensated for by the interference term.
Collisions lead to the appearance of a qualitatively new interference
structure (interference collisional resonance) which is described by the
factor [(2у,- + у, —— ie)]-1. The corresponding term in the expres-
sion (5.31) reaches its maximum under e = 0, where it is exactly equal to
the band amplitude. Figure 5.6 shows the form of an interference
collisional resonance compared with the band. This resonance breaks into
two Lorentzian contours with halfwidths 2y, + v, and I}.
In general the non-linear part of the work done by the field includes
four sharp terms, i.e. two profiles with halfwidths 2ym + vm and 2y„ + v„
Fig. 5.6. Homogeneous saturation band (1) and interference collisional resonance (2).
284 Probe field spectroscopy under large Doppler broadening [Ch. 5
which are determined by the lifetimes at the levels m and n up to the first
velocity-changing collision and two profiles with halfwidths Гт and Г„,
each of them being determined by the total lifetime at a level.
The presence of narrow spectral components due to the total lifetime at
a level can be accounted for in the following way. Unlike non-diagonal
elements of the density matrix the populations are not influenced by
processes of the phase shift type. Therefore, over the total lifetime at a
level, in spite of collisions oscillations of the level j population arise at a
difference frequency e which lead to the appearance of a spectral
component with an appropriate width in the work done by the probe field.
On the contrary, in the considered model of collisions the population is
characterized by two relaxation times: the time (2y, + vz)-1 of the state
relaxation with a given velocity and the total lifetime Tf1 at a level. They
are both reflected in PM(£2M), being responsible for the halfwidths 2y; + v,
and 7} of individual components.
From formula (5.31) the ratio of amplitudes of narrow and wide
non-linear resonances can be found. For one of the terms of the sum this
ratio is given by (j = m)
Г- (2ym + vm)(2y„ + v„) 2y„ + v
я--------ЙЕ-----------= v* d + »")
nm=^, (5.32)
where nm is the number of collisions over the total lifetime at the level m
(see section 4.3). If nm is large the narrow resonance amplitude may
notably exceed that of the wide resonance.
Formulae (5.30) and (5.31) show that for countertravelling waves the
broadening of the non-linear resonance is caused solely by the relaxation
of the non-diagonal element of the density matrix (the relaxation constant
is у + v')- On the contrary, in the case of kfl ff к the non-linear resonance
is broadened in conformity with the relaxation of populations only.
This interesting fact is, however, typical only of the model of strong
collisions and does not take place under other types of collisional
relaxation*.
Consider the model of selective scattering. The velocity distribution of
populations in the model of the one-dimensional difference kernel is given
by the formula (4.125). On employing it in the expression (5.26) for the
* Even in the model of relaxation constants under fffc there is a resonance with a halfwidth of 2Fin
the profile of (see eq. (5.21a)).
§5.2]
The probe field method in two-level systems
285
work done by the field Рц one may obtain after simple manipulations
U*:
vit Г / £2
PfI=2ft<ofI(Nn-Nm)|GJ2-^exp - (-*) X
kv L \kvJ J
1 2 |G|2 Rey=?,„ \l7j + v,2(y + v) - i(fl„ + fl) +
11. f53
_<2(y + v)-i(fl^ + fl-jkC)Jj’
х/тг " / £2 \
Рц =2hto^Nn-Nm) |GJ2 — exp - (—г) x
kv L \kv/ J
1- 2|G|2Re 2 [tz------------------7 х
,-m.n L(2ym + vm)(2y„ + v„)
2y, + V/ , Г Щ + Ще) /
2y, + v, -i£ J_oo2(y + v) - i(e - k£)
(5-34)
Here is the regular part of the Green’s function in the formula
(4.115). The function e) is obtained from by the substitution
Г, - ie, 2y,- + Vj 2y, + v, - i£.
Comparison of formulae (5.33) and (4.126) shows that on substituting
flM—» — kv, 2(y + v)—> у + v in eq. (5.33) the work done by the field
becomes proportional to the population difference pmm — pnn- Thus the
line shape PM(flM) for countertravelling waves is in this model as well
similar to the velocity distribution of pmm — pnn. Consequently, the results
of the analysis of the non-equilibrium part of the velocity distribution
performed in section 4.3 can be extended to PM(flM).
Consider the expression (5.34) corresponding to the case of unidirec-
tional waves. The non-linear part of consists of three types of terms.
The terms with resonance factors l/(2y; + v, — ie) have complete ana-
logues in the relation (5.31): they correspond to an atom-field interaction
from the time of excitation up to the first collision. The other components
result from the field interaction with atoms which had already undergone
collisions. Terms of the second type include integrals with a regular part
of the Green’s function ^(£); they describe a collisional non-linear
resonance, the form of which is the same as that of the countertravelling
waves. Finally, terms of the third type include Fjj(Z, c); they are due to
NIEFs and are called collisional interference resonances. Amplitudes of
the resonances of the second and third types (i.e. their values under £ = 0)
are the same. All the components (terms) are sharp functions of flM. The
286
Probe field spectroscopy under large Doppler broadening
[Ch. 5
widest of them is a collisional dip with a halfwidth exceeding 2(y + v') and
the narrowest of them is the collisional interference resonance. The
characteristic interval e where a corresponding term is changed is
determined by the total lifetime at the level 1/fJ.
If the width s/Vl 4- ny of functions ^(£) is significantly smaller than
2(y+v')/& the expression (5.34) changes over to the relation (5.20)
(model of relaxation constants). In the opposite limiting case 2(y 4- v')«
foyVl 4- rtj the collisional dip shape is exponential (see eq. (4.129)) and the
contour of interference collisional structure is given by (the exponential
kernel (4.120) is assumed)
Re 2yy 4- V, - ie V(f} —ie)(2yy 4- vy — ie)' 35^
It is clear from the comparison of the given limiting cases and also from
the relation (5.34) itself that the broadening of the non-linear resonance
under is due, in particular, also to the relaxation of the non-
diagonal density matrix element (dependence of integral terms in eq.
(5.34) on у 4- v). The resonance width under кц fl, к in its turn depends not
only on у 4- v' but also on the type of population relaxation (collisional
dip).
Figure 5.7 illustrates characteristic forms of the spectral components of
the non-linear resonance. This figure shows one term of the sum taken
Fig. 5.7. Non-linear resonance components: (1) dip with a halfwidth 2yy + vy; (2) collisional dip; (3)
interference collisional resonance.
§5.3]
Spectrum of spontaneous emission
287
over j in eq. (5.34). The total number of spectral components in
non-linear resonance is, consequently, six. Their widths for the general
case are different.
Still more components appear in the model when both selective and
isotropic scattering are allowed for. Combining the results of both the
models, one can easily see that in the general case the number of various
spectral components in PM(£2M) under tf Ar is nine (including the band of
homogeneous saturation).
Therefore probe field spectra provide a great detal of information on
relaxation processes taking place in a system of colliding atoms. On the
contrary, the abundance of information makes interpretation of the results
difficult; therefore, when a concrete investigation is being carried out, the
conditions must be chosen so that one particular effect might have a
dominant role.
5.3. Spectrum of spontaneous emission
In section 3.4 the method of describing spontaneous emission by atoms by
means of an effective classical field was given on the basis of a quantum
electrodynamical approach (see eq. (3.248)). It proved to be possible to
evaluate the spectrum of spontaneous emission on separating the part of
the work done by a classical probe field corresponding to emission. In
addition, the probe field intensity must be chosen in accordance with eq.
(3.246) or (3.247).
When spontaneous emission at a transition adjacent to a transition
perturbed by an external field is analysed, this separation can be readily
performed using the considerations of excitation processes. If, for
example, we are interested in a spontaneous emission at the transition
m-l in the scheme of fig. 3.10 (Em>Et) the terms proportional to the
excitation rate of levels m and n must be retained in the expression for the
work done by the probe field. On the contrary, when Et> Em the
spontaneous transition I —> m is connected with the excitation of the level I
only (see eqs (3.252) and (3.253)).
Therefore the effects of the motion of atoms and collisions on the
spectrum of spontaneous emissions at adjacent transitions are similar to
those considered in section 5.1 for the work done by the probe field.
The situation is quite different when the spontaneous emission occurs at
288
Probe field spectroscopy under large Doppler broadening
[Ch. 5
a transition m-n resonant with a strong field. The point is that the atom
after excitation of the lower level n makes a transition under the action of
a strong field to the upper level m and then can take part in spontaneous
decay. Therefore in this case the work done by the probe field is broken
up into absorption and emission in a different way (see eqs (3.254)) not
connected with excitation processes. The results of section 5.2 are not
directly related to spontaneous emission and in order to extend them to
the latter additional analysis is necessary.
The expression for the spectral density of the spontaneous emission
intensity (—PM) of atoms at rest interacting with a monochromatic
travelling wave is obtained in problem (13). For the system of moving
atoms, as found in section 4.1, the substitution Q—>Q- k’V,
Sip— kp'V and velocity averaging must be carried out. As the general
formula is very awkward only the most striking illustrative cases will be
treated here.
In the approximation of first non-linear corrections and large Doppler
broadening (kv»r, |fi| <<kv) it can be found from the formula of
problem (13) that
= —k:
k, = k:
VJt Г / О \2"
—= 2й coM | |2-^7 exp -(-£) x
kv L \kv/ J
N.-2(N.-N.)|GpARe2r_.^ + fl)];
(5.36)
x/jt Г / £2 \2"1"
-PM = 2ft<OJGJ2^Texp -(-£) Nm-2(Nm-Nn)\G\2x
kv L \kvJ JL
1 У2Г-Гт-Гп
2Г — Г„\. Гт(2Г — ie)
(5.37)
The linear part of in the relations (5.36) and (5.37) depends on the
population of the upper level m only, which would be expected for
spontaneous emission by atoms in the absence of a strong field. The
difference of the non-linear part of the spectrum (5.36) and (5.37) from
the non-linear part of the total work done by the probe field lies in the
fact that in the sum taken over j = m, n in formulae (5.19) and (5.20) one
term is retained for each expression because for the spontaneous emission
spectrum the Bennett distribution at the level m and the beats of
populations at the level n are of importance.
If Nm = N„, the plots of PM(fiM) given by the formulae (5.36) and (5.37)
§5.3]
Spectrum of spontaneous emission
289
contain no non-linear resonances at all. Under Nm #= N„ the spontaneous
emission spectra qualitatively follow the probe field spectra.
The greatest qualitative difference of the spontaneous emission
spectrum from the spectrum of the work done by the probe field is
reached in the limiting case |G|»kv. Рц is calculated from the general
formula of problem (13) by employing a procedure analogous to that of
deriving the formula (5.23) (see problem (29)). If Гт = Г„ = Г and
|G|»|fi| then Рц is given by
_f> - |G,F Re(ly(„){r _ i[e _ (Jfc> _ t). v] +
1/2
Г- i[e - - (кц - 2Mxk) • v] +
____________1/2____________1\
Г- i[e + Qx - (*„ - 2M2k) • v] J/’
fi1«2|G| + i22/4|G|, 2M^=\±Q/2\G\. (5.38)
Figure 5.8 shows the characteristic form of a spectrum under кц = к and
кц = —к. The spectrum is almost symmetric about the point e = 0. The
side components are close to e = ±Д. The integrated intensity of the side
component is half as high as that of the central component. With кц = — к
all the components have approximately the same Doppler widths (2kv and
2*v(l±O/2|G|)).
For the case when кц = к the central component is characterized by a
Lorentzian shape and halfwidth Г. The side components of the triplet are
Fig. 5.8. Spontaneous emission spectrum under |G| » Arv, |S2|: (1) = k; (2) кц = —k.
290 Probe field spectroscopy under large Doppler broadening [Ch. 5
given by the Voigt function (the convolution of Lorentzian and Gaussian
functions) and their shape and halfwidth are, therefore, governed by the
relation between Г and kv|l —2ZW1>2| = kv |$2|/2 |G|. If r»kv |$2|/2 |G|
they have a Lorentzian shape and the halfwidth equals Г. In the opposite
case the side components have a Gaussian shape and their halfwidth is
kv |fi|/2|G|.
Comparison of fig. 5.8 with figs 5.5 and 5.6 shows that the shape of the
spontaneous emission spectrum is strikingly different from the spectrum of
the work done by the probe field. Note also that the expression (5.38) is
proportional to a half-sum (Nm + V„)/2, i.e. to the value of the level m
population when the populations of the two levels have been fully
equalized by the strong field.
The spontaneous emission spectrum of atoms moving in the field of a
standing monochromatic wave has some interesting peculiarities. In the
system of an atom at rest such a wave is bichromatic (of frequencies
(o±k'v) and the atomic levels split into an infinite number of equi-
distant sublevels (Гт = Г„ = Г, co = see relation (3.122) and sub-
sequent discussion):
-Em + hsk • v, E^ = En + Os'к • v, (5.39)
The spontaneous emission spectrum caused by transitions ms—*ns' has
the following form (compare eq. (3.165) and problem (30)):
D ^mn >- V D /_________________C/W(v)__________\
Рц ~ 16л2 ft<0 Re\r - i[fiM - (кц + Ik) - v]Z’ ( ,40)
the term кц • v allowing for the Doppler effect in passing to the laboratory
system. It follows from the expression (5.40), in particular, that when the
observation is taken along the к direction the term Z = — 1 specifies the
spectral component whose halfwidth Г is independent of the external field
intensity*.
The spectrum at an adjacent transition must have a structure similar to
that of eq. (5.40). However, because |kM| and |£| are different the Doppler
shifts are not completely cancelled. If |ЛД| = 2|Л| (or |kM| is generally a
multiple of |Л|) the cancellation in the term I = — 2 (or I = —|ЛД|/|Л|) is
complete and the halfwidth of this term is also Г.
The methods described in the aforegoing sections may be employed
while investigating the influence of collisions on the contours of non-linear
resonances in spontaneous emission spectra. Both physical and method-
* For details see ref. [8].
§5.4]
Polarization phenomena
291
ological aspects of this problem are in many respects similar to those
considered in earlier work (see, for example, ref. [9]).
The present section was devoted to the spontaneous emission spectrum
at transitions between excited states. The results obtained above are not
fully applicable to the case when the lower level of the transition is a
ground state. Characteristic features of transitions involving the ground
state are due to its infinite lifetime, which is peculiarly reflected in
spontaneous emission spectra. This subject is not treated in the present
book and we can only refer the reader to refs [10-15].
5.4. Polarization phenomena
In sections 4.2 and 4.3 attention was drawn to the dependence of the
amplitudes and shapes of non-linear resonances on field polarization. The
number of polarization phenomena increases significantly in the spectros-
copy of the probe field and they assume special importance. The point is
that the polarization states of the strong and probe fields may be chosen
arbitrarily and independently of each other. Therefore field polarization
can prove to be an effective means of modifying experimental conditions.
Moreover, there are components in the probe field spectra due to NIEFs
which are especially sensitive to field polarizations.
By way of an example illustrating the above statement, consider
two-level systems. In the model of non-degenerate states (see section 5.2)
NIEFs are manifested in beats of the level populations and bring about
spectral structures with widths characteristic of the relaxation of popula-
tions. In systems with degenerate levels the polarization moments of
higher orders (к = 1,2,... ) also experience beats and the widths of the
corresponding spectral structures depend on the relaxation rates of those
moments. On the contrary, when the polarizations of the strong and
probe fields are different, the work done by the latter is to be determined
by several polarization moments, not only the weights of moments, of
different ranks being varied but also the moments relating to the upper
and lower states. Because of this the shape of the non-linear resonance
can be radically modified with changing polarization conditions. The same
is true for three-level systems.
Now we proceed to an estimation of the work done by the probe field.
First, consider a two-level system assuming that atoms interact with a
292
Probe field spectroscopy under large Doppler broadening
[Ch. 5
strong and a weak (probe) field at a transition between states m and n
degenerate in the directions of angular momenta Jm and J„.
In eqs (2.67) for the density matrix in the representation of polarization
moments it must be taken that
Uj = —(!> ₽-(«-* т)_ Qi p.-HQpl-k,, r)
(5.41)
where the matrix is responsible for the interaction with a probe
field*. For the matrices G{k and G^k the relations (4.30) hold true.
As in the model of non-degenerate states the solution of eqs (2.67) is
sought in the form of**
Pn(t,r) = pjj + rne-iv + rjiev,
Pmn(t, r) = e-i(ar“* ’ r)(pm„ + rmne-i<p + rm„ei<₽),
(p = Et — (£M — k) • r, e = — Q, (5.42)
where the matrices p,7, r,7 and r,7 are independent of coordinates and time.
The quantities p;7 and pm„ in eqs (5.42) constitute a density matrix in the
absence of a probe field and satisfy the following equations (compare with
eqs (4.26) and (4.158)):
G™nP„m),
2УиРии — —I == 4” S«„ — i(GXnPnm — G’HmPmn),
I пгпп nn x^fnnrnm
(y i£2 )pmn Smn i(Gm„pmm Gm„p„„),
Q' = Q-kv, y = ym + y„,
pnm(^) = (-1)7"-л+’рХ»(^ - q)-
(5.43)
Here and henceforth for simplicity the radiative decay m —» и is not taken
into account.
The equations for quantities r,7 and r,7, obtained in the same way as in
the model of non-degenerate states, have the following form (compare
* The suffix p indicates that the characteristics belong to the probe field and are unrelated to level
numbering.
** It must be recalled that the quantities pv are columns with components pv{Kq).
§5.4]
Polarization phenomena
293
with eqs (5.16)):
(2ym - ie')rmm = smm - i(G^,r„„, - G™„rnm) + iG™„„p„ra,
(2yn ic )гяя S„„ i(Gmnrn/„ Gnmrm„) iGf4mnpnm,
(y- iX2' - ie')rm„ = Sm„ - i(G^,rmm - G" „r„„) - i(G^,„pmm - G^,„p„„),
(y + iQ’ - ie'Xnm = S„m - i(G"mr„„ - G7„,rmm). (5.44)
The matrix of the system of equations (5.44) under e = 0 coincides with
the matrix of the system of eqs (5.43) (if for the latter we add an equation
for p„ra). The solution of eqs (5.43) serves as a “right-hand side” in eqs
(5.44).
From now on we shall mostly be seeking the solution of eqs (5.43) and
(5.44) by iteration with respect to the strong field amplitude. The
frequencies and kernels of the collision integrals are assumed to be
diagonal in к and q and independent of q (see the relation (4.159)).
The solution of eqs (5.43) and (5.44) will be expressed in terms of
Green’s functions which in turn satisfy the following equations:
Fj^Kqv I K^iVi) = dKK,dqqiFjK(y |
Fmn(icqv | = dKKydqqiFK(v |
(2y, + Vj)FjK(y | v') = f Ajr(v | Vi)F/K(vi | v') dvt + <5(v - v');
(y + v - iQ')FK(v | v') = J AK(v | v1)FK(v1 | v') dv, + <5(v - v');
X(v | vO = A„(xv | JcvO; AK(v | vO = Лт„(кг | rv,);
Fnm(Kqv | К1^1Ц) = FZ^Kqv | кдм). (5.45)
I KWh) = 8KKJqqJjK(y | vO;
fmn(Kqv | K^iVi) = dKKldqq}fK(v | vt);
(2y, + у,- - iE')fjK(y | v') = AjK(y | Vi)^(v! | v') dv, + <5(v - v');
(y + v - iQ'JfK(y I v') = j AK(v | Vi)/,(vi | v') dvj + <5(v - v');
v = vm„; Vj = Vjj', Q' = Q-k-v; Q'^= Qfl - kfl -v,
e’ = Q'^ — Q' = e — (k^ — k) ‘ v; e = Qfl — Q. (5.46)
294
Probe field spectroscopy under large Doppler broadening
[Ch. 5
The Green’s functions FjK and FK correspond to the equations for p;; and
pm„ and eqs (5.45) are equivalent to eqs (4.161). To the equations for r;;
and r„,„ there correspond the Green’s functions and fK. The Green’s
function of the equation for r„m is not presented as we shall not need it
subsequently.
Note that under =k and e = 0 the equations (5.46) become identical
to eqs (5.45). Consequently, the solution of eqs (5.46) under ft A: may
be obtained from the solution of eqs (5.45) by means of a formal
substitution:
2y;—> 2y; - iE, у-> у - iE. (5.47)
The aforegoing remarks will also be of use subsequently, since the
solutions of equations for the Green’s functions FjK and FK have already
been discussed in section 4.3 for different models of collisions. On the
contrary, the Green’s functions fjK and fK will be necessary only in the case
AM = k (the strong and probe fields have unidirectional waves).
On solving eqs (5.43) the iteration relations (4.162) may be employed
where the Green’s functions and Fmn satisfy eqs (5.45). By analogy,
iteration relations may also be formulated for r;7, rm„ and r„m.
Let us confine ourselves to the approximations of first non-linear
corrections and calculate the quantity rmn(la) proportional to the dipole
moment induced at the frequency of the probe field and necessary when
deriving the work done by the probe field P*. For pm„(K^,v) and
pjj(Kq,v) the expressions (4.164) and (4.165) can be used (the Green’s
function F~„ must be omitted in the latter expression).
In the aforementioned approximation the quantity rnm in eqs (5.44) may
be ignored. Then the same procedure that was used in obtaining formula
(4.157) (the method of successive approximations) yields the following
expressions for (rm„(lor)) and the work done by the field
(rm„(lor)) = -i4m(GMaam„ - 2 (—l)1"'<Icrl - or, | Kq) x
Kqoi
2 [G^Kq^ + G^KqjB^]},
Nnm = Nnlgn-Nmlgm, & = 2J, + 1, к = 0,1, 2. (5.48)
Рц = 2ha)flNnm Ref |GM|2am„ - 2 S [1Л^)12Д> +
j=m,n Kq J
(5-49)
§5.4]
Polarization phenomena
295
The following notations were introduced into the formulae (5.48) and
(5.49):
(Xmn = (fl(V | 1Ц)1¥(ц)),
B}K = (A(v | I «г)[Л(«21 »h) + fi(v21 v3)]W(th)),
= (fi(v | | Ц-Л/Хц-1 th) + Л*(«21 th)]W(ih)>,
J(xq) = S (-ly-^lal - <h | KqjG^Gt,
<xn
I^q) = S (-1)1_<”<lal - о. I Kq)G,aG*m,
0O1
I^q) = X (-l)1-a,(lal - о. I Kq)GaGZ
<7<71
GIUJ = dmn^al2h, Ga = d„M2b,
Г1 1 Jcl2 Г1 1 jrl2
aL = 9 ; a2nK = 9 . (5.50)
Jm Jn' '‘"'n ** n
Here the angular brackets denote integration with respect to all the
velocities. The quantity J(xq) is introduced by analogy with the polariza-
tion tensors 4 and I of the probe and strong fields (see relation (4.34))
and has the meaning of a “crossed” polarization tensor as J(Kq) includes
bilinear combinations of circular components of the two fields. We shall
concentrate our attention on the non-linear part, because the linear part
proportional to amn has been already analysed in ch. 3.
The dependence of the non-linear part of PM(X2M) on the frequency is
concentrated in the functions B;K(£2M) and B/K(X2„). The terms containing
B;K(£2M) are due to the field-induced changes in the velocity distribution of
the levels’ polarization moments, as is clear from the definition (5.50) for
BjK and from the expression (4.165). The functions ДЛ(Х2М) according to
their definition produce interference changes in the line contour PM(X2M)
(interference terms). The specific form of BA(X2M) and ДЛ(Х2М) is
determined by the collision model.
Equation (5.45) for the Green’s functions FjK(v | and FK(y | Vj)
formally are not different from eqs (4.63) and (4.64) of the model of
non-degenerate states. Because of this, the terms are similar to
the population terms of the work done by the probe field, treated in
section 5.2.
As to the terms Д/К(Х2М), they become zero under = -k as a result of
integration with respect to velocities, i.e. for the same reasons as the
296
Probe field spectroscopy under large Doppler broadening
[Ch. 5
interference terms in the model of non-degenerate states. Provided that
кц=к, when the terms are non-negligible, the Green’s functions fjK
and fK can easily be found from the Green’s functions FjK and FK by means
of the substitution (5.47). If this is taken into account the quantities
may be shown to be obtained from BjK by the substitution 2y;—»2y; — ie.
Thus specific expressions for the quantities and ДК(Х2М) may be
obtained on the basis of the results of section 5.2. For example, in the
model of relaxation constants, by analogy with the relations (5.19) and
(5.20) we have
„ Vit
BjK = 2 — exp
kv
kv) J 7}r2r-i(X2M TX2)’
(5.51)
„ „ VJt
^k-2 —exp
rjK — \е2Г—\е
(5.52)
The upper sign in the formula (5.51) corresponds to unidirectional waves
(кц = £); the lower sign corresponds to those travelling in opposite
directions (&M = —k).
The polarization dependence of the probe and strong waves in the
expression (5.49) for the work done by the field is represented by
polarization tensors J(Kq), I(Fq) and I^Kq). A change in the wave
polarization state is formally manifested in changing coefficients of
Byr(i2M) and /3;K(i2M). The functions BA(X2M) and /3;K(i2M) themselves
remain unchanged under varying polarization conditions and, as has
already been shown, are fully determined by relaxation processes.
Note that interference terms in the relation (5.49) contain a “polariza-
tion” factor S, |J(*^)|2 different from its analogue 7„(кд)7*(кд) in the
population terms, i.e. in the general case interference terms enter into the
non-linear part of PM(X2M) with a weight different from that of the terms
due to the velocity distribution of polarization moments. For the
particular case of waves with the same polarization the equality |У(|2 =
Ip(Kq)I*(Kq) holds good and the weights of these terms are the same*.
We now discuss some concrete examples. When кц = — к there are
no interference terms and non-linear resonance by eq. (5.51) has a
Lorentzian shape independent of polarization with a halfwidth 2Г. The
change in polarization conditions affects only its amplitude. Thus in the
* In the model of non-degenerate states (see section 5.2) the weights of the interference and
population terms are the same, since this model is physically applicable just to the same polarization
of two waves.
§5.4]
Polarization phenomena
297
model of relaxation constants the case = — к presents no special interest
from the viewpoint of polarization phenomena.
When waves are unidirectional each function ДЛ(Х2М) apart from the
Lorentzian with a halfwidth 2Г common to all of them includes also a
Lorentzian with a halfwidth Г/К. Since the values of Г)К, generally
speaking, differ for different j = m,n and k = 0,1,2, the non-linear
resonance in the plot of PM(X2M) includes as a result seven spectral
components.
Let the strong and probe waves be characterized by the simplest
polarization states: similar linear (TT), similar circular (++), orthogonal
linear (J-») and orthogonal circular (+ —) polarizations. The correspond-
ing transitions are shown schematically in fig. 5.9.
The expression (5.49) for the work done by the field will be presented
as
rx/jt Г /Х2 \21 1
= 2fttoMN„m |GM|2j —exp - (—*) - |G|2Re tp(X2M) , (5.53)
ч KV \Ku/ J j
where the non-linear resonance is described by the function Re q> (X2M).
Fig. 5.9. Transitions caused by the strong (-----------) and probe (“"«) fields with the simplest
polarization states.
298
Probe field spectroscopy under large Doppler broadening
[Ch. 5
On calculating the polarization tensors J (icq), I^xq) and I (icq) for the
above-cited cases, we obtain
ft: <p(4) = i S (Bj0 + 2Bj2 + pi0 + 2fif2), (5.54)
;=m,«
++: = i 2 (^0 + 2^1 + iBj2 +
+ + (5.55)
H: <p(4) = i S (Bj0-Bi2 + ipn + tfj2), (5.56)
j=m,n
+-: <p(4) = i S (Byo-^i+ ^ + 3/^). (5.57)
The expressions (5.54)-(5.57) hold true for other models of relaxation
processes as well (expressions (5.50) for BjK and Дк); nevertheless, to be
specific we shall proceed from the formulae (5.51) and (5.52) correspond-
ing to the model of relaxation constants.
As can be seen from formulae (5.54)-(5.57) only the coefficient of Bj0 is
unaffected by polarizations; the weights of the other quantities BjK and fljK
vary over a wide range depending on the polarization of fields. The
functions Bj! and under some polarization conditions may be absent.
According to the relation (5.51) all factors BjK as functions of are
similar and their weights in the function <p(X2M) influence only the
Lorentzian amplitude with the width 2Г. Most significant is the result
that in eqs (5.54)-(5.57) the weights of the interference factors @jK
containing spectral components with halfwidths rjK vary.
The complete set of functions fl,K appears in the case of similar circular
polarizations of the strong and probe waves. When the linear polarizations
are the same (eq. (5.54)) the pair /3;0 and flj2 emerges; this results in
spectral components with halfwidths equal to the relaxation constants of
populations (rj0) and alignment (7^)- If the waves have orthogonal linear
polarizations (eq. (5.56)) the pair and fi}2 gains importance and
contours with halfwidths 2Г, and Г/2 emerge (Tfi is the relaxation
constant of orientation). In the case of orthogonal circular polarization
(eq. (5.57)) only the functions fij2 are of importance.
It is characteristic of the model of non-degenerate states that the
interference and population terms of the non-linear resonance have the
same weights (they are equal under e = 0; see section 5.2). If the levels
are degenerate, this correlation of weights of BjK and fljK is retained when
the field polarizations are the same and is violated under dissimilar field
polarizations. In the first case the result can easily be understood as the
§5.4]
Polarization phenomena
299
schematic diagrams of figs 5.9a,b are totalities of two-level subsystems for
each of which the relation between the weights of the corresponding terms
equals unity. When field polarizations are orthogonal (see figs 5.9c,d) the
complete system of levels and transitions effectively breaks into sets of
three-level systems. This is, however, complicated by the fact that both
states of some transitions induced by the probe field are perturbed by the
strong field. This complication is, nevertheless, negligible when we deal
with the approximation of first non-linear corrections. The main thing is
that here specific properties of NIEFs of three-level systems are mani-
fested, consisting in the emergence of polarization at forbidden transitions
mM-mM' and nM-nM'. Here transitions between magnetic sublevels of
one of the levels j = m,n act as forbidden transitions. In other words,
NIEFs under orthogonal field polarizations are due to the coherence of
magnetic sublevels. It is because of this that there are no До terms in
formulae (5.56) and (5.57) into which the relaxation constants of level
populations enter and, on the contrary, these formulae contain /3yl and
Pj2, which include the relaxation constants of sublevel coherence. In
particular, under orthogonal circular polarization (fig. 5.9d) the coherence
of sublevels differing by \M — M'\ = 2 which emerges in the polarization
moments of rank к = 2 and higher is important. Accordingly, only fi]2 out
of all the functions fijK (к ®s2) appears in expression (5.57).
As for the terms BA(X2M), their role in relations (5.54)-(5.57) is
interpreted in a different way: each of them is present to the extent that the
strong field affects the polarization moments of the corresponding rank. For
example, if the strong field is linearly polarized it modifies the population
of levels (B}0 term) and produced alignment (B/2) which is reflected by the
formulae (5.54) and (5.56) (see also eqs (4.32) and (4.36)). A field that is
circularly polarized affects populations, and induces orientation and align-
ment. Because of this, all the B;K(£2M) appear in relations (5.55) and (5.57).
Each of the terms BjK and fiiK includes the coefficient a]K which depends
on к and on the momenta of both levels m and n (see relations
(5.50)-(5.52)) and affects the weights of contours with halfwidths Г/к. If
the angular momenta of the levels m and n are equal (Jm = J„) then
^тк
Let us write the coefficients a]K in the explicit form. For the transition
from eq. (5.50) we obtain
2 = 2 3 2 = 2 =_________3_______.
am0 an0 2J + i, ami anl 2J(J + 1)(2J + 1) ’
2 = 2 = _3 (2J-l)(2J + 3)
am2 an2 10/(/ + 1)(2J + l)'
(5.58)
300
Probe field spectroscopy under large Doppler broadening
[Ch. 5
If Jm ^Jn then for the definite case Jn=J and Jm = J — 1 we find
2 _3 7-1 2 _3 7 + 1
J(2J-1)’ fl"1—2 J(2J + 1)’
= 3 (J —1)(2J —3) . 3 (J + l)(2J + 3)
m2 10 J(2 J - 1)(2 J + 1) ’ "2 10 J(2J — 1)(2 J +1) v '
For the transitions with AJ = 0, from the relations (5.58) it follows that
аД/а?2 = 5/(2J — 1)(2J + 3). As J increases, this ratio falls off proportion-
ally to 1/J2 and for J = 2 it amounts approximately to 1/4. This means that
when values of J in formulae (5.55)-(5.57) are large (J > 2) the functions
and Ду1(Х2м) can be ignored in comparison to Bj2 and pj2. In the
particular case J = 1/2, on the contrary, Bj2 and fij2 vanish in conformity
with the fact that there can be no alignment at the levels with J = 1/2.
Therefore, in the limiting cases of large and small values of J in formulae
(5.54)-(5.57), additional selection of terms takes place and in the
non-linear resonance there is additional selection of contours.
For transitions with |AJ| = 1 the coefficients a2mK and a2K differ and the
contours with characteristics of the m and n levels in non-linear resonance
have different weights. As can be seen from expressions (5.59) this is
characteristic mostly of the transitions Jm = = 1 and Jm = 1/2—» Jn =
3/2. In the first case J = 1, «^1 = «^l2 = 0, and consequently in the
formulae (5.54)-(5.57) all the terms with j = m except Bm0 and fim0 are
eliminated from the sum over j = m,n. As for the other terms, for the
given transition alt0/a20 = 3 and the functions Bm0 and fim0 prevail over Bn0
and Д„о. In the second case (J = 3/2) the functions Bm2 and fim2 are absent
from the formulae (5.54)-(5.57).
When values of J at the transition with AJ = 1 are different the relative
role of the terms with j = m and j = n is determined by the factors ailK/a2K.
From the expressions (5.59) we have
a2m0_2J + l a2ml (J - 1)(2J + 1)
a20 2/-Г a2nl (J+1)(2J-1)’
a2m2=(J-l)(2J-3)
a22 (J + l)(2J + 3)-
(5.60)
§5.4]
Polarization phenomena
301
As J increases, the ratios (5.60) tend to unity, i.e. resonances relating to
the levels m and n possess equal weights when the J values are large. At
the same time, under J = 2, 3, 4 the ratio a2m2la22 equals respectively 1/21,
1/6 and 3/11 and for these values of J the functions Bm2 and fim2 can be
practically ignored in comparison with Bn2 and fin2. The ratios a2mK/a2nK
with к = 0 and к = 1 are not small even when J = 2.
Unlike the case for the transition with AJ = 0, the quantity аД with
J »1 for transitions with |AJ| = 1 is not small compared with a20 and a22.
The proportions of these values under J » 1 are
a?0 : afi: «/2 = 1:2: ю- (5.61)
According to this the contribution of the functions Bj2 and f}j2 in
expressions (5.54)-(5.56) is small and in some cases may be ignored. It
must be recalled that for the transitions with AJ = 0 under J »1, on the
contrary, the values of and are negligibly small.
There is quite a peculiar case of relaxation constants independent of к
(TjK = IJ) which occurs either when relaxation is purely spontaneous or if
collisions lead only to quenching of levels. In this case BjK and fi)K under
different values of к differ only in the coefficients a2K and they should be
grouped. If account is taken of the fact that for the aforegoing relaxation
models the relation
Г + П = 2Г
* tn И
(5.62)
holds true and use is made of expressions (5.58) and (5.59) for the
coefficients a2K the formulae (5.54)-(5.57) may be reduced to the
following forms:
Vn 2 Vjt
Л = (a2m0 + 2a2m2) = (a20 + 2a2n2)-,
kv 3 kv
ф(Я) = Az\exP
-Гт\Г„ if) Г„\Гт ie).
(5.63)
(5.64)
302
Probe field spectroscopy under large Doppler broadening
[Ch. 5
A2 = -^ (ia2mo + a2mi + 3^2)
K-U
= V? (20*0+ a*i + 3^2);
K-U
+ -:
ф(Я) =
g»i + a»2 a2mi + a2m2 I Vn
.Гт(Г„ - ie) Гп(Гт - ie) J kv eXp
Q \21
-Jjf I
kv) .
(5.65)
ф(Ч) = 2
a22
Гт(Г„ - is)
a2m2
ул
— —exp
e)J kv
Q \21
_2f I
kv) J'
(5.66)
In the model under consideration with кц=к the Lorentzian with
halfwidth 2Г vanishes and the non-linear resonance consists of two
Lorentzians with halfwidths Гт and Г„. Resonance with a halfwidth
2Г = Гт + Гп is absent under an arbitrary polarization of field and not only
for the polarizations of eqs (5.63)-(5.66). This can be proved by the
identity (see problem (31))
2 Цкд)1*(кд)\ *} - |J(k$)|2| T , П =°-
Kq Ul J2J W2 J2 •'P J
If the probe and strong fields are equally polarized, the amplitudes of
these Lorentzians are equal (see relations (5.63) and (5.64)), and the
specific state of polarization affects only the common multiplier in <p(DM).
Lorentzians with halfwidths Гт and Гп also possessed the same amplitude
in the model of non-degenerate states (compare with the equation (5.21)).
Under orthogonal polarizations of fields the ratio of amplitudes of the
Lorentzians remains the same for transitions with Jm = J„ when a2mK = a2r
and is changed if Jm^Jn. In the latter case the Lorentzian with a halfwidth
equal to the relaxation constant of a level with a greater value of J has a
large amplitude. Thus, for the transition Jm=J—1—»J„=J the ratios of
amplitudes of the Lorentzians contained in formulae (5.65) and (5.66) are
respectively
a2i + a22_(J-l)(6J + l). a22 JJ-l)(2J-3)
^1 + ^2 (J+1)(6J-1)’ a22 (J+l)(2J + 3)'
(5.67)
The second of these relations has already been discussed and regarding
the first it should be noted that if J = 2 the ratio between the amplitudes
§5.4]
Polarization phenomena
303
of Lorentzians under orthogonal linear polarizations is 13/33, i.e. the
weight of a Lorentzian with a halfwidth Гт is not large.
Collisional disorientation and phase modulation violate the conditions
of interference suppression of the resonance 1/(2Г — ie). As a result, the
number of Lorentzians under кц tf к is not two as in formulae (5.63)-
(5.66) but seven, the amplitude of resonance 1/(2Г —ie) being given by
the expression
c = аЪ 7-4(^У*(^) -9r j, r 1Л^)12 •
jxq Llyn- ljK
When this quantity is measured we obtain data on phase modulation and
disorientation (2Г &Гт + Гп, к dependence of riK). For 2Г» riK we have
" jtcq ljK
which is comparable with the total amplitude cA of resonances 1/(Г)к — ie):
1 Д?
jKq -ijK
Collisions are similarly manifested within the scope of the model of
non-degenerate states* (see discussion of formulae (5.21) and (5.21a)).
The coefficients a]K and polarization weights in the relations (5.53)-
(5.57) are independent of the properties of a particular relaxation
mechanism. Therefore most of the conclusions drawn above hold true not
only for the model of relaxation constants but also for any other model
leading to formulae (5.49) and (5.50).
Polarization phenomena in three-level systems have much in common
with those discussed above. Let the probe field be in resonance with the
transition m-l (see fig. 3.10; Em>Ez). The matrix element of the
interaction U7m/ at the transition m-l is given by
U, = -G7,,exp[-i(D„t - кц г)]. (5.68)
The density matrix elements at the m-n transitions are, as before,
governed by eqs (5.43). For the additional elements required for
evaluation of the work done by the probe field the equations following
* Similar phenomena take place in four-photon parametric processes [16, 17]. The results of studies of
such phenomena and references (except to the work of Soviet authors) may be found in reft [18-20].
304
Probe field spectroscopy under large Doppler broadening
[Ch. 5
from eqs (2.67) presented below hold true:
(Ym + Yi - = SmZ + iGLr^ - iG^pmm,
[y„ + Yi - Ж = s„/ “ iG^p„m + iGJ,mrm„
Pm/ = rm/ exp[-i(D„t - кц • r)],
p„, = r„z exp[-i(DM - Q)t + \(k^ - k) • r], (5.69)
Q' = Q - к • v, - k^ • v.
The collision integrals in these equations, as before, are assumed diagonal
in Kq, i.e. having the structure of formula (4.159). By analogy with eqs
(5.45) and (5.46), introduce the Green’s functions of the collisional part of
eqs (5.69):
fml(Kqv | = dKKidqqtfmlK(v | vt),
f„i(Kqv | Ki^jVi) = dKK,dqq,f„lK(v | vt),
(Ym + Yl+Vml~ i^)fmlK(V | V ')
= J AmlK(v | Vi)/mZK(vi | v’) dvt + <5(v - v'),
[Yn + Yi + Vni ~ ЦЦ. - Q')]fniK(v I v')
= f AnlK(v | v1)/rfK(v11 v') dv, + <5(v - v'). (5.70)
When calculating the dipole moment ((rmZ(lo))) induced at the
frequency of the probe field and the work done by the field in the
approximation of first non-linear corrections we take into consideration
formulae (4.164) and (4.165). Iterations similar to those employed in the
derivation of the formulae (4.167) and (5.48) yield the following expres-
sions for (rmZ(lo)) and the work done by the probe field PM:
(rmi(lo)) = ilNml^mlG^ -Nnm^ ( - 1~( 1 - Oil Kq) X
*• Kqaj
[GaiJ(Kq)& + GwI(k<]}; (5.71)
Д RekA|GJ2-N„mS[l-/(^)l2/t + /,(^)Z*(^)^]j,
Kq >
§5.4]
Polarization phenomena
305
с,а=^„/2й, igj2=s m2,
0
N- N
(Xml = {fmntv I v,)W(Vi)), Nv = —j— - —t
BK = aBK{fmll(v | Vi)FmK(v! | v2)[Fi(i»2 11^) + Fi*(i»2 | v3)]W(v3)>,
Рк = aPK (fmii(v I v1)/„/K(v1 I V2)[fmll(.V2 I «b) + F*(«2 I v3)] W(v3)>;
fl к l]2 . , f 1 К 11(1 К
.^ = 9 , , , , ,
(5-72)
(5.73)
1 1
n
n
The quantities J(xq), 1ц(кд), I(xq), FmK and FK have the same meanings
as in formulae (5.48)-(5.50).
The functions BK are expressed in terms of the velocity distribution of
the level m polarization moments. The terms of eq. (5.72) containing fiK
are due to NIEFs and contain the Green’s function f„iK(y | v') connected
with the forbidden transition n-l. Thus the physical meaning of the
functions and BK in the expression (5.72) is the same as that of the
functions fi)K and B1K in the two-level system (formula (5.49)). The
similarity of expressions (5.49) and (5.72) is increased as 0K and BK are
multiplied by the same combinations of field polarization tensors as each
of the functions fl,K and B1K of the sum over j = m,n of relation (5.49).
On account of the aforegoing circumstances the formal manifestation of
polarization phenomena in three-level systems is the same as that in the
two-level systems. Let us represent the work done by the probe field
(5.72) by analogy with eq. (5.53) as
2-
+ |G|2Re<?1(£>1)
Л = 2йсо„ |G„ |2[Nlmexp -
I kv \kuv/
(5-74)
Formulae (5.54)-(5.57) given for the simplest polarizations of strong and
probe waves hold true also for Ф1(Х2М) if the substitution
S BjK-+BK, PiK-^PK (5.75)
j—m,n j=m,n
is performed. The general conclusions drawn in the course of the analysis
of relations (5.54)-(5.57) are also valid if the forms of BjK and fi)K are not
specified.
In the three-level system the non-linear resonance described by the
function Ф1(Х2М) in the expression (5.74) seems to be of a less complicated
306
Probe field spectroscopy under large Doppler broadening
[Ch. 5
structure thfln that of the two-level system (instead of the functions BmK,
BnK and fimK, finK only BK and appear).
Let us obtain the explicit forms of BK and /ft in the model of relaxation
constants*. From eqs (5.70) in this model it follows (the collisional shifts
are neglected) that
, , . 4 6(v - Vl)
/-Л»1",)
f . X _______________d(v - Vl)__________
Jn/K(v|v,;
rijK = Yi + Yj + ~
(5.76)
Integration over velocities in formula (5.73) under conditions of large
Doppler broadening results in the following values of the functions
B^Q^) and /ft(£ft) (compare with eqs (5.6) and (5.7)):
B ^2V^CX Г (Q*Vl __________________________J_______________.
k,tv eXpL UMv/ J Гтк rmll + (кц/к)Г} - i[D„ T (k,Jk)Q\ ’
(5-77)
__ 2 Ул / \ __________Орк__________________1________
k)tv PL Xk^v/]Гп1к + [(кц-к)/к]Г1-1ЕГтП + (кц/к)Г1-1£
e = Qi1—^Q; r^r,^, кц>к. (5.78)
The upper sign in formula (5.77) corresponds to unidirectional waves
(A^tfA). The expression (5.78) for /ft corresponds to the case к„ ftfc. In
the opposite case (кц ft k) the interference terms vanish (J3K = 0).
When the quantities BK describe in the non-linear resonance a
spectral component of Lorentzian shape with a halfwidth ГтП + Цк^/к
independent of field polarization (of к). The interference terms fiK bring
about the appearance of components with halfwidths Г„1К + Г[(кц - к)/к
where Г„,К is a relaxation constant of the polarization moment (of rank к)
of the forbidden transition n-l. When polarization conditions alter, the
weights of individual terms fiK in the non-linear resonance characterized
by the function g?i(£2M) are changed by formulae (5.53)-(5.57) (if in these
formulae the substitution (5.57) is performed). Accordingly, the weights
of spectral components with «--dependent halfwidths also change.
* For the model of a difference kernel the functions and p„ are derived in ref. [21].
§5.4]
Polarization phenomena
307
Under some conditions the Green’s functions in eqs (5.73) may prove to
be independent of к. For example, Гтк and Гп1к do not depend on к if the
relaxation is radiative. In such cases the relation between sums (over Kq)
of interference terms and sums of population terms is independent of the
polarization of the field; polarizations affect only the total amplitude of
the resonance. This statement immediately follows from the identity (see
problem (31))
S аРк |/(^)|2 = % аВк1ц(кд)1*(кд).
кд Kq
Thus the change of field polarization cannot suppress or intensify the
interference term compared with the population term. In this respect
resonances of the probe field in three-level and two-level systems differ
greatly. Physically, this can be explained by the following: in the case of
three levels, transitions between magnetic sublevels effectively break into
a set of three-level subsystems, whereas in the case of a two-level system
both three-level (figs 5.9c,d) and two-level (figs 5.9a,b) subsystems
appear.
When analysing the propagation process of a probe wave, i.e. in
essence non-linear optical phenomena, we deal with some interesting
events. The polarized strong field induces anisotropy of the medium. The
medium becomes uniaxial (under linear polarization of the strong field) or
gyrotropic (when polarization is circular) and it features such properties as
dichroism and double refraction characteristic of anisotropic media.
It is known that in anisotropic media waves with a quite definite
polarization state, the so-called normal waves, propagate without polari-
zation changes. Under anisotropy induced by a strong external field
normal probe waves are polarized either as the strong field or or-
thogonally. If a strong field is linearly polarized the normal probe waves
also have linear polarization, parallel and perpendicular to the strong
wave polarization. If the strong wave is polarized circularly the normal
probe waves are characterized by left-hand and right-hand circular
polarization. It is for normal waves that it is possible to speak about such
characteristics as the index of refraction and absorption (amplification)
factor. Probe waves polarized in a way different from the normal waves
change their polarization states as they propagate.
A normal wave propagating along the x axis is characterized by the
following space and time dependence:
ЕЦТ = ехр[-1(сом/ - k^x) + i &kT x]. (5.79)
308
Probe field spectroscopy under large Doppler broadening
[Ch. 5
Here r is the index of the normal wave and denotes the amplitude of
the electric vector; the value AfcT is a complex part of the wavevector as a
result of the interaction of the probe wave with the medium. The dipole
moment (r(lr)) induced at the frequency of the normal wave and the
quantity A£T are proportional to each other:
<r(lt)> = AG,T AkT = iAG^A + AT);
= ^Td/2ft; d = dmn, dmr, (r) = (rmn), (rml). (5.80)
In the relation (5.80) there is a component A that is responsible for probe
wave interaction with the medium in the absence of a strong field and is,
consequently, isotropic. The quantity AT containing the normal wave index
is assumed to be proportional to the non-linear part of the dipole
moment.
Let AT be so small that the condition
ATl«l (5.81)
is fulfilled where I is the geometrical path for the probe wave. The electric
field of a normal wave as it propagates in a medium of length I and its
intensity under the condition (5.81) have the forms
E„T(t,1) = exp[-i(<M - кц1) - A/](l - ATl), (5.82)
U0 = |^|2e-M''(l-2/A;),
A' = Re А, Л; = Re kT. (5.83)
The factor exp(—Al) in eq. (5.82) describes absorption and dispersion in
the absence of the strong field. The strong field accounts for the
appearance of additional components proportional to respectively AT and
Л' in the relations (5.82) and (5.83). The non-linear part of the work done
by the probe field, which has already been calculated, is obviously
proportional to the quantity A’T. Experimentally it may be recorded on
separation of the part of the intensity of the radiation which has passed
through the medium and which is due to the strong field (it is proportional
to 2lk'T). This separation is usually performed using the so-called modula-
tion method. The strong field is amplitude modulated and the probe field
radiation is registered by a synchronous detector at the modulation
frequency.
If the probe field polarization differs from that of the normal wave, to
describe the propagation process one may expand it in normal waves,
each of which changes in the fashion given by relation (5.82).
§5.4]
Polarization phenomena
309
Let us consider for a definite example a strong field that is circularly
polarized and a probe field that is polarized linearly; the wavevectors кц
and к are collinear. Normal waves are the waves of circular polarization
(t= ±1). In this case their amplitudes are the same.
The total intensity of the probe wave as it leaves the medium is
W = I Ve-2A"[1 - 2/(Л; + (5.84)
The intensity change due to the strong field is proportional to A J + AL,.
When the probe field polarization is arbitrary, linear combinations (with
coefficients of the same sign) of the absorption factors A] and AL, of
normal waves are recorded.
It has already been noted that, as the probe wave passes through an
anisotropic medium, not only is the total intensity changed but also other
characteristics: the absorption of normal components of the wave takes
place in different ways, and the polarization state changes. For instance,
under linear polarization of the probe field and circular polarization of the
strong field, the intensities of the normal components are equal at the
input (before the interaction with the medium) and are different at the
output. The difference of intensities of normal components is according to
relation (5.83) given by
Ul) -|^|2/(A;-AL1)e-2A', (5.85)
i.e. it is proportional to the difference of логта! wave absorption factors.
The polarization at the same time changes from linear (before the
interaction with the medium) to elliptic with the simultaneous rotation of
the ellipse’s axes. The parameters of the ellipse (angle of rotation and
axial ratio) can be obtained from formula (5.82). The corresponding
results are summarized in Table 5.1. The axial ratio of the ellipse is
defined as the ratio of the minimum and maximum intensities of
radiation which passed through the analyser as its axial orientation was
being varied.
If the axis of an analyser is oriented orthogonally to the polarization
plane of radiation at the input (polarizer and analyser are perpendicular
or crossed), then in the absence of a strong field the radiation fails to pass
through the analyser. However, when the medium is affected by a strong
field then as a result of the effects of dichroism and double refraction the
signal after the analyser differs from zero and is proportional to
е"2л/|Ai - A-J2/2. (5.86)
Similar considerations can be useful when discussing linear polarization
310
Probe field spectroscopy under large Doppler broadening
[Ch. 5
Table 5.1
Polarization states Gt GMZ G+ GJ
Non-linear part of absorption e-^ + Al)/ е~2ЛУ(А; + AL,)/
Difference in intensities of normal waves е-2л''(Л[|-Л;)/ е 2Л '(А; - AL,)/
Angle of rotation of the polarization ellipse i(An-Al)/ iW-A'-iX
Degree of ellipticity l(Afi-A’x)2/2 -1(a; - al,)2/2
Signal after the crossed analyser е-2А''|Ац-Ах|2/2 е-2л /|А, - A_,|2/2
of the strong field. Normal waves will be waves with electric vectors ЕцП,
Ец± directed in parallel with or orthogonally to the electric vector of the
strong wave: the symbols Ah and Ax respectively are employed.
The calculated results for various characteristics of the probe radiation
discussed above and obtained using the formulae (5.82) and (5.83) are
given in Table 5.1.
Thus, on measuring the above-mentioned radiation characteristics (see
Table 5.1), various combinations of the absorption factors A' and
refractive indices A" of normal waves can be recorded.
The relation between the quantities At, A_15 AH and A± and the atomic
characteristics may be obtained by using the expressions (5.48) and (5.71)
for dipole moments induced at the probe field frequency. Thus, when we
deal with a two-level system, the quantities A„, A,, A± and A_t are
proportional to <p(DM) defined by the formulae (5.54)-(5.57) (in the same
order) with the proportionality factor |G|2M. For three-level systems the
relations between the quantities AT and fiK, BK are similar.
Table 5.2 shows the connection between characteristic combinations of
the macroscopic quantities AT which are pronounced in certain measure-
ments (see Table 5.1) and the microscopic characteristics of the medium
BK and /V It is clear from the results presented that under different
polarization conditions entirely different sets of quantities BK and are
measured. The most interesting case is when only one term (Bx, B2)
assumes significance. The broadening of the corresponding non-linear
resonances is, consequently, caused by the relaxation of one of the
polarization moments of rank к = 1 or 2.
§5.5]
Recoil effect
311
Table 5.2
Polarization of the strong field Macroscopic characteristics M* Mt*
^11 Bo + 2B2 Bo + 2B2 + 0O + 202
A± Bo — B2 Bo ~ B2 + 201 + 202
T Ац-А± 3B2 3B2 + 00 ~ 201 + 202
Ay + A j. 2Bo + B2 2B0+B2 + 00 + 201 + 202
B0 + 2^1 + 2&2 B0 + lBi + iB2 + 00 + 201 + 202
A-i Bq ~ 2B1 + 2B2 Bo — 2.B1 + 2B2 + 302
Ai — A_! 3B. 3Bj + 00 +101 ~ 202
A] + A_i 2BO + BZ 2B0 + Bo + 00 + 201 + 202
The fact that some quantities BK and or their minimum sets are
recorded considerably simplifies the problem of investigating microscopic
properties of the medium. In this respect the use of polarization
phenomena seems to be an important element of future studies.
5.5. Recoil effect
The radiative transition of an atom between the states m and n is
accompanied by change in its velocity because of the finite value of the
photon momentum hk. By the law of conservation of momentum
mv =mvi + hk, (5.87)
where v and Vj are atomic velocities before and after emission of the
photon. Keeping in mind the law of conservation of energy at the
transition of an atom between states m and n, i.e.
Em+mv2/2 = E„ + mv2/2 + hco, (5.88)
we obtain the resonance value of the frequency
co = comn +k • v - д = го,™ + к • Vj + д, d = hk2/2m. (5.89)
As well as the Doppler shift of the frequency к • v there is an additional
shift d which is called the shift due to the recoil effect. The numerical
312
Probe field spectroscopy under large Doppler broadening
[Ch. 5
value of 8 is given by
<5 = 1.25 X 106A-2A/-’ s-1 = 0.62Л-2Л/-’ MHz, (5.90)
where the wavelength Л is expressed in microns and M is the atomic
weight. In the X-ray region and for у quanta shift 8 is greater than the
line width and the recoil effect can easily be recorded. In the optical region
of spectrum (Л = 1 pm) for M — 10 we have d = 105s-1 which is con-
siderably less than the Doppler width. However, at low pressures the
lifetimes of molecular states may be large enough for the shift of
non-linear resonances due to the recoil effect to be of the order of their
widths. Under such conditions the recoil effect markedly affects the shape
of non-linear resonances.
Consider the interaction of an atom with a plane travelling monochro-
matic wave. In the model of non-degenerate states the equations for the
density matrix in the Wigner representation have the form (see eqs (2.53);
compare with eq. (4.7))
rmpm(v) = -2 Re[iG*p(v - kk/2m)] + QmW(v), G = -dE/2k,
Г„р„(у) = 2 Re[iG*p(v 4- kk/2m)] 4- QnW(v),
[Г— i(Q — к • u)]p(v) = —iG[pm(v 4- hkl2m) — pn(y — A4/2zn)],
(5.91)
Pmn{r, v, t) = p(v) exp[-i(£2r - к • r)], Q = <o - <omn.
From eqs (5.91) we obtain the velocity distribution of the atoms for upper
(m) and lower (n) levels:
pm(v) = NmW(v) — — |G|2
NmW(v) — N„W (y — 2if)
r3 + (Q-k-v + <5)2 ’
pn(v) = N„W(v) + — \G\2
n
NmW(v + 21?) - N„W(y)
r3 + (Q-k-v -<5)2 ’
hk
4 = 2^’
(5-92)
(5.93)
(5-94)
Let us consider the definite case Nm> N„. According to relations (5.92)
and (5.93), at the levels m and n a Bennett hole and peak with a width Ц
appear. The hole is displaced from the peak by 281к because of the recoil
effect as shown in fig. 5.10. The arguments of the functions W(y ± 2tj) in
§5.5]
Recoil effect
313
Fig. 5.10. The distribution of the atoms in velocities v at the levels (a) m and (b) n.
the relations (5.92) and (5.93) reflect the change in atomic velocity under
the radiative transition.
For the work done by the field we obtain the following formula:
P(Q) = -2й<о Re(iG*p(v))
= 4ft<o|G|2T^
N„W(v)
r? + (Q-k-v- <5)2
NmW(v)
I? + (Q - к • v + <5)2
(5.95)
The resonances of the two terms of the expression (5.95) corresponding to
absorption and induced emission are displaced owing to the recoil effect
by a distance ±<5. In the optical spectrum kv » d and the recoil shift can
be neglected. The only exception is the case of very close population
values when \Nm-N„\<z(Nm+N„)d. However, this being so, P(Q) is
very small. Therefore, the shape of the spectral line in the case of a
travelling wave is insignificantly affected by the recoil effect, and
non-linear phenomena consisting in the |G|2 dependence of Ц are
consistent with this conclusion.
Non-linear resonances in the case of a standing wave have not a
Doppler but a spontaneous or an impact width. Therefore they are of
great interest for the detection of recoil-induced shifts.
314
Probe field spectroscopy under large Doppler broadening
[Ch. 5
We shall confine ourselves to the analysis of first non-linear corrections.
It was shown in section 4.2 that in this approximation countertravelling
waves forming a standing wave contribute independently to py(v).
Therefore, from the formulae (5.92) and (5.93) we obtain
pm(v) = W(v){^ - (Nm - N„) x
Г p2 p2 11
.Г2 4- (Q - Л • V 4- <5)2 + Г2 4- (Q 4- Л • V 4- <5)2 J J ’
p„(v) = W(v)(y. + (Nm - N„) x
г Г2 Г2 'll
.Г2 4- (Q - Л • v - <5)2 + Г2 4- (Q 4- Л • v - <5)2 J J' (5’97)
The standing wave leads to the emergence of two holes in pm(v) and two
peaks in p„(v) (if Nm > N„). The holes and peaks are symmetric about the
point к • v = 0, but the distance between the holes differs from that
between the peaks by 48.
At certain values of the field frequencies holes (or peaks) take up the
same position. This frequency value for the holes is £2г = — 8 and for the
peaks it is Under such values of the field frequency the population
change caused by one travelling wave is resonant for the other travelling
wave. Therefore, work done by the field as a function of the frequency £2
must possess two minima. Assuming <5, r«kv we can arrive at the
relation
P(Q) = 4Vn^(W„-W„,)|G|2lexp ~(Q,~7,|G|2Z(^)I,
/С17 v L \ /С (/ ' J Л. J
(5.98)
z . Г /Щ2Ц\ г2 т„ г2 i
*(fi)~exp[ ][1+ т r2 + (Q+<5y+ гГ2 + (д_^]’
(5-99)
These formulae show that P(£2) has two minima at the frequencies
£2= ±8. The minima are caused by the coincidence of either Bennett
holes or peaks in accordance with the aforegoing. The depths of the
§5.5]
Recoil effect
315
minima are proportional to the lifetimes rm and r„ of atoms at the levels m
and n. This is quite natural as the population change of a level j = m, n is
greater for a more prolonged interaction of the field with an atom at the
level j, i.e. for greater r,-.
In deriving the formula (5.98) all the Doppler factors in the non-linear
term were assumed equal to exp[—Г22/(А:и)2], since this term is a small
correction. In the linear term the shift of the Doppler contours due to the
recoil effect was allowed for. As a result, the maximum of the linear
absorption proves to be displaced by a value d,, and <5i may considerably
exceed d if Nm + N„ » |Nm - N„\.
According to the analysis of section 4.2, the non-linear resonance of the
work done by the field P(Q) causes the appearance of a non-linear
resonance in the plotted frequency dependence of the output power of a
single-frequency gas laser (the so-called Lamb’s dip). The latter is
described by a function !/%(£?) and its shape depends on recoil effect. If
Г<д, the Lamb’s dip is split into two components with different
amplitudes (тт^т„). At Г>д the recoil doublet components merge to
form a single asymmetric dip whose centre Q' is displaced by the value
(5.100)
In existing gas lasers the halfwidths of the active medium lines
considerably exceed d and the recoil effect is not manifested. Neverthe-
less, in gas lasers with absorbing cells the linewidth of the absorbing gas
may be much smaller and the conditions are better for recoil effect
observation.
For a laser with a non-linear absorbing cell the function %(£?) in
formula (5.98) has the form
Z(<2) =
' Л2тШ1/т, Г?т„1/Т1
. П + ^ + д,)2 П + ^-д,)2-
। Тг^иг/ т2 ти2/ т2
П + (^2 + <52)2 п + (fi2 - <52)2-
(5.101)
Here the indices 1 and 2 denote respectively the characteristics of the
amplifying and absorbing gases. The coefficient a depends on many
parameters (the ratios of the lengths of the absorbing and amplifying
316
Probe field spectroscopy under large Doppler broadening
[Ch. 5
media, the ratios of the values of the non-saturated absorption coefficients
etc.) and its particular value is of no importance for us at present.
The first term in eq. (5.101) describe non-linear resonance of the work
done by the field and the Lamb’s dip. The absorbing gas (the second term
in eq. (5.101)) leads to a doublet resonance of the opposite sign which is
close to the Bohr frequency of the absorption line and the halfwidths of its
components are equal to Г2. Let the conditions Ц » and Ц » <52 > Г2 be
satisfied. Then the laser radiation power as a function of frequency is
characterized by a doublet resonance as shown in fig. 5.11. If Д » Г2 > д2
the plotted laser radiation power has a single peak which, generally
speaking is asymmetric at tm2 & rn2.
Non-linear resonance splitting due to the recoil effect was theoretically
predicted in 1968 [22,23] and experimentally observed in 1975 [2,24]. In
subsequent papers the influence of different factors on the recoil effect was
studied, i.e. the transit effects, spontaneous transitions, high radiation
power etc. (see ref. [25]).
The ratio of recoil-induced shift to frequency co is one-half the ratio of
the photon energy to the rest energy of the atom:
д/co = hco/2mc2.
Fig. 5.11. The work done by the field versus the frequency for the system with an absorption cell.
References
317
In optical spectra д/(о ~ 10-10. Historically, the recoil effect was the first
very refined effect for optical spectroscopy. Its analysis demonstrated the
possibilities of laser spectroscopy and gave a stimulus to studies requiring
a resolution power of 1010 or higher. In particular, the spectral line shift
has been found to be due to a quadratic Doppler effect. This shift is given
by [2]
A<o/<o = -Time2 = -0.9 x 1(T13T7M
(T° is the temperature in kelvins and M is the atomic weight), i.e. it is less
by a factor of 102 than the shift due to the recoil effect.
Light pressure is a mechanical manifestation of the recoil effect. Under
sufficiently high intensities standing wave nodes can be treated as potential
holes which under certain conditions can trap atoms. In this case the atom
is localized in space and its emission (or absorption) spectrum has no
Doppler broadening. If the frequencies of countertravelling waves are
different the nodes of the corresponding quasi-standing wave move. In
this way neutral particles can be accelerated. The above-mentioned
phenomena as well as many others associated with recoil effects in a
powerful light field are discussed in refs [26,27] (see also section 7.2).
References
[1] S.G. Rautian and A. A. Feoktistov, Zh. Eksp. Teor. Fiz. 56 (1969) 227 [Sov. Phys. JETP 29
(1969) 126].
S.G. Rautian, G.I. Smirnov and A.M. Shalagin, Zh. Eksp. Teor. Fiz. 62 (1972) 2097 [Sov. Phys.
JETP 35 (1972) 1095].
[2] V.S. Letokhov and V.P. Chebotaev, Nonlinear Laser Spectroscopy (Springer, Berlin, 1977).
[3] K. Shimoda, in: Laser Spectroscopy of Atoms and Molecules, ed. H. Walther (Springer, Berlin,
1976), p. 197.
[4] G. Grynberg, B. Cagnac and F. Biraben, Multiphoton resonant processes in atoms, in: Coherent
Nonlinear Optics, eds M.S. Feld and V.S. Letokhov (Springer, Berlin, 1980).
[5] T. Hansch, R. Keil, A. Schabert, Ch. Schmeizer and P. Toschek, Z. Phys. 226 (1969) 293.
[6] A.K. Popov, Zh. Eksp. Teor. Fiz. 58 (1970) 1623 [Sov. Phys. JETP 31 (1970) 870].
[7] T.Ja. Popova, A.K. Popov, S.G. Rautian and R.L Sokolovski, Zh. Eksp. Teor. Fiz. 57 (1969)
850 [Sov. Phys. JETP 30 (1970) 466].
[8] S.G. Rautian, Tr. Fiz. Inst. Akad. Nauk SSSR, 43 (1968) 3.
[9] V.P. Kochanov, S.G. Rautian, E.G. Saprykin and A.M. Shalagin, Zh. Eksp. Teor. Fiz. 70
(1976) 2074 [Sov. Phys. JETP 43 (1976) 1082].
[10] W. Heitler, Quantum Theory of Radiation (Oxford University Press, Oxford, London, 1960).
[11] R.B. Mollow, Phys. Rev. A 2 (1970) 76.
[12] F. Schuda, S. R. Stroud, Jr., and M. Hercher, J. Phys. В 7 (1974) L198.
[13] H. Walther, in: Laser Spectroscopy, eds. S. Heroche, J.C. Pebay-Peyroula, T.W. Hansch and
S.E. Harris (Springer, Berlin, 1975).
318
Probe field spectroscopy under large Doppler broadening [Ch. 5
[14] R.E. Grove, F.Y. Wu and S. Ezekiel, Phys. Rev. A 15 (1977) 227.
[15] P.L. Knight and P.W. Milonni, Phys. Rep. 66 (1980) 21.
[16] A.K. Popov and G.Kh. Tartakovski, Opt. Commun. 18 (1976) 499.
[17] Im Tkhek-de, O.P. Podavalova, A.K. Popov and V.P. Ran'shikov, Opt. Commun. 30 (1979)
196.
[18] N. Bloembergen, A.B. Bogdan and M.W. Downer, Laser Spectroscopy, Vol. V (Springer,
Berlin, 1981), p. 157.
[19] L.J. Rothberg and N. Bloembergen, Phys. Rev. A 30 (1984) 820.
[20] Y.H. Zou and N. Bloembergen, Phys. Rev. A 34 (1986) 2968.
[21] S.G. Rautian and D.A. Shapiro, Zh. Eksp. Teor. Fiz. 94 (1988) 110 [Sov. Phys. JETP 67
(1988) 2018].
[22] A.P. Kol’chenko, S.G. Rautian and R.I. Sokolovski, Zh. Eksp. Teor. Fiz. 55 (1968) 1864 [Sov.
Phys. JETP 28 (1969) 986].
[23] F.A. Vorob’ev, S.G. Rautian and R.I. Sokolovski, Opt. Spektrosk. 27 (1969) 728.
[24] J. L. Hall, C. J. Bordi and K. Uehara, Phys. Rev. Lett. 37 (1976) 1339.
[25] M. Demtroder, Laser Spectroscopy (Springer, Berlin, 1982).
[26] A.P. Kazantzev, Usp. Fiz. Nauk 124 (1978) 113 [Sov. Phys. Usp. 21 (1978) 58].
[27] P. Meystre and S. Stenholm, eds, The Mechanical Effects of Light, in: J. Opt. Soc. Am. 2B(11)
(1985) 1706.
6
Light-induced drift of gases
6.1. Qualitative description of the light-induced
drift
The equations of gas kinetics derived in section 2.7 show that radiation
may be a source of a non-equilibrium state with respect to the transla-
tional degrees of freedom of gas particles. Such non-equilibrium velocity
distributions lead to some new phenomena of gas kinetics. The most
interesting phenomenon is light-induced drift (LID) [1] when under certain
conditions in the radiation field the components of the gas mixture drift
with respect to each other.
Whereas a systematic description of LID based on the equations of
section 2.7 will be given later, we shall first qualitatively consider the
physical causes of this phenomenon.
Suppose that a gas system consisting of two components is irradiated by
a travelling quasi-monochromatic wave. The radiation frequency (o is
close to Bohr’s frequency <om„ of the transition of one component from the
ground state n to the first excited state m, and the second (buffer)
component does not interact with the field. As is known (see section 4.1),
interaction with radiation is selective in velocities because of the Doppler
effect, i.e. the radiation causes transitions to the excited state only of the
particles with velocities close to the “resonance” values satisfying the
condition к • v = (o — a>mn = Q. The effective range Av2 of the velocity
projections vz on the wavevector it-is given by the homogeneous half width
of the absorption line Г: Avz ~ Г Ik. The change in atomic velocity due to
the finiteness of the photon momentum (recoil effect) will be neglected,
i.e. photon absorption causes the particle transition to the excited state
leaving its velocity unaltered. As long as collisions are not pronounced,
radiation produces non-equilibrium velocity distributions in the ground
and excited states (see fig. 6.1) which add up to the initial Maxwellian
distribution.
Now we shall concentrate our attention on the possible consequences of
a non-equilibrium velocity distribution caused by the radiation at different
319
320
Light-induced drift of gases
[Ch. 6
Fig. 6.1. Velocity distribution of particles in ground (p„„(vz)) and excited (pmm(vz)) states in the
absence of recoil effect and collisions.
levels (Bennett hole and peak in fig. 6.1). Under the non-
equilibrium distribution at the level m is asymmetric with respect to
vz = 0. This implies, in particular, that there is a non-zero flow of excited
particles jm. Therefore, when £?=/=(), radiation induces opposing flows of
excited and unexcited particles that are equal in magnitude: jn = -jm. This
situation holds until the collisions with buffer particles manifest
themselves.
Evidently, the flow of one gas component with respect to the other must
be decelerated as a result of collisions. Therefore the flows of excited and
unexcited particles in the buffer gas medium must be decelerated, the
forces of deceleration being proportional to the flows: Fib =
(i = m,n) (see eq. (2.362)). Let us now direct our attention to the fact
that particles in different quantum states may be treated as different
particles, and for them any physical characteristics in the general case has
different values. In particular, excited and unexcited particles have
different transport cross-sections of collisions with buffer gas particles.
This leads to the different proportionality coefficients v” and v" of the
flows and the corresponding friction forces. Since radiation induces
opposing flows equal in magnitude, the friction forces Fmb and Fnb have
different magnitudes owing to the difference between v™ and v". Their sum
§6.2]
General laws of light-induced drift
321
<
<-
e
e
<-
<
<-
Fig. 6.2. Illustration of the initiation of LID motion. The curved arrows indicate radiative transitions.
F = Fmb + F„b is the resulting force acting on the absorbing particles from
the buffer particles and it is not equal to zero. By the force F the gas of
absorbing particles as a whole is set in motion with respect to the buffer
gas. This is the physical reason for the LID phenomenon. Figure 6.2
illustrates the formation of LID. The difference between v™ and v" is
schematically shown as different dimensions of the particle in the states m
and n.
The principle role of the buffer gas must be specially emphasized.
Suppose it is absent; then absorbing particles may collide only with each
other. The radiation-induced flows jm and jn at the same time may be
decelerated only mutually. By the law of conservation of momentum, the
absorbing gas as a whole in this case is at rest. Therefore LID is possible
only in the presence of the buffer component. The law of conservation of
momentum manifests itself in the oppositely directed motion of the buffer
and absorbing components.
The direction of the motion (drift) of absorbing particles depends on
the relation between v” and v" and on the sign of Q. For example, if
v” > v" and Q < 0, drift takes place along the radiation direction; at Q > 0
the drift has the opposite direction. With Q = 0 (absorption line centre)
there is no drift, which can be attributed to symmetry.
6.2. General laws of light-induced drift
Consider the principal features of LID on the basis of the macroscopic
kinetic equations obtained in section 2.7. For simplicity, take the
322
Light-induced drift of gases
[Ch. 6
concentration of buffer particles to be much greater than that of the
absorbing particles (Nb » N) so that collisions of absorbing particles with
each other can be neglected. In addition, we shall restrict ourselves to the
model of velocity-independent transport frequencies of collisions where
the relationship between friction forces and particle flows is the simplest:
Fib = -mv‘Ji, i = m,n. (6.1)
Here the v\ are the transport frequencies of the collisions of absorbing
particles in the state i with buffer particles.
From general considerations as well as directly from eqs (2.337) after
substituting there expressions (6.1) for Fib it is clear that the flows relax or
reach the steady state over a time of the order of the mean free time
r, ~l/vi. As a rule, this time is much less than the change time of
external conditions and macroscopic characteristics of the medium.
This circumstance enables us to neglect in eqs (2.337) the terms
containing the derivatives of flows with respect to time as usual when the
transport and some other problems of hydrodynamics are being analysed
[2, 3]. Equations (2.337) in this case take the form
(rm + vT)jm+-^‘Pm = NQ,
m
— V • P = (v? - vT]jm - v"j,
m
V-(P+Pb) = 0, j=jm+jn. (6.2)
From these equations the existence of LID immediately follows. Indeed,
suppose that the gas system in its initial state was spatially homogeneous.
At early stages of the process the terms with space derivatives in eqs (6.2)
may be neglected. From the first equation it follows that
jm = NQ/(rm + vT), (6.3)
i.e. the flow of excited particles arises if Q #=0. The latter holds when the
v dependence of p(y) is asymmetric, i.e. under optical excitation
asymmetric in velocity. As is known, this very picture is typical of a gas
interacting with a narrow-band radiation. The latter of eqs (6.2) shows
that as soon as jm #= 0 and the values of v” and v" are different the flow J
of the absorbing gas as a whole is also non-zero:
.* v? - vf . V" - V? NQ
J v" Jm Vl rm + vT' ( )
§6.2]
General laws of light-induced drift
323
(6-5)
To the flow j corresponds the drift velocity и of absorbing particles:
= J v? - vT Q
U N vl rm + vT'
On emergence of the drift motion the process development depends on
the boundary conditions. For example, in an absorbing cell with open
ends stationary flow is established with spatially homogeneous partial
concentrations. The initiated flow (6.4) will not be changed further, so the
expression (6.5) for the drift velocity also holds true for the stationary
mode. If there is an obstacle to the flow, i.e. if at least one end of the cell
is closed or connected with a limited volume which is not connected with
the second end of the cell, LID brings about a non-homogeneous
distribution of the absorbing particle’s concentration.
According to eq. (6.2), as the concentration is redistributed, the general
pressure tensor P + is spatially homogeneous. Now we shall ignore the
effect of pressure anisotropy (see section 7.4) as this effect is of a higher
order of smallness. This means that the tensors P and P reduce to scalars
(partial pressures). Therefore under such conditions the LID-induced
spatial separation of the components of the gas mixture takes place
without violation of total pressure homogeneity. In this respect LID is
similar to the diffusion process which has, however, a reverse sequence.
Indeed, an ordinary diffusion process consists in the fact that the
concentration gradient of one of the components due to external
conditions creates the diffusion flow of this component which, in its turn,
equalizes the concentrations. For LID in the cell with an obstacle the
component flow is typically initiated by radiation in the originally
homogeneous mixture and then the drift of the component brings about
its concentration gradient. In both cases the total pressure is practically
unperturbed.
The above analysis of the causes of LID shows that the effect is greater
for larger relative differences between the transport frequencies of
collisions v” and v" because of the potential difference in interaction of
excited and not excited particles with buffer particles. On the contrary, it
is the difference of these potentials that causes the phase shift of an atomic
oscillator during collisions. Consequently, from the point of view of the
LID effect, the particles which are characterized by the absence of phase
memory during collisions are the most interesting. Under these condi-
tions, the in-term of the non-diagonal collision integral may be dropped.
In this approximation the equation (2.314) for the non-diagonal element
of the density matrix in the case of a travelling monochromatic wave takes
324
Light-induced drift of gases
[Ch. 6
the form
pmn(v) = p(v) exp[—i(X2r - к • r)], Vmn = -G exp[-i(£r - к - r)],
+ ® • V + у + v - i(O-*- v)lp(v) = iG[p„„(u) - pmm(v)].
Lot J
(6-6)
It is convenient to analyse LID as well as other light-induced phenom-
ena of gas kinetics on the basis of the macroscopic equations (2.335)-
(2.339). The source of the non-equilibrium state in these equations is the
quantity p(v) which, nevertheless, cannot be obtained within the scope of
these equations. To find p(v) one must refer to the initial equations
(2.314) for the density matrix. It may seem that this makes macroscopic
equations utterly useless, since if eqs (2.314) are successfully solved all the
medium characteristics can be obtained in a trivial way. However, direct
solution of eqs (2.314) entails great difficulties and is often impossible and
our suggestion really simplifies the problem. The point is that when gas
interacts with radiation processes of two types can be singled out whose
time scales differ significantly. The processes of the first type are fast:
establishment of polarization at the transition between internal states of
particles (according to eqs (6.6) the characteristic time of such processes is
l/(y + v) and less) and formation of non-equilibrium level and velocity
distributions. The processes of the second type are the time and space
changes of the macroscopic characteristics of the medium and radiation.
Such processes, as a rule, are much slower. For a description of the slow
processes the macroscopic equations (2.335)-(2.339) are especially con-
venient. In order to describe fast processes (actually only the calculation
of p(v) is implied) it is reasonable to assume that the problem is
stationary and spatially homogeneous. This means that p(v) is a function
of medium and radiation characteristics local in time and space.
Therefore, in order to obtain p(v) in eqs (2.313), (2.314) and (6.6) the
space and time derivatives may be neglected and we may use the system
of equations
^pmm(v) = sm(v) + Np(y),
Sm(v) + S„(v) = 0, Sb(v) = 0- (6-7)
2 IGI2(v + vl
^p(v) = v)2 + (Q_jk.v)2 [P««(v) “ P—(v)]. (6.8)
It must be recalled that these equations are written in the system of a
buffer gas at rest. The collision frequency v here is a real quantity. The
§6.2]
General laws of light-induced drift
325
imaginary part of v may be taken into account when necessary by means
of the substitution Q—» Q — v".
The above assumptions leading to eqs (6.7) and (6.8) are similar to
those used in ordinary gas dynamics and hydrodynamics when the local
equilibrium approximation is introduced. The difference consists only in
the fact that eqs (6.7) and (6.8) imply the possibility of a highly
non-equilibrium velocity distribution; therefore we should speak not
about local equilibrium but about local stationarity and homogeneity. It
will be readily seen that under p(v) = 0 (absence of radiation) relations
(6.7) go over into a formally written condition of local equilibrium.
We continue with the discussion of the LID effect itself. Note the
specific dependence of the drift velocity on the frequency detuning Q.
Populations Pn(v) do not change when the signs of both Q and v alternate
because of the symmetry of the problem (the velocity distribution of
buffer particles in the absence of radiation is isotropic and the absorption
line is symmetric in D). Taking this circumstance into account on the basis
of the definition (2.344) of Q and the expression (6.8) for p(v) we may
conclude that sign change of Q changes the sign of Q and subsequently
that of the drift velocity u. Such (antisymmetric) dependence of и on the
detuning Q is the distinguishing feature of the LID effect compared with
most other known effects associated with optical excitation (light pressure,
thermodiffusion etc.). It provides a good foundation for the experimental
separation of the LID effect from other effects even if this effect is
relatively small.
The general analysis of the role collisions play in radiation-gas
interaction which was carried out in the previous chapters shows, in
particular, that collisions with a buffer gas lead to a reduction of the work
done by a field of a given intensity. Consequently, the maximum value of
the work done by the field is attained under the minimum concentration
of the buffer gas. This evidently also holds for the quantity p(v) which is
connected with the drift velocity и (see the relation (6.5)). The coefficient
of proportionality between и and Q also increases as the buffer gas
concentration decreases, reaching in the limit the value (vf - v”)/v”f^.
Thus, according to the relation (6.5) the drift velocity и increases as the
concentration of the buffer gas decreases. At first sight there seems to be a
contradiction: when the concentration of the buffer gas formally tends to
zero the drift velocity not only fails to fall but attains the maximum value
whereas in the absence of the buffer gas there must be no LID effect.
Actually there is no contradiction if we take into account the assumptions
preceding the expression (6.5). First we assumed that the buffer particle
326
Light-induced drift of gases
[Ch. 6
concentration is greater than that of the absorbing particles and in the
second place we imposed the condition of local homogeneity which, in
particular, implies smallness of the mean free path with respect to the
minimum size of the gas container. When either of these conditions is
violated the formula (6.5) is modified (see refs [4-7]) and leads to the
obvious limiting result: there is no LID in the absence of the buffer gas.
Thus, the maximum value of the drift velocity is given by the expression
(6.5) when the concentration of the buffer gas in this expression formally
tends to zero. The value p(v) required for calculating Q must be obtained
by proceeding from eqs (6.7) and (6.8). In this case it is obvious that the
collision integral Sm(v) in the first equation (6.7) may be neglected.
However, strictly speaking, it is impossible to neglect completely the
collisions in calculating p(v). However small the concentration of buffer
particles may be (within the scope of the applicability of the equations, of
course), the collisions “connect” the velocity distributions pmm(v) and
p„„(y) by the second of the equations (6.7), forming a non-equilibrium
total distribution function.
Assume that the non-equilibrium part of the total distribution function
is not great. This holds if the collision characteristics of the levels n and m
do not differ significantly or if the radiation intensity is not too high. Then
calculating p(v) we may take
Pmm(v) +pnn(v) = NW(v), (6.9)
after which p(y) is uniquely obtained from the first of the equations (6.7):
, x - rm|G|2W(v) _ 8 |G|2
P(V) (rm/2)2(l + K) + (Q-k-v)2’ K It ' (' }
Hence, according to the definition (2.344), we obtain Q and then the drift
velocity:
к _ vl - v? к
“ = kV~2^-TTiy^lzW^]’
z=x + iy, x = Q/kv, у = Гту/Т+~к/2кь. (6.11)
The tabulated (e.g. in ref. [8]) function w(z) with z = i£ is given by the
formula (4.150).
This particular case shows that the expression for the drift velocity can
sometimes be obtained without rendering the collision mechanism con-
crete and direct solution of kinetic equations. The velocity drift in this
case is expressed in terms of the phenomenological parameters v" and v™,
§6.2]
General laws of light-induced drift
327
which in their turn have a simple relation with the well-known phenomen-
ological characteristics, the diffusion coefficients. Indeed, let us refer to eqs
(6.2), assuming that radiation is absent. When the state is weakly
non-equilibrium it may be assumed that
P = NT = Nmv2H. (6.12)
Then it follows from the second of the equations (6.2) that
jn = -D„VN, Dn = v2/2vt, (6.13)
i.e. the well-known diffusion relation. The quantity v" in this case is
immediately expressed in terms of the diffusion coefficient Dn of particles
in the state n. By analogy, v? may be shown to have the same relation
with the corresponding diffusion coefficient Dm.
From the relation (6.11) it is easy to estimate the value which may be
attained by the drift velocity. The coefficient k/(1 4- к) under a sufficiently
high radiation intensity approaches unity. The coefficient у Re[zw(z)] by
a suitable choice of the intensity and frequency of the radiation may be
made of the order of unity. If we now take that |v? — v”|/v" ~ 1 (large
relative difference of transport collision frequencies), then from the
expression (6.11) it follows that
u~v, (6.14)
i.e. the drift velocity under suitable conditions may attain a value of the
order of the characteristic velocity of the thermal motion.
Another simple case makes it possible to express the drift velocity in
terms of v" and v7; this requires no concrete choice of collision
mechanism. This case refers to a low radiation intensity but an arbitrary
concentration of buffer particles. Assume that radiation causes a weak
change in the ground state population. In the expression (6.8) forp(v) the
population pmm(v) may be neglected and the distribution p„„(v) may be
considered close to the Maxwellian distribution, so that
p(v) = ----2-1^1 ^+V?—£ W(v). (6.15)
(Г+ v)2+(Q-k-v)2 y v '
One more phenomenological collisional parameter (v) which charac-
terizes impact line broadening is included. The drift velocity in this case is
k.Vi-vT 2\G\2
и = - v-------—------;— Re[zH’(z)l, (6.16)
к vt rm + vTkv 1 v n K '
z = [Q 4- i(y 4- v)]/kv = x + iy.
328
Light-induced drift of gases
[Ch. 6
Practically, it is often useful to relate the drift velocity to the
independently obtainable quantity (p) (absorption probability or number
of absorption acts per unit time and per particle). This characteristic
coincides with the work done by the field (discussed in previous chapters)
to an accuracy of the coefficient htoN. On the basis of eqs (6.5) and
(2.344) this may formally be written
к v" — v™ (p)
- v---------------a>,
к vl Гт + vT
(6-17)
(6.18)
The factor <p introduced here (the normalized first moment of the function
p(v)) characterizes the degree of asymmetry and velocity selectivity of the
function p(v). This factor reflects the antisymmetric Q dependence which
is typical of LID.
The relation (6.17) is sure to hold also for the particular cases defined by
the formulae (6.11) and (6.16). The factor <p common to these cases is [9]
< p = Re[zw(z)]/Re[w(z)],
z = x + iy = (Q + irs)/kv, Fs = (y + v)Vl + к • (6.19)
Here Д is the homogeneous line halfwidth comprising the impact and
field-induced broadening.
Under large Doppler broadening (kv » Д) we have
< p = Q/kv (|Q|<jlv). (6.20)
For the case Fs » kv,
Qkv
<p~ri+Q2'
(6.21)
If |Q|» Fs, kv then, practically irrespective of the relation between Г$ and
kv, for <p we obtain
< p = kv/Q. (6.22)
Figure 6.3 shows the Q dependence of tp for several values of the
parameter у = rs/kv.
In the expression (6.17) for the drift velocity the factor <p and the factor
(v" — v”)/v7 are typical of the LID effect whereas (p) is a commonly used
optical characteristic. The type of the dependence of <p on the medium
and radiation parameters, particularly the collision mechanism, is not very
sensitive to their variation. On the contrary, the quantity (p) is highly
§6.3]
Light-induced drift as the realization of Maxwell's demon
329
Fig. 6.3. Plots of the function (1) rs/kv =0.02; (2) rs/kv =0.1; (3) rs/kv = 1.
sensitive to the type of relaxation process and in order to obtain it very
often the kinetic equations (6.7) and (6.8) need to be solved. Since,
however, {p) may be experimentally obtained from independent measure-
ments, the relation (6.17) assumes great practical importance.
Finally, note that the relation (6.17) including the factor tp according to
formula (6.18) is exact if the effective mean transport frequencies of
collisions of the form (2.303) appear as v" and v™.
[/ 6.3. Light-induced drift as the realization of
Maxwell’s demon
Attention is drawn to two unique properties of the system of equations
(2.335)-(2.339) describing the kinetics of the gas system interacting with
330
Light-induced drift of gases
[Ch. 6
the radiation field. Term-by-term addition of the two last equations of the
expressions (2.336) yields
d
— (mJ + mbJb) + V • (P + Pb) = 0, (6.23)
at
where the sum mj + mbjb is the total momentum per unit volume of the
gas mixture and P + P is the total pressure tensor.
Summation of two last equations of the expressions (2.338) yields
d
- (W + Ю + div(9 + qb) = 0, (6.24)
at
where W + Wb is the energy of translational (thermal) motion per unit
volume of the gas mixture and q + qb is the total flow of the translational
energy.
The equations (6.23) and (6.24) reflect the laws of conservation of
momentum and energy. It is interesting that there is no source of external
forces in eq. (6.23) and no external source of the energy in eq. (6.24), i.e.
the form of these equations is the same as that of the equations for the
free gas system. This means that the total momentum of the gas system
and the total energy of translational motion of gas particles remain
unchanged*.
Thus the equations (2.335)-(2.339) describe a specific class of phenom-
ena which can occur under conditions when the momentum and transla-
tional energy of the particles of the gas medium are motion integrals. In
particular, if the LID phenomenon is considered, radiation “provokes”
the momentum exchange between the gas components which results in the
motion of the components with respect to each other. The energy of this
relative motion is evidently drawn from the thermal energy of the gas.
Qualitative considerations also bring us to the conclusion that the gas
momentum and translational energy of the gas particles will be conserved.
In our description of the radiation-gas interaction the recoil effect (light
pressure) was neglected and consequently the action of the radiation force
on the gas was excluded. This suggests the conservation of the momentum
of the gas system. Furthermore, we confined ourselves to the considera-
tion of elastic collisions only, which implies that there are no processes of
radiation energy dissipation into heat. Absorption of a quantum of
* In the general case we should speak about the conservation of momentum and energy not only of
the gas but also of its container whose walls determine the boundary conditions for eqs (6.23) and
(6.24).
§6.3]
Light-induced drift as the realization of Maxwell's demon
331
radiation accompanied by a spontaneous emission is a process of
scattering. The energy of the quantum in this process remains practically
unchanged. The radiation field including the incident and scattered
radiation therefore loses no energy. The radiation energy only temporarily
stays in the gas medium as the excitation energy, but as a result of the
elasticity of collisions it is not converted into translational energy of gas
particles.
Therefore the phenomenon of LID may be treated also as a specific
radiation effect on the motion of gas particles which causes gas com-
ponents to move and leads to their separation, no momentum or radiation
energy being expended on this motion. In this respect the action of
radiation in the LID effect is absolutely similar to that of the hypothetical
Maxwell’s demon.
Indeed, let us have two vessels connected with each other and
containing mixture of two types of gas. Under equilibrium the concentra-
tions of the two components are homogeneous. Before LID was dis-
covered, no process in the course of which one component could be
accumulated in one container and the other component in the second
container had been considered possible. Such a situation used to be
associated with Maxwell’s demon which “acts upon” the point of
connection of the vessels (see fig. 6.4a). Recall that the demon would by
means of the shutter stop the particles of one type coming into one
container, at the same time not letting the particles of the other type into
the second vessel. It is considered that ideally the demon could do this
without energy expenditure.
Now that we understand the nature of LID we can state that such a
process can exist without diabolic interference. That is, radiation could
easily act as Maxwell’s demon, separating the gas component by means of
the LID effect (see fig. 6.4b).
The relative motion and separation of the gas mixture components is
accompanied by decreasing gas entropy (“chaos” is replaced by “order”).
In the closed system of gas plus radiation field, according to the second
principle of thermodynamics the entropy must not decrease. The decrease
in gas entropy is automatically compensated by the increasing radiation
entropy: as it is scattered the ordered directed radiation is transformed
into isotropic, i.e. disordered, radiation, which is reflected by the entropy
increase. Therefore the only exchange possible between radiation and
medium in LID is the entropy exchange.
Under experimental conditions radiation-gas interactions can certainly
be accompanied by some dissipation of radiation energy. This is especially
332
Light-induced drift of gases
[Ch. 6
Radiation
(Ы
О О О О
oj ° о°
° © О
о ° ° л
О* ° ©
о ° ° о
о • О
Fig. 6.4. (a) A Maxwell demon is sorting out particles; (b) the same result can be achieved when a
mixture is exposed to radiation causing the LID effect.
characteristic of the excitation conditions of vibrational states of mole-
cules. Energy dissipation, nevertheless, is an. accompanying process
'entirely unrelated to the LID mechanism.
The model adopted here (neglecting recoil effect and inelastic colli-
sions) exhibits the special nature of the LID effect and other light-induced
phenomena of gas kinetics. Furthermore, this model also has practical
significance. As a rule, the manifestation of the LID effect is a few orders
§6.4]
Drift velocity dependence on medium and radiation characteristics
333
of magnitude stronger than the light pressure (see section 7.2); therefore
disregard of the recoil effect is practically always justified. There is also a
class of objects (e.g. atoms under excitation of electron transitions) for
which inelastic collisions play a vanishing role, i.e. for these objects at
least the above simplifying assumptions are acceptable.
6.4. Drift velocity dependence on medium and
radiation characteristics
The drift velocity и is the most important characteristic of the LID effect.
Some significant relations were obtained for и in section 6.2 without direct
solution of the kinetic equations. The dependence of и on the medium
and radiation characteristics can be adequately analysed only when it is
possible to solve eqs (6.7) and (6.8) for a wide range of radiation
intensities and buffer gas pressures. There is, however, a particular case
when concretization of the collision model is not necessary. This will be
considered first.
Obviously, under certain conditions (e.g. under a sufficiently high buffer
gas pressure) interaction of the absorbing particles with the radiation is
weakly selective in velocities. In other words, radiation results in smooth
non-equilibrium corrections to velocity distributions with small ampli-
tudes; hence the populations p„„(v) and pmm(v) may be represented as
[10]
p„(v) = (ty + ^v (6.25)
The conditions of the validity of such a representation will be discussed in
more detail and now we shall make use of the relation (6.25) in eqs (6.7)
and (6.8) in order to obtain the drift velocity in stationary spatially
homogeneous conditions. On substituting p„(v) of the type (6.25) into the
collision integral we easily obtain
5,(v) = - vtfu) v • jtW(v). (6.26)
v
Here the property (2.246a) of the kernel of the collision integral has been
used as well as the connection of the transport frequency of collisions with
334
Light-induced drift of gases
[Ch. 6
the kernel. It follows from the second equation of the expressions (6.7)
that
v"(v)jn = - vT(v)jm. (6.27)
Relations (6.26) and (6.27) indicate that eqs (6.7) and (6.8) with the
functions p„(v) of the type (6.25) may yield a consistent solution, strictly
speaking, only on the assumption that transport frequencies are inde-
pendent of velocity. Let us make this assumption. On substituting eq.
(6.1) into the first equation of the expressions (6.7), making use of the
relation (6.27) we obtain
Гм[(у + v)2(l + k) + (Q - к • v)2pVm +
2
(Гт + vr)[(y + v)2(l + к) + (Q - к • v)2] — v • jm
V
= 27V |G|2 (y 4-v), (6.28)
к = 4 |G|2/rm(y + v), к = 2 |G|2 (1 + vr/v?)/(y + v)(Fm + *7*).
The effective range of velocity variation is restricted to the value of v;
therefore, if the condition
(kv)2«(y + v)2(l + *•) + Я2 (6.29)
is fulfilled, we may confine ourselves in eq. (6.28) to the terms with low
orders of the parameter к • v. Hence
Nm = 2 |GI2 7V(y + v)/[rM[(y + v)2(l + k) + Й2], (6.30)
j =k#Q_________________2\G\\y + V}N________________
Note that the condition (6.29) is actually also the condition for interaction
with radiation weakly selective in velocity. The condition (6.29) in
combination with the assumption that transport frequencies of collisions
are independent of velocities ensures the validity of the approximation
(6.25).
Making use of relations (6.4) and (6.5), which are valid in this case as
well, we obtain the drift velocity, the expression for which is reduced to
the universal form (6.17) (see also ref. [10]):
к v" — vf {p)
и = — v--------------<p,
к v?
2|G|2(y + v)
(y + v)2(l + k) + Я2 ’
Ф = r2V.Qo2’ П = (У+ v)2(1 + k). (6.32)
J. g 1 ie*
§6.4]
Drift velocity dependence on medium and radiation characteristics
335
By virtue of the condition (6.29) the parameter <p is in this case small so
that the drift velocity when the formula (6.32) is valid is much less than
the thermal velocity v. As the radiation intensity increases the magnitude
of и first rises proportionally to the intensity, then under к = ктах attains
its maximum value, i.e.
|v?-vV| kvQ
U = V----------------Г7~
2(Гт + v?)
Гт(1 + vr/v?)
I2 + Я2 [Гт(1 + уГ/у?)Г1/2
(6.33)
(y + v)2 L 2(rm + vr) J
and afterwards decreases and under sufficiently high radiation intensity is
inversely proportional to the intensity:
IvS-vfl £v|£2|
u == V-----------------T .
_,n i i ..\Z
(6.34)
The dependence of drift velocity on Q is antisymmetric, as would be
expected. Under small Q the drift velocity is proportional to Q\ under a
certain value of Q (|£2| = (y + v)V(l + k)/3, if к and k do not differ by
too much) и takes on its maximum value:
_2. -IvT-vTl kv к
U 16 V Vl + V? у + V (1 + к)3/2 ’
In the range of high values of Q (Q2»(y + v)2(l + к), (у + v)2(l + £)),
we have
|v? - vd kv(y + v)2
U = V—— --------777^---K,
(6.35)
(6.36)
T |й|3
i.e. the value of и is decreasing rather rapidly («|£2|-3).
The decrease in the drift velocity under high radiation intensities and at
large Q is quite natural. As the intensity rises, the field-induced line
broadening grows in importance, which results in the fact that interaction
with the radiation becomes less and less selective in velocities. This
happens also when |£2| increases, starting from a certain value of |£2|.
Furthermore, at large Q the radiation absorption decreases. Those are
two facts responsible for the rather sharp decrease (<х|£2|-3) in the drift
velocity in the range of large values of Q, reflected by the relation (6.36).
The absolute maximum of the drift velocity as a function of Q and к,
according to the formula (6.33), is
|vf—vd kv
umax = v —
1 r 1 /
nm(l + vr/v?)
(6.37)
2v" у + v .
336
Light-induced drift of gases
[Ch. 6
If the transport frequencies of collisions v™ and v" differ substantially,
umax differs from the thermal velocity v practically by the factor
kv/iy + v). This implies that under “unfavourable” conditions (6.29) the
drift velocity may also constitute a significant part of the thermal velocity.
Consider the effect of the buffer gas concentration Nb on the drift
velocity. The quantities v and v\ are proportional to Nh. The value of umax
given by the formula (6.37) rises monotonically as the concentration falls.
The limit of this increase is, evidently, beyond the scope of the condition
(6.29).
In a particular case (6.36) corresponding to excitation in the absorption
line “wing” the drift velocity dependence on Nb has the form of an
increasing curve with saturation. Indeed, the concentration-dependent
multiplier of expression (6.36) is
y + v / vT\
(y + v)2*-= 2—7------(1+ —) |G|2,
7 rm + vT\ v?/1 ”
The cross-section of line broadening which determines the collision
frequency v always exceeds the transport cross-section. Therefore, as Nb
increases, the ratio (у + v)l(Tm + v?) also increases, varying from 1/2 to
the asymptotic value v/v™.
Representation of the distribution functions in the form (6.25) is a
simplified version of the method of kinetic equation solution based on the
expansion of the distribution function in terms of moments (Grad’s
method [11]). In refs [4,12,13] the drift velocity is obtained by Grad’s
method with 13 moments. The corrections due to the distribution function
moments higher than the first turned out to be small to the approximation
(6.29) (they contain a numerical parameter of the smallness). The main
contribution to и is from the zeroth and first moments and the
2 f (k • vl2
vi = T5 *i(v)W(v) dv (6.38)
V J ft
appear as the transport frequencies of collisions (compare with the
relation (2.312)). Therefore, for the formula (6.32) to be valid, only one
essential condition (6.29) needs to be satisfied, whereas the requirement
that transport frequencies of collisions should be independent of velocity
is not fundamental if the quantities entering into eqs (6.32) are defined by
the relation (6.38).
When LID is analysed in detail within the range of parameters where
the interaction with the radiation is sharply selective in velocities the
collision mechanism must be given concretely. We shall use the simplest
§6.4]
Drift velocity dependence on medium and radiation characteristics
337
model of strong collisions in which eqs (6.7) take the form
(Tm + vm)pmw(v) = vmNmW(v) + Np(v),
VmPmmiy) + v„p„„(v) = [(vm - v„)Nm + v„2V]W(v). (6.39)
vm and v„ denote here the frequencies of strong collisions which, as can
easily be seen, coincide with transport frequencies. From eqs (6.39)
combined with eq. (6.8) we find the quantity p(y):
p(v) = y(v)/(T1 + т2(У», <p) = <У)/(Т1 + т2(У)), Vn + Vm _ 2 2 |G|2 T1 (6.40)
fl , _ v ч T? ~ _ ^19 К • У„(Гт + vm) Гт y + v (У) = [ y(v)dv = Viry Re[w(z)] , J A 1 К
z = x + iy, x = Q/kv, у = rs/kv. (6.41)
On the basis of the formula (6.5) we obtain the drift velocity [9]:
к _ v„ - vm Vn у Re[zw(z)]
к vn + vm 1 + Цк + х/я (тг/т^у Re[w(z)]'
(6.42)
Note that the expression for и may be represented as eq. (6.17) where the
factor <p has the form (6.19) with the saturation parameter к from formula
(6.40).
Let us now turn to the analysis of the result obtained. In the range of
small pressures of the buffer gas
vn«Tm (6.43)
we obtain
Тг/b = (v„ - vm)/(v„ + vm). (6.44)
The term in the denominator of eq. (6.42) containing the factor t2/t1 and
responsible for homogeneous saturation does not vanish when the buffer
gas pressure tends to zero as could be expected. This implies that,
however seldom collisions may take place, considerable changes in the
velocity distribution of particles can be accumulated over a sufficiently
prolonged period. Note that the formula (6.42) fails to go over into
formula (6Л1) at v„—>0 because of this very term in the denominator of
eq. (6.42). For the relation (6.11) to hold true there must be either
338
Light-induced drift of gases
[Ch. 6
|v„ ~ Vm|/(vn + Vm) « 1 ОГ К « 1, which was already mentioned on deriv-
ing eq. (6.11).
The factor т2/as can be seen from the expression (6.44) may be also
negative and with v„ « vm, Гт it tends to —1. Under these conditions the
maximum possible drift velocity umax is attained. Using the known values
for w(z) [8], we obtain
umax==0.5v (к» 1,x = 0.5, у = 2), (6.45)
i.e. it is about one-half of the thermal velocity v.
The condition v„ « Гт can easily be satisfied by decreasing the buffer
gas pressure, but the relation between v„ and vm is independent of the
experimental conditions and is determined by the nature of the object. In
the case of electronic transitions of atoms the difference between v„ and
vm may be of order of the values of vm and v„ [14,15] themselves. Let
vm = 2v„; then, taking again vn«rm and k»1, we find that the drift
velocity maximum is reached at x == 1, у == 1:
Mmax“V/15. (6.46)
Note that the condition у = (у + v)Vl + к/kv == 1 at к »1 can be
fulfilled only when у + v «kv. In other words, the velocity drift
maximum in the field of monochromatic radiation is attained in systems
with Doppler line broadening, and provided that the homogeneous width
reaches the value kv as a result of field-induced broadening (к»1).
Subsequently (see section 6.5) it will be shown that still greater values of
the drift velocity than those of relations (6.45) and (6.46) are possible.
The model of strong collisions enables one to study the drift velocity
variation over the entire range of the buffer gas pressure whatever the
non-equilibrium scale in the velocity distribution of particles.
Under the following condition of large Doppler broadening,
7^ « kv,
(6-47)
we obtain from the relation (6.42)
k _ vn - vm
u = — V---------
k vn + vm
1 +
T2Vjt Д ~
Tj kv 1 + к
(6.48)
Here we see that as 7^ increases (as a result of either radiation intensity
or pressure) the drift velocity also rises. This is quite understandable: the
values rs determine the width of the Bennett structure (the effective
§6.5]
Light-induced drift in the field of non-monochromatic radiation
339
interval of “resonance” velocities); an increase in is accompanied by an
increasing number of particles interacting with the radiation and until the
condition (6.47) is violated selectivity in velocities is retained. A rise in
radiation intensity leads also to increasing amplitudes of the Bennett
structures and of the homogeneous saturation band (component propor-
tional to t2/Ti in the denominator of expression (6.48)). This also leads to
an increasing drift velocity until saturation occurs (к»1). If the buffer
gas pressure is such that
vm»rmkv/rs (6.49)
and the condition (6.47) is still valid, it follows from the formula (6.48)
that
к _ v„ - vm Tj Q
— v---------------.
к vn + vm t2 kv
(6.50)
The maximum value of the drift velocity in the formula (6.48) as a
function of Q is attained when Q ** kv:
_vn + vmrs к
и ~ v------— ------.
v„ + vm kv 1 + к
(6.51)
This shows that a higher velocity can be reached only beyond the scope of
the condition (6.47).
In the limiting case opposite to the inequality (6.47) or, to be more
exact, when the condition (6.29) is fulfilled, the result following from the
formula (6.42) fully coincides with relation (6.32) if the obvious substitu-
tion v, is performed.
The above analysis and numerical estimations show that the LID effect
is exhibited to the greatest extent at the boundary between impact and
Doppler broadening. This is quite natural, since under these conditions
the proportion of particles interacting with the radiation is at a maximum
and the selectivity in velocities is at the same time not small.
6.5. Light-induced drift in the field of non-
monochromatic radiation: highest attainable
drift velocity
So far we have been analysing the LID effect by assuming the radiation to
be monochromatic. Intuitively, we can say that under interaction with
340
Light-induced drift of gases
[Ch. 6
radiation the degree of velocity selectivity is determined by its spectrum.
Therefore it would be especially interesting to analyse the LID effect in
the case of non-rtionochromatic radiation.
With regard to the statistical properties of the radiation we take the
model of the splitting of the correlation between atomic and field random
variables (this model is discussed in more detail in refs [16-18]). In this
model for p(v) instead of relation (6:8) we have
A7 z x _ r ( \ z xi f ДО)(У + v) z, „X
Afc(v) [p-(v) P_(v)] J ^ + vy + ^Q_k.vy’ (6-52)
where Bm„ is the second Einstein coefficient, I(Q) is the spectral density of
radiation and Q = a) — a)mn.
For monochromatic radiation, i.e. I(Q) = - Qo), the relation
(6.52) goes over to eq. (6.8) with the value |G|2 = BmM2n. If is
independent of Q (so-called “white” radiation) then interaction with such
radiation is evidently not selective in velocity and the LID effect is absent.
On the contrary, velocity selectivity is inevitable when Z(X2) is non-
horriogeneous in the vicinity of the absorption line (Q = 0).
We shall obtain the dependence of the LID effect on the radiation
spectrum, confining ourselves to the simplest model of collisions—the
model of strong collisions. The equations (6.39) hold good and are com-
plemented by the relation (6.52). The connection of the drift velocity
with the quantity p(v) is still given by the formula (6.5) and for p(v) the
formula (6.40) is valid where the function Y(v) is modified as follows:
Y(v) =
Т1У(у)
1 + T,y(v)
W(v),
, . = f (y + v)Z(Q) dQ
я J (y + v)2 + (Q-k-v)2'
(6.53)
To obtain the final result the type of the dependence I(Q) must be
specified. Consider a radiation spectrum of Lorentzian shape:
In this case formula (6.53) yields
Y( x Ду +v + yR)2yiy(v) Bmnlo Tt
V rl + (Q0-k-v)2 ’ л y + v + yR’
rs = (y + v + yR)Vl + « • (6.55)
§6.5]
Light-induced drift in the field erf non-monochromatic radiation
341
The functional form obtained for Y(v) is the same as in the formula (6.40)
corresponding to the monochromatic radiation. The finite width of the
radiation spectrum enters into the homogeneous linewidth as an additive
term and the detiming Qo of the central radiation frequency from the
absorption line centrum serves as Q. Thus, in the specific case of a
Lorentzian radiation spectrum (6.54), LID characteristics are described by
the same relations as for monochromatic radiation but with the homoge-
neous absorption line halfwidth increased (by the radiation spectrum
halfwidth). This implies, in particular, that the quantity Z^ becomes an
independent parameter which can be governed by varying the width of the
radiation spectrum. This circumstance may be used, for example, to
increase the drift velocity under the given radiation intensity. As we have
already seen, the drift velocity increases as the buffer gas concentration
decreases. However, if the concentration is very low, the effective range
of “resonance” velocities determined by the quantity Z^ becomes small
compared with v and in the case of monochromatic radiation can be
increased only by a substantial increase in intensity. For non-
monochromatic radiation this range can be increased to the optimum
value with the help of the radiation spectrum width yR. This implies that
the maximum drift velocity is attainable at a lower radiation intensity [18].
If the radiation spectrum in the vicinity of the absorption line is
smoothly non-homogeneous so that I(Q) may be represented as
I{Q) = Z(0) + Z'(0)G, Г = , (6.56)
ds2
then
r(")=J7^[1 + lTrS’ (657)
Here Kef is the effective saturation parameter for the “white” radiation.
From this relation, expression (6.40) for p(v) and the general formula
(6.5) we obtain the drift velocity:
к vn-vm Kef 1 fcuZ'(O)
к v„ + vm 1 + Kef 1 + 2ке1/ГтТ1 2Z(0)
(6.58)
The drift velocity in this case is proportional to kvI'(0), i.e. to the change
in spectral density of radiation over the range ~kv.
Of particular interest from the viewpoint of studying LID properties is a
radiation spectrum with a sharp edge. Consider the specific case of a
342
Light-induced drift of gases
[Ch. 6
“step-shaped” spectrum
Q^ Qo;
Q> Qo-
(6.59)
Suppose also that the condition of large Doppler broadening is fulfilled
(y + v « kv). Then it follows from the formula (6.53) that
Y(v) = <
*ef
1 + Kef
W(v),
к • v < Йо, — Bm„Z0Ti;
(6.60)
Ю,
kv
The drift velocity in this case is described by the expression
к v vm-vn_______________exp[- (Йр/Аги)2]___________
U к vm + v„ 1 + 1/Kef + (t2/2t!)[1 + </>(Q0/kv)] ’
(6.61)
where ф(г) is the probability integral.
Note a remarkable feature of the obtained result. The expression (6.61)
in principle admits any drift value, however high. Indeed, at v„« vm, Гт
we have x-Jxx —>-l. If we put T2/Ti = -1 and neglect the term 1/кеЬ
assuming Kef»1, the expression (6.61) takes the form
lr jj 1
и = «-1*-*(“)! <6-62>
which reveals unlimited growth of и as й0 increases. In particular, the
value of и may exceed the value of the thermal velocity v.
The basic possibility of obtaining an abnormally high value of the drift
velocity in a radiation field with the spectral shape (6.59) may be proved
also by qualitative treatment of this case (see fig. 6.5). Then all the
particles with velocities vz < Q/k interact with the radiation. If у 4- v «
kv, particles within the velocity range vz > Q/k practically fail to interact
with the radiation. In the velocity range vz < Q/k both induced transitions
between the levels m and n characterized by the probability (per unit
time) BmnI0 and spontaneous transitions m—>n with the probability Гт
occur. Within the range vz > Q0/k only the transition m —»n exists
because of spontaneous decay of the level m but the reverse transition is
absent. A collisional exchange exists between those velocity ranges with
frequencies v„ and vm at levels n and m respectively. Assume that v„<vm,
Гт, Втп1ц. Under these conditions particles are intensively accumulated at
the level n in the velocity range vz > QJk. Indeed, their transition there
from the interval vz < Q0/k is a result of three successive processes, i.e.
§6.5]
Light-induced drift in the field of non-monochromatic radiation
343
1(0)
о o0 n
Fig. 6.5. Radiation spectrum I(Q) as a “step” and stationary velocity distributions of the population
at у + v « kv:------------------------------, initial Maxwellian distribution.
optical excitation to the level m, collision at the level m accompanied by
transition to the range vz > Q0/k, and spontaneous transition to the level
n. The reverse transition to the range vz < QJk is only possible as a result
of collisions at the level n. Therefore, the less frequent these collisions
are, i.e. the smaller is v„, the more effective is the accumulation of
particles in the range vz > Q0/k. The stationary velocity distribution of the
344
Light-induced drift of gases
[Ch. 6
particles takes the form shown in fig. 6.5. It is evident from this figure that
velocity drift cannot only reach the value of the thermal velocity v but, in
principle, can even exceed it if Qo is increased to some value Qo > kv.
More exact estimations based on the formula (6.61) show that in order
to attain the condition u > v it is necessary to have vm^102v„. Such a
relation is hardly possible in any real object. Therefore the problem of
drift with a velocity exceeding v (supersonic LID) remains, it seems,
entirely theoretical.
The radiation spectrum has been shown to influence the LID effect
strongly. A significant increase in the drift velocity can be achieved by
specially forming the radiation spectrum. From this viewpoint the best
result is obtained with a “step-shaped” radiation spectrum. Analysis of
the LID effect in the case of non-monochromatic radiation is given in
more detail in refs [18,19].
6.6. Spatial distribution of the concentration
A rigorous approach to the analysis of the LID effect under spatially
non-homogeneous conditions presents a much more complicated task than
the determination of drift velocity in the case of spatially homogeneous
conditions. Under an arbitrary deviation from equilibrium it is impossible
even to obtain an equation describing the concentration variation of
absorbing particles. Therefore various simplifying circumstances and
approximations will be used.
The main approximation assumed here is that of local homogeneity and
stationarity formulated in section 6.2 which enables one to find a solution
for p(v) from the spatially homogeneous equations (6.7) and (6.8) and to
use it in the macroscopic kinetics equations (2.335)-(2.339). On the basis
of this approximation we obtain an equation describing the variation in
the concentration N of absorbing particles. The initial equation is the
equation of continuity
dN
— + divj = 0, (6.63)
where the flow J must be expressed in terms of concentrations. To this
§6.6]
Spatial distribution of the concentration
345
end, let us turn to eqs (6.2) for flows. It follows from these equations that
j = JlAD
. V? - vy Q
JuD vl rm + vT ’
jd =----— V • Pn---—rm + v"v . pm. (6.64)
Jd mv" nmvirm + vT
The total flow is the sum of the light-induced (jLID) and diffusion (Jd)
flows. The latter is caused by spatial inhomogeneity of partial pressures.
In accordance with the assumed approximation the quantity Q is supposed
given, i.e. obtained from eqs (6.7) and (6.8).
The difference in coefficients of the terms V • Pn and V • Pm in the
expression for diffusion flow brings about a new light-induced phenome-
non which will be discussed in detail in section 7.3. Here we shall only
note that the concentration change due to this phenomenon is significantly
less than that caused by LID and for simplicity the difference between
these coefficients will be neglected. This yields
Л=-^Т-Р. (6.65)
The relation (6.65) formally follows also from eqs (6.2) if the term
(l/m)V • Pm is neglected. The approximation (6.65) is valid at least when
the difference between v" and v™ is insignificant or when the proportion of
perturbed particles is small.
In the general case the pressure P has a tensor structure, but for the
time being we shall ignore this fact as its effect is not so important and
take the relation between pressure and concentration to be P = NT. As a
result we obtain the following equation for the concentration:
dN
— + div(uA - D VA) = 0 » (6.66)
at
where, under the stipulated conditions, the drift velocity и is given by the
formula (6.5) and D is the diffusion coefficient of unexcited particles. In
the case и = 0 the equation (6.66) becomes the ordinary diffusion equation.
In order to estimate the possible error which may arise because of the
assumed approximation of local homogeneity and the stationarity ap-
proximation consider a specific case of homogeneous broadening:
(kv)2 « (y 4-v)2 4-Q2. (6.67)
346
Light-induced drift of gases
[Ch. 6
In this case we can obtain the equation for the concentration without this
assumption. Provided that the condition (6.67) is fulfilled, the velocity
distributions of the populations may be represented similarly to the
relation (6.25). On substituting the expression (6.8) for p(v) into eqs (6.2)
and making use of the smallness of the Doppler shift к • v, we obtain the
following equations for the flows:
V2 v2
(Tm + + У Л — g'(Nn - Nm) + g(J„ -Jm),
v2
-^N = (vni-vT)im-vnJ,
_2|G'|2(y+ v) , = dg
8 (y+v^+D2’ 8 dQ'
Formal solution of these equations with respect to j yields
j ~ JlID + jdi
v2 v" — V? g'
Jlid ~k^ vl rm +vT + g(l +vr/v?)(Nn “ Nm)’
Jd = ~Dn VNn - Dm VNm,
D 62 rm + v^ + 2g
n 2v?rm + vr + g(l + vr/v7)’
D 62 rm + v1 + 2g
m 2v’{ Гт + v? + g(l + v"/ v") ’
(6.68)
(6.69)
where Dm and Dn act as diffusion coefficients of the absorbing particles in
states m and n. The total flow j naturally breaks up into light-induced and
diffusion parts.
The concentation Nm of excited particles is described by the first
equation of expressions (2.335) where the terms with derivatives with
respect to time and coordinates are also present. The conditions of their
truncation are quite clear. The external factors which influence the
variation in concentration Nm must not significantly change over the
lifetime 1/Гт of the particle at the level m. In addition, the distance
passed by the excited particle must not be such as to change the
concentration Nm significantly. These conditions correspond to quite a
wide range of problems and we shall consider them fulfilled. Then
Nm = N{p)/rm, (6.70)
wherein coordinates and time enter as parameters.
§6.6]
Spatial distribution of the concentration
347
In the approximation of homogeneous broadening (6.67) the concentra-
tion Nm of excited particles is described, as can be easily shown, by the
formula (6.30). As a consequence, for the light-induced flow jLID we
obtain from eq. (6.69)
• - *-jyv"~v” к(у + у)2 Qkv
JuD к v7 + vr(y+у)2(1 + к) + й2П+^22’
where the notation used is the same as in formulae (6.28)-(6.32). Direct
comparison of the formulae (6.71) and (6.32) shows that the light-induced
flow jYid calculated under stationary spatially homogeneous conditions
remains the same when conditions are non-homogeneous.
As can be seen from the comparison of eqs (6.64) and (6.69) the main
errors of the local homogeneity and stationarity approximation are
associated with the diffusion flow jd. Indeed, if in eq. (6.64) the tensor
structure of the pressure is neglected and the relations Pt = N,T are
assumed to hold true, then formulae (6.64) and (6.69) for jd coincide only
under the condition of low intensity (g « Гт + v”, Гп + v"). Therefore, the
violation of the local homogeneity and stationarity approximation leads to
the dependence of Dm and Dn on the intensity of radiation. Under high
radiation intensities Dm and Dn are equalized. This is one more justifica-
tion of the introduction of a common diffusion coefficient D on deriving
eq. (6.66). The introduction of such a diffusion coefficient is justified,
therefore, in the following cases: when the difference between v" and vf
is small; under a low concentration of buffer gas; when v?, v" « Гт 4- 2g
(the diffusion coefficient D in this case may depend on the radiation
intensity); at a low radiation intensity when the proportion of excited
particles is small; under a high radiation intensity when Dn = Dm =
u2/(v? + v?). Equation (6.66) reflects the basic properties of LID and of
other cases and it must be defined more exactly only for the analysis of
fine details.
Under stationary conditions eq. (6.66) describes the space distribution
of the concentration. For the concentration to become spatially in-
homogeneous there must be a barrier to the motion of the drifting
particles. Let, for example, drift take place in a long thin cell with
radiation passing through the cell. One of the ends of the cell towards
which the particles are drifting is closed. The drift will result in particles’
accumulating near this end until the opposing diffusion flow appears and
compensates the drift flow. After that the motion of particles will stop and
a stationary distribution of the concentration will be established, satisfying
348
Light-induced drift of gases
[Ch. 6
the equation
dz D
(6.72)
This equation has an especially simple solution for an optically thin
medium when the drift velocity is independent of z. In this case for и > 0
we have [10]
u
N(z) = N(L) exp|^ - - (L - z) j ,
(6.73)
(6-74)
where L is the coordinate of the closed end of the cell. This formula
describes the exponential dependence of concentration on coordinates.
The characteristic scale of space non-homogeneity is
и 2u v"
where I is the mean free path. According to estimates (6.45) and (6.46) for
the drift velocity we see that under conditions optimum for LID particles
may collect in a layer about several mean free paths in extent.
If a reservoir of particles exists from which they can enter the cell, the
concentration will be increased near the closed end. This process
obviously dies out when the medium is no longer optically thin, i.e. when
radiation is completely absorbed in the range of increased concentration.
In the case of an optically dense medium eq. (6.72) alone is not
sufficient for a description of the concentration distribution. One more
equation must be added to it, describing the radiation intensity variation:
^=-ha){p)N, S=-^~\E\2.
dz on
(6.75)
Here 5 denotes the density of radiation energy flow.
Equations (6.72) and (6.75) are a system of essentially non-linear
equations. Their analytic solution in a general form is impossible.
However, from these equations one can derive a practically useful
relation, which connects easily experimentally obtained characteristics
with those of LID.
The drift velocity will be represented as in eq. (6.32) expressed in terms
of the quantity (p) which, in its turn, will be replaced by the relation
(6.75). Then eq. (6.72) will be written as (with the z axis directed along
§6.7]
Optical pistons and “solitons”
349
the propagation of radiation)
dN_v"~ v? 2cp d5
dz Гт + v? faov dz ’
(6.76)
In all particular cases considered above the factor is described by the
formula (6.19). We can expect that this approximation for q> will be useful
for other situations as well. The factor <p is connected with the radiation
intensity by the value rs = (y + v)Vl + *•. This dependence is closely
associated with field-induced broadening of the absorption line and it may
be neglected if the field-induced broadening is not large (either Doppler
broadening is large or the radiation intensity is not too high). In this case
the coefficient of dS/dz in eq. (6.76) may be considered independent of z,
after which the equation is easily integrated:
vr-v?2A5
AN = —--------(p.
Гт + v™ hojv
(6-77)
Here AN denotes the concentration change over the length of the
absorbing cell and A5 is the density change of the radiation energy flow.
The quantity AS can easily be obtained by measuring the radiation power
loss in the cell. Other non-trivial characteristics in the relation (6.77) are
specific for LID and they can easily be obtained experimentally. Charac-
teristically, the formula (6.77) is valid for a medium of arbitrary optical
density.
6.7. Optical pistons and “solitons”
The present section is devoted to the consideration of the two most
pronounced LID manifestations in optically dense media. The first of
these, i.e. the optical piston, appears in a medium initially at equilibrium
with a high concentration of absorbing particles as a result of drift motion
along the radiation propagation direction [10].
Suppose the radiation propagating along the z axis enters a cell with a
gas of high optical density. At first, the radiation does not penetrate
deeply, being absorbed not far from the input of the cell. From here
absorbing particles begin their drift into the cell. Radiation penetrates into
the region without any absorbing particles. This results in the appearance
of a moving interface (see fig. 6.6), on the left of which there is radiation
350
Light-induced drift of gases
[Ch. 6
Fig. 6.6. Spatial distribution of radiation intensity /(z) and concentration of absorbing particles N(z)
in the vicinity of a moving optical piston.
but there are no absorbing particles and on the right there is no radiation
but absorbing particles are present. In the intermediate layer whose
thickness is determined by the absorption coefficient the LID effect
“makes” the interface move to the right.
We now have an evident analogy with a piston moving the gas of
absorbing particles. If the concentration of buffer particles is much greater
than that of the absorbing particles, the piston is practically penetrable for
the buffer particles. Absorbing particles previously collected by the piston
in front of it and diffusing freely propagate to the region where there is no
radiation. Figure 6.6 illustrates the qualitative dependence of the con-
centration N on the coordinates when the piston is in motion.
If the far end of the cell is closed the gas of absorbing particles will be
compressed by the piston up to some limiting state and a concentration
distribution similar to that shown in fig. 6.7 will be established. Under
stationary conditions the concentration N in the region where radiation is
absent is spatially homogeneous. This concentration is easily obtained
from the formula (6.77). In the case of an optical piston
N =
v" — v? 2cp
Гт + vTha>v
S,
(6.78)
where the quantity S characterizes the incident radiation.
According to this formula, the concentration maintained by the piston is
greater for higher radiation intensities. However, from some value of S,
§6.7]
Optical pistons and “solitons”
351
N(i)
0
Fig. 6.7. Spatial distribution of radiation intensity /(z) and concentration of absorbing particles N(z)
under the conditions of a stationary optical piston.
the factor depends on the radiation intensity and formula (6.77) can no
longer be used. The intensity dependence of must be included when the
field broadening of the absorption line becomes large, i.e. under
homogeneous broadening when the approximation (6.25) can be used and
the expression (6.32) holds for q>. In this case eq. (6.76) takes the form
_ Adz
1 + к + [$2/(y + v)]2 ’
v” - v? fiv? / Q \2
v“ + vr2jtd™ \y + v/
W^,(l + vr/v") s
ft2c(y + v)(rm + v“)
(6-79)
where is a matrix element of the dipole moment. This equation has
the following solution:
Л7 - A In 1 + * + + v)]2
/V Nq — i41n г л / / \ ♦
1 + K0 + [£2/(y + v)]2
(6.80)
In the case of a light piston No = 0 and if we take к = 0 the formula (6.80)
describes the value of the concentration of absorbing particles in front of
352
Light-induced drift of gases
[Ch. 6
the optical piston:
N = |A| ln{ 1 + t + + y)]2) . (6.81)
With k0« 1 + [Q/(y + v)]2 this formula, as would be expected, coincides
with the formula (6.78).
According to the expression (6.81) the particle concentration main-
tained by the optical piston increases with increased radiation intensity
according to a logarithmic law. This implies that there is no limiting
degree of particle compression by the optical piston. However, for
practical purposes the estimate Nmax ~ A is quite sufficient.
Numerical estimation of Nmax will be carried out for some specific values
of the parameters. Allowing for electronic transitions of atoms we take
Iv" — v“|/(vf + v“)«1 and d„„ =« 1 D = 3.34 x 10“30 C m. Then at Ql(y +
v) = 1, v" = IO-8 s-1 we have
Nmax«1016crn-3. (6.81a)
The dimension of the intermediate layer /p (the thickness of the optical
piston) is estimated from the condition of approximately equal diffusion
and drift flows, i.e.
lP~Dlu, (6.82)
and for the estimation of the drift velocity a radiation intensity of the
order of the initial intensity is employed. The thickness of the intermedi-
ate layer according to formula (6.74) may be a fraction of the absorbing
cell length, i.e. the optical piston may be very thin.
It is not easy to describe the dynamics of the optical piston analytically
and only numerical calculations have been reported [6,20]; these fully
agree with the qualitative picture described above.
Most attractive is the statement of the problem when a specially formed
spatially limited cloud of absorbing particles can drift in the radiation field
[21,22]. In this case the velocity drift can be directly measured ex-
perimentally and also a special regime is possible when the shape of the
propagating cloud does not change (of a soliton type).
First consider the conditions under which the cloud of particles is
optically thin so that the drift velocity и in any of its parts is the same.
This makes it possible in eq. (6.66) to change over to a coordinate system
moving with velocity u. Equation (6.66) in this system takes the form (one
§6.7]
Optical pistons and “solitons”
353
dimensional, motion along the z axis)
3N d2N
— -D — = 0, £ = z-ut, N = N^,t), (6.83)
dt aC,
i.e. it becomes the usual equation of diffusion. Therefore the cloud of
particles moves with constant velocity и and at the same time experiences
diffusion spread. In the simplest case of a Gaussian initial particle
concentration distribution in the cloud, i.e.
N(t, 0) = N(z, 0) = M-, exp(— Г/4), (6.84)
the solution of eq. (6.83) is
^Z’^ = a(0eXP ~^a*(r) ’ fl2(0 = «о + 4£>t. (6.85)
At any instant the concentration distribution in the cloud is Gaussian with
a characteristic width increasing with time and obeying the diffusion law.
A qualitative change in the cloud propagation takes place if radiation
passing through it is significantly absorbed. Let, for instance, the cloud be
drifting in the direction of the propagation of not too intense radiation
(i.e. the intensity increase is accompanied by a corresponding increase in
drift velocity). Then at the trailing edge of the cloud where radiation is
more intense the drift velocity of particles is higher than at the leading
edge. Therefore the cloud must be compressed as it propagates. The
compression will take place until diffusion process hinders it. As a result,
the shape of the cloud becomes stable and in further propagation no
change of the shape will take place [23]. In the cloud system of
coordinates the concentration distribution will be described by a station-
ary equation which follows from eq. (6.66):
dN и — й
= t = z-ut, N = N(£), u = u(£). (6.86)
Here й is the propagation velocity of the cloud. This equation is
essentially non-linear since the drift velocity u(£) is determined by the
concentration’s coordinate distribution.
Consider the solution of eq. (6.86) in two limiting cases. The first case is
that of weak radiation intensity change within the cloud. Then the
quantity S from eq. (6.75) may be represented as
S(£) = 5(0) - Лш(р)£^(Г) dr, (6.87)
354
Light-induced drift of gases
[Ch. 6
where point £ = 0 will be chosen under condition u(0) = й. The drift
velocity may be written as follows using the smallness of A5 = 5(£) —
5(0):
u(£) = u + ^A5
d5
= й-^й(о{р)[ N(£')d£'. (6.88)
do Jo
Therefore eq. (6.86) takes the form
— -AJv|\(£')d£'> Л = Б^ЙШ^>- <6-89)
d£ Jo ZJ d*S
This equation is fairly simple and has the following solution:
N(£) = 2V(0)/{cosh[VA/V(0)/2£]}2. (6.90)
In this case the concentration in the cloud is distributed symmetrically
(about £ = 0) with characteristic width V2/A/V(0). The cloud becomes
narrower as concentration in it increases. This case is considered in more
detail in ref. [23].
The other limiting case corresponds to a high optical density of the
cloud such that radiation is fully absorbed over the small part forming the
trailing edge of the cloud. In the remaining part of the cloud it may be
assumed that и = 0, so that according to eq. (6.86) there is the following
simple coordinate dependence of concentration:
N(£) = N(£o) exp[ -%(£- £0)] , (6.91)
where £0 is the coordinate where the drift velocity и is sufficiently small.
The characteristic thickness of the leading edge is
li-D/H. (6.92)
In the cloud characterized by a high optical thickness practically all the
particles are concentrated in its leading edge. For this cloud the following
condition
u«umax (6.93)
is characteristic, the characteristic thickness of the trailing edge being
estimated as
l2~D/Umax. (6.94)
§6.8]
Light-induced drift of molecules and multilevel atoms
355
The quantity N(to') appearing in relation (6.91) actually approximates the
maximum value of the concentration at the point where и = й (see eq.
(6.86)). The value of N(to') may be estimated from the formula (6.78)
since at small velocities (й«итах) its trailing edge is formed under
conditions close to those of the stationary optical piston.
Let us find the velocity of the cloud. On integrating the formula (6.91)
we obtain
u = DN(C0)/M, M=\ N(£)d£. (6.95)
J—oo
The greater M, the slower the cloud is. It is useful to compare й with the
maximum drift velocity
Here S and (p) correspond to the initial characteristics of the radiation.
We have, therefore, obtained a universal relation for the parameters of an
optically dense cloud: the ratio between the velocities й and umax and the
sizes of the cloud edges equals the ratio between the radiation power and
the work done by the field of all the cloud particles under the initial
radiation intensity.
6.8. Light-induced drift of molecules and multilevel
atoms
So far the LID effect has been analysed in the model of two-level
particles. This model proves to be justified only in cases when it is
impossible for the particle interacting with the radiation to make a
transition to other levels as a result of either collisions or radiative
processes. Most real objects (atoms, molecules) involve many levels, and
it is necessary to consider multilevel systems. Atoms are multilevel
because of the fine and hyperfine structure of levels and their degeneracy.
Molecules are multilevel to a great extent because of rotational and
vibrational states.
The character of the LID strongly depends on levels not directly
perturbed by radiation but to which collisional or radiative transitions are
possible. Each multilevel object can have individual properties since there
356
Light-induced drift of gases
[Ch. 6
is a great variety of energy characteristics of levels and the processes of
transition between them.
In the present section, not claiming to give a complete picture we shall
study how the fact that particles are multilevel affects the character of the
LID effect. First consider the following scheme. Suppose there are two
groups of levels (fig. 6.8) which will be called states m and n. Levels
within the state will be designated by numbers Jm and Jn. Collisional
transitions Jm^>J'm, Jn^J'n are allowed within each state. In addition,
spontaneous and collisional transitions are possible from each level Jm of
the upper state m to the Jn levels of the ground state n. The radiation is
resonant with the isolated transition between the levels Jn0 and Jm0 of
states n and m. This scheme of levels may reflect both the rotational
structure of the vibrational states of molecules as well as the fine and
hyperfine structure of the electronic states of atoms.
The equations for the density matrix corresponding to the level system
shown in fig. 6.8 have the form
"3 1
— + v • V + rm(Jm) pm(Jm, v) = Sm(Jm, v) + Np(v)dJmlm0;
Lot J
I d \
I — + V • V)p„(J„, v) = Sn(Jn, v) +
\at /
S rm(Jn | v) - Np(v)d7^0;
Jm
2|G|2(y + v) Г 1 1
P(V) = (y + v)2+(fl-fc-v)2 Рй(/й0’ V) ” V)
(6-97)
where p,(J,, v) -denotes the population of level J, of state i. Non-diagonal
elements of the density matrix for the transitions Ji^J\ are not excited by
radiation and do not appear during collisions. The quantity rm(Jn | Jm) is
the rate of spontaneous transition (first Einstein coefficient) from level Jm
to level J„; the resultant rate of spontaneous decay of level Jm is
The statistical weights g(Jn0) and g(Jm0) of levels Jn0 and Jm0 were
introduced in the expression for p(v) in order that the degeneration of
levels should be, even if roughly, allowed for. The symbols 8JlJm reflect the
fact that radiation interacts with the isolated transition Jn0-Jm0. The
collisional integrals in eqs (6.97) in conformity with the assumed model
§6.8]
Light-induced drift of molecules and multilevel atoms
357
JmO
m
J
Fig. 6.8. Schematic diagram of levels with an isolated radiative transition.
have the following structures:
v) = X dVi[A(/nJmv | vO -
| rnJmv)pm(Jm, v)] -
X dv, A(nJnlVi | rnJmv)pm(Jm, v),
Л1 J
Sn(J„, v) = XI dvt [A(nJ„v | nJniVi)p„(J„i, Vi) -
Л1 J
A(nJniVi | nJ„v)p„(Jn, v)] +
X d^ A(nJnv | rnJmXvx)pm(JmX, Vi).
Jmi J
(6.98)
The kernels of collision integrals A(iJ,v | iJiXvx) describe collisional
transitions within level groups of state i, and the kernel A(nJnv | mJmvx)
specifies collisional transition from level Jm of state m to level Jn of state n.
Reverse transitions from state n to state m are neglected on the
assumption that the energies of levels Jm substantially exceed the thermal
energy. The designation of collisional integral kernels introduced here is
358
Light-induced drift of gases
[Ch. 6
the abbreviated notation of section 2.4:
| jJjV) = A{UtUtVx | jJjjJjV). (6.99)
Following the routine procedure we may obtain from eqs (6.97) the
macroscopic kinetics equations formally appearing as the corresponding
equations in section 2.7. However, the specific meaning of the macro-
scopic characteristic is different. In particular, for pressure tensors and
friction forces there is a natural generalization:
P“p = X j mvavpPi(Ji, v) dv,
Ft = X [ , v) dv. (6.100)
Ji -
Allowing for the structure of the collisional integrals (6.98), the expres-
sions for friction forces Fm and F = Fm + Fn may be reduced to
Fm = -m X [ 1Ш, v) + v)]vpm(Jm, v) dv,
Jm J
F = ~m^ J [vT(Jm, v) + vr(7™, v)]vpm(Jm, v) dv -
m X f v1(Jn, v)vpn(Jn, v) dv. (6.101)
л J
The collisional characteristics v'i, v7m and vnm introduced above are
related to the kernels of the collisional integrals as follows:
v) = X J dv, (1 - v • vJv2)A(iJavt | i//v),
V) = X f dvi (1 - v • Vi/v2)A(nJ„v11 mJmv),
л J
V) = X dVi A(nJnVi I (6.102)
л
The physical meaning of these characteristics is quite clear: v“(Jm, v) and
v?(J„,v) are transport frequencies of collisions during which the particle
remains in states m and n; the quantity is the transport
frequency of inelastic collisions involving quenching of the state m; finally,
v) is the total frequency of collisions quenching state m.
§6.8]
Light-induced drift of molecules and multilevel atoms
359
The relation between transport frequencies of collisions and charac-
teristics of scattering is now presented:
и С и • v
vKJi, v) = — u —— tfjXJjJbU, &E)pb(Jb, v-u) du,
tn J V
— с и* / и • «Л
a£(J/bu, AE) = 2 — (1----------— )a(iWbi«i d«!
ЛЛ.1 J u ' u '
v‘i(Z, = v); Uj = Vu24-2 АЕ/д. (6.103)
Here a(iJlJbiUi\iJ)Jbu) is the differential cross-section and ai is the
transport cross-section of scattering (with inelastic processes allowed for),
AE is difference between the energies of initial and final states, and и and
Ui are the relative velocities before and after collision.
All the above-introduced collisional frequencies in the general case are
functions of velocity v and Jt. The v dependence has already been
discussed in section 2.6 and the model neglecting this dependence has
been proved to be useful. The Л dependence is contingent on the specific
object. If it is a rotational structure of vibrational states m and n of
molecules, a weak dependence of collision frequencies on rotational
number should be expected, at least weaker than the dependence on
vibrational state.
Now a model will be taken where collision frequencies depend neither
on v nor on Jt. It will be also assumed that rm(Jm) is independent of Jm.
Then the friction forces (6.101) take the form
Fm = -rn(v?+vnm}jm, F = -m[(v? + vTn)jm + vnJn], (6.101a)
and hence we have the following equations for particle flows:
(Em 4- v~" 4- vTVm +-V • Pm = NQ,
m
vrnym + упд,
m
Q = f VP(V) (6-104)
Thus we have obtained equations similar to eqs (6.2) for the two-level
system, where the generalization involves taking the quenching collisions
into account. As a result of the quenching collisions v™ is increased by the
quantity v?m and Гт is increased by the quantity v~" — v"m. The sum
у». + ym» js the total transport collision frequency for the state m and the
360
Light-induced drift of gases
[Ch. 6
difference vnm — v"m describes the quenching of level m causing no
significant change in particle velocity. Therefore the physical meaning of
the difference v™ — v?m is similar to that of the quantity Гт.
The similarity of eqs (6.104) and (6.2) makes it possible for the results
obtained in sections 6.2 and 6.6 for two-level systems to be used for
multilevel systems. Specifically, in the assumed model (6.98) expressions
(6.17) and (6.77) for drift velocity and concentration change hold true
after the substitution
vT-> vr + v?m, Гт -» Гт + vnm - v?m.
The complexity of relaxation processes in a multilevel system is reflected
mainly in the absorption probability {p) and in the value of the absorbed
power AS, which can be measured independently. As for the factor <p,
expression (6.19) is still a fairly good approximation for it. This is so, at
least, in the approximation of weak saturation and in the model of strong
collisions.
Only in the model of strong collisions can eqs (6.97) be solved
analytically under conditions of arbitrary saturation. In the most general
case of the model of strong collisions eqs (6.97) have the following form:
/a \
» +v.V + l^ + vm )pm(Jm, v) = vmipm(v)WB(Jm) +
\at /
Vm2pmUm)W(v) +
vm3Nm WB(Jm)W(v) + Np(v)dMlM,
/ a \
I + v . V 4- V„ )pn(J„, v) = vnlp„(y)WB(Jn) +
\dt /
v„2p„(J„)W(v) + vn3N„WB(Jn)W(v) +
^mNmWB(Jn)W(y) +
S rm(J„ | Jm)pm(Jm, v) - Np(v)8
Jm
p,(y)= S Р/(Л, v)> р,(Л) = I Pi(Ji, v) dv, i = m, n\
j, '
Ni: = S [ рД, V) dv, S ВД) = 1, rm = S rm(Jn I Jm),
Ji ~ Ji Jn
Vm = vmi + vm2 + vm3 + vnm, V„ = v„i + vn2 + v„3. (6.105)
The following processes have been taken into consideration: collisions
changing Jt but changing neither the velocity nor the state i (corresponding
§6.8]
Light-induced drift of molecules and multilevel atoms
361
collision frequencies vn), velocity-changing collisions which do not alter J,
(collision frequencies v,2), collisions simultaneously changing velocity and
Jt but leaving the state i unaltered (collision frequencies v,-3) and, finally,
collisions quenching state m and establishing at the same time equilibrium
distributions in J„ and v in state n (collision frequencies v"m). In
accordance with the model of strong collisions each collision is taken to
cause an equilibrium distribution of the characteristics involved (Max-
wellian velocity distribution W(v) or Boltzmann distribution WB(J,) in the
Ji levels). Equations (6.105) take into account also spontaneous transitions
from levels Jm of state m to the set of levels Jn of the ground state n. The
values of Гт will be assumed the same for all the levels Jm.
Let us find solutions of eqs (6.105) under stationary and spatially
homogeneous conditions. Proceeding from the structure of the equations
(6.105) it is easily shown that the velocity distribution of the m state
populations as well as their distribution in Jm levels may be represented as
pm(Jm, v) = N[Tim(Jm)p(y) + ЪДЛЛООЩ»)]. (6.106)
The factors Tim(Jm) and t^^) can easily be obtained by comparing
relation (6.106) with eq. (6.105) for pm(Jm, v) as well as the results of its
integration over v and summation over Jm:
(6.107)
The distribution pm(Jm,v) in velocities and levels Jm contains a part
selective in both v and Jm, a term selective in v and equilibrium in Jm, a
part selective in Jm and equilibrium in v and, finally, a part equilibrium
both in v and in Jm. Weight factors of each of the parts are proportional to
the lifetimes of the particle in the corresponding states. In particular, the
weight factor of the term selective in v and in Jm is the quantity
1/(1^, + vm), i.e. the lifetime in state m with given values of v and Jm. The
term selective in v in the summed (over Jm) velocity distribution is
proportional to the factor 1/(Гт + vm — vml), i.e. the lifetime of a particle
in state m with a given velocity. The term selective in Jm in the integrated
population distribution over levels Jm is proportional to the lifetime
l/(Tm + vm — vm2) of a particle at level Jm. The total population of state m
is proportional to the total lifetime 1/(Гт + v~") of this state. Physically,
therefore, the results can be interpreted easily.
362
Light-induced drift of gases
[Ch. 6
Expressions (6.106) and (6.107) will be substituted into the second
equation of relations (6.105) which for the distribution v) yields the
following stationary spatially homogeneous equation:
vnpn(Jn, v) = vnlpn(y)WB(J„) + vn2p„(J„)W(y) +
vniNnWB(Jn)W(v) + N{p)WB(Jn)W(v) ~
^{[<5ЛЛо
- S rm(Jn I p(v)~
^rm(Jn\Jm)T2m(Jm)(p)W(v)j,
(6.108)
the solution of which can be represented similarly to expression (6.106):
p„(J„, v) = N[WB(J„)W(v) - Tln(J„)p(v) - T2„(Jn)(p)W(v)]. (6.109)
Using the same procedure as when obtaining the coefficients rlm(Jm) and
Ът(/т) in formula (6.106), we find the coefficients т1л(7л) and Т2ЛЛ). This
yields
т1л(Л) = -[«,(/„)- <n1>WB(J„)] + -<-fll>WB(7"),
v„ V„ - V„1
Ml.) “ W7-) - <“»>
Vn ^n2
ai(Jn) ~ S F,n(Jn | Лп)Т1т(Лп),
Jm
a2(Jn) = ~ S Лп(/п I •An)[Tim(Jm) + T2„,(Jm)],
Jm
{аг)=^а^п) =
Vm - Vml
rm + vm- vmi’
{a2)^a2(Jn) = ^+-m.
(6.110)
The radiation-induced change of distribution p„(J„,v) is similar to the
distribution pm(Jm, v) but also has some specific features. As in state m,
the distribution p„(J„,v) contains parts selective in v and Jn, selective
only in v or only in Jn and parts equilibrium both in v and in J„. Each part
is proportional to the particle lifetime in corresponding non-equilibrium
state. Characteristic features of the distribution p„(J„,v) are due to the
fact that the source of the equilibrium state is in this case not only the
§6.8]
Light-induced drift of molecules and multilevel atoms
363
direct radiation effect on level Jn0 but also transfer from levels Jm due to
spontaneous transitions. This transfer on the one hand reduces the
amplitude of the non-equilibrium structure at the level J„o and, on the
other hand, produces at other levels Jn a non-equilibrium structure of the
opposite sign in the v and J„ distributions.
From relations (6.106) and (6.109) as well as immediately from eqs
(6.105) we obtain the relation between the drift velocity and the quantity
p(v) in the following way:
V — V
r Л r J
и =
V
r n
____1
гт + V
V*n = V«2 + V„3, Vm = Vm2 + Vm3 + vnm.
(6.111)
As expected, the factor (v‘ — vj^/vj, responsible for LID effect includes
only frequencies of the velocity-changing collisions. The collision fre-
quencies vnl and vm> do not influence this factor and the frequencies vn2
and v„3 as well as vm2, vm3 and vnm are additively combined. The latter
implies that only velocity-changing collisions are important to the LID
effect, whatever the change of the internal state of the particle (if any).
The quantity p(v) will be obtained by using the difference of popula-
tions, which in turn is taken from relations (6.106) and (6.109). This
results in the following equation for the quantity p(v):
2 |G|2(y + v)/g(J„0)
P (? + v)2+(fi-Lv/WBWW()
[Ttp(v) + r2<p>W(v)]},
(Jmo)g(Jno)/
T2= T2„(J„0) + T2m(Jm0)g(Jn0)lg(Jm0').
Hence the final result for p(v) and {p) is
/ x (y + v)2kW(v) . r---
y(v) “ Fs+(O-*.»f - (Г + v)Vf^,
k._2|G|2 Tj
У +vg(J„0)'
(6.112)
(6.113)
364
Light-induced drift of gases
[Ch. 6
The obtained expressions functionally depend on the velocity v and
radiation characteristics in the same way as in the model of two-level
particles (see formula (6.40)). The result of eq. (6.113) differs from that of
eq. (6.40) only in the additional factor WB(Jn0) and the specific meaning of
the parameters b and t2. Therefore, in the assumed relaxation model
(6.105), the fact that we deal with a multilevel problem is reflected only in
redefined relaxation parameters tx and t2 which had already appeared in
the two-level model and in the emergence of a new multiplier of the
quantity p(v).
Proceeding from the formulae (6.111) and (6.113), we obtain an explicit
expression for the drift velocity:
к _ v*-< <p)
kV vj, Гт + vlm
* _ - Vm ^в(Ло)________Улу Re[zw(z)]________
к V vjji Гт + v'„ 1 + 1/k + Ул (т2/r,)y Re[w(z)]'
(6.114)
The factor tp and the function w{z) are given by the formula (6.19), the
saturation parameter к being taken from the formula (6.113).
The main difference between the formula (6.114) and the corresponding
relationship (6.40) for two-level particles consists in the appearance of the
multiplier WB(J„0), reducing the drift velocity. Physically, this fact can be
readily interpreted. Only a part of the particles characterized by the
Boltzmann factor WB(Jn0) interacts with the radiation. Drift motion at the
same time involves all the particles. Consequently, the drift velocity is
lower when the portion of particles at the “working” level under
equilibrium conditions is smaller.
The existence of levels unexcited by radiation is exhibited in the value
of the relaxation parameters Tj and r2. As a rule the ratio t2Itx is in this
case greater than in the two-level model. Especially significant is the
increase in the ratio t2/ti under the conditions of so-called “optical
pumping”. Optical pumping takes place at vnl—>0, i.e. when collisional
exchange between levels J„ is hardly possible. In this case ^(/„o) and then
also the ratio t2/tj tend to infinity. As a consequence, the medium
becomes transparent and the velocity drift tends to zero. Optical pumping
results in the level Jn0 population’s tending to zero, and all the particles’
making transitions to other J„ levels of the state n. Optical pumping may
be manifested also when there are only two levels in the state n.
Spontaneous transitions Jm0—*Jn ^Jn0 evidently favour the appearance of
optical pumping.
§6.8]
Light-induced drift of molecules and multilevel atoms
365
Under optical pumping drift velocities are given by
WB(Jn0)
и = — v---------------<p.
к v* т2 Гт + у'т*
(6.114a)
In this case the drift velocity’s dependence on the radiation characteristics
is determined only by the factor <p.
If we consider molecular vibrational-rotational transitions then m and n
are the vibrational states of a molecule and Jm and Jn are rotational levels.
Under normal conditions spontaneous relaxation of an excited vibrational
state is as a rule small compared with collisional relaxation. Therefore the
parameters ти(тю) and т^СЛо) take the following forms:
UU = - [1 - Ш-Ы] + ,
Vn 1
b.tt>) - —V [1 - Hitt.)] + " b.
v„ - V„2 V
V_ V™ — V„i
r m ’tn ’mi
M4o) = —3---------[1 - %(Лю)] + - Tlm. (6.115)
The Boltzmann factor WB(Jn0) for molecular systems is usually very
small. In addition, for vibrational transitions a large value of the factor
(y'n ~ У1т)/Уп is hardly likely. These are the two main circumstances
leading to a small value of the drift velocity compared with that in the
two-level model. If we take |v‘ - v^,|/v‘ = IO-2 and WB(J„0) = 10-3 and
assume all the other conditions to be optimum, according to formula
(6.114) we obtain
u~10-5v, (6.116)
which implies u — lcms1. This value may seem too small, but its
consequences may prove to be rather significant. For example, for a
concentration of the buffer gas Nb = 2 x 1017 cm-3, a drift velocity и =
1 cm s-1 leads in accordance with formula (6.74) to a concentration change
of the order of the concentration itself at a distance Z.= lv/2u ~ Iffcm
(atr« 3 x 10 15 cm2).
Molecules are characterized by a low rate of vibrational relaxation
compared with translational and rotational relaxation (v"'"«v,1, vi2, v,3).
This leads to an increasing factor r2/Ti and, consequently, to a reduced
366
Light-induced drift of gases
[Ch. 6
drift velocity. However, when molecules whose vibrational frequency has
a similar value are used as the buffer gas the value of v™" increases
because of the excitation transfer to the buffer gas. In the long run, a
value of t2/t! not exceeding 1 may almost always be reached.
When the formula (6.114) is applied to the description of the LID of
atoms, state n denotes the ground electronic state of an atom and Jn
designates components of its fine or hyperfine structure. Normally, the
number of Jn levels is not large (one or a few levels) and therefore the
factor WB(J„) is not small. Also, excited electronic states m of an atom
relax mostly because of spontaneous transitions. The factor |v^ — vjj/vj,
for electronic transitions may be of the order of unity. Therefore, when
the adverse effect of optical pumping can be avoided, the drift velocity of
atoms may be close to that in the model of two-level particles.
The model of strong collisions for multilevel systems studied in the
present section is a natural generalization of the different versions of this
model employed for LID description in refe [9,24,25]. Each of the
models in refs [9,24,25] is a particular case of the model under study; the
general form of the solution (6.114) remains unchanged, only the
parameters Ti and t2 being modified.
In the multilevel model analysed above we took the radiation to be
resonant with one isolated transition. This assumption holds when
absorption lines do not overlap. Such conditions are, however, not always
attainable and radiation may turn out to be absorbed at several transi-
tions. In particular, the frequency separation of the hyperfine structural
components of atomic states is, as a rule, less than or of the order of the
Doppler linewidth. Radiation can thus simultaneously affect several
components of the hyperfine structure. If in the ground state of an atom
orbital moment is zero, collisional exchange between components of the
hyperfine structure is strongly impeded (e.g. alkali metals). The manifes-
tation of the optical pumping effect can be most pronounced in such
objects.
Consider a model scheme where monochromatic radiation induces
optical transitions between each of the n, components of the ground
electronic state and an excited electronic state which consists of a single
level m with degree of degeneracy gm. Each of the n, levels has a degree of
degeneracy (statistical weight) gj,. Take the model of strong collisions with
respect to velocities, collisional exchange between levels n,- being neg-
lected (condition of strong optical pumping). The change in level
populations in this model is described by the following equations
$6.8]
Light-induced drift of molecules and multilevel atoms
367
(stationary, spatially homogeneous conditions):
(I'm + Vm)pm(v) = VmNmW (v) + N £ P/(«),
4pi>(«) = vj,N'„W(v) + Fmpm(v) - Npi(y),
О) Гт
(6.117)
Strictly speaking, the appearance of coherence between levels n, should
have been allowed for as well as the non-linear interference effects
involved. However, in such a general presentation the problem becomes
too complicated and we confined ourselves to consideration of the
situation when the frequency distance between any pair of levels n, and nk
equal to Щ significantly exceeds homogeneous linewidths. This
implies that in any atom radiation may be absorbed only at one of the
transitions n, — m. Consequently, no coherence appears between levels n,
and the manifestation of optical pumping is at a maximum.
From eq. (6.97) by standard procedures we find
p(y) = N Tlm^pi(y) +r^p^W^v) ,
= г 1(6.118)
rm + vm t
The distribution pm(v) is a set of selective Bennett structures on a
background of the homogeneous saturation band. Each of the Bennett
structures described by the function p,(v) is due to radiation absorption at
the transition n,-m. The distribution pm(v) is shown qualitatively in fig.
6.9 for the case of three hyperfine components.
Allowing for relation (6.118), we obtain from eqs (6.117) the following
equation for p'„(v):
pj,(v) = N„W(v) - 7v[ T$p,(v) - (-7 - rft) S P*(«) +
L \v„ /
T2m(p>W(v)l,
-I
,0 1 Л \
Tn ' *m * rm'
(6.119)
368
Light-induced drift of gases
[Ch. 6
Fig. 6.9. Velocity distribution of populations of levels m, nt, n2 and n3 under conditions of
overlapping absorption lines and large Doppler broadening.
On integrating this equation over velocities we obtain the important
relation
(6.120)
which implies the mutual proportionality of absorption probabilities
This relation is one of the most significant manifestations of optical
§6.8]
Light-induced drift of molecules and multilevel atoms
369
/
pumping. Its physical meaning is quite clear. Irrespective of intensity and
frequency of radiation over a sufficiently long time a dynamic equilibrium
is attained under which the transition from the level n, due to radiation
absorption (the quantity (p,)) is compensated by the transition /n—
due to spontaneous processes (the quantity which, in its turn, is
equal to Гт{р )/Гт).
Let us represent the populations p„(v) as
p‘(v)=^M¥(v)-
8n
4- (-7 -
g„ = Sgi, (6.121)
I
where the term in square brackets is due to the action of the radiation.
The velocity distribution pj,(v) includes Bennett structures in the form of
holes due to the immediate action of radiation on the population of level
и, (the term p,(v) and peaks due to the transfer of non-equilibrium
structures from level m under the spontaneous transition m —> и, (see fig.
6.9b,c,d). These peaks are the results of radiation absorption at transi-
tions nk-m under к =# i.
For the weight factors which determine the amplitude of the
homogeneous saturation band in the distribution p'„(v) the following
relation is valid:
г* 1 Л v
= (6.122)
i *m i *-m
which is obtained on integrating relation (6.121) over velocities and
summing it over levels и, allowing for particle balance (N = Nm + S, N'„).
The velocity distribution of all particles is
Pm(v) + S Pn(y) = NW(v) - NTlm[X Vm , V" +
/ 1 1 L v„
2 l7-7irihi+S w w(,4
к x yn yn' J i yn J
(6.123)
The velocity distribution of particles contains in the general case non-
equilibrium parts proportional to the functions p,(v). Weight factors are
370
Light-induced drift of gases
[Ch. 6
non-zero both because the collision frequencies v‘„ differ from vm and
since the v‘„ are different from each other.
From the formula (6.123) we can obtain the expression for the drift
velocity in terms of absorption probability:
к rm + vm^
Vn V / 1 1 \ Г*
--i—+2J —- —)г« 7ГФ.
v'„ k \V„ vkJ J Гт
(6.124)
The factors <p, and absorption probability {p) are still not defined. To this
end, let us write down equations for p,(v) using formula (6.117) and
expressions (6.118) and (6.121) for populations:
Pi(v) =
2 |G,|2 (y, + v,)/#*„
(y, + v,)2 + (Д - к • v)2
— W(v)-
gn
- (-7- 2 Pk(y) + T?*(p,) W(v)
1 / F \ £' 1
r</> = — (1 - —,
v* \ Г + v / a Г + v
rn ' лт 1 rm' on 1 rm
gml m
(6.125)
We obtain a system of coupled equations. However, our previous
assumption of the smallness of homogeneous linewidths of absorption as
compared with the hyperfine splitting enables us to neglect terms pk(v) at
k^i in eqs (6.125) under large Doppler broadening. Indeed, in this case
Bennett structures due to the functions p*(v) do not overlap. Therefore,
in the range of velocities where the quantity p,(v) significantly differs from
zero, other functions pk(v), к =# i, approach zero. The system of equations
(6.125) is thus diagonalized and its solution is
gn Yj(v)
Pl(v> g.^' + ^W
z (У. + у<)2*.^(у)
'( } Г2 + (Ц-ЛтУ
П = (у, + к.)2(1 + к,).
gn
gn ^0+ ^(X) ’
2 |G,|2 tP
У, + v, g‘„ ’
(6.126)
Each of the functions p,(v) has the same structure as in the case of an
isolated absorption line.
§6.8]
Light-induced drift of molecules and multilevel atoms
371
Proceeding from relations (6.126), we can obtain explicit expressions
for the factors <p, entering into formula (6.124):
Re[z<H'(z,)]
Re[w(z/)] ’
Zi = QJkv + IFst/kv.
(6.127)
The dependence of the factors <p, on the radiation characteristics is,
therefore, ordinary with respect to the transition n-m.
Formulae (6.126) are not final results, since the parameters remain
unknown. Now the property of eq. (6.120) and relation (6.122) must be
used. It follows from eq. (6.120) in particular that
1 1
7* rmg„ <p,) {p)'
(6.128)
Substitute here the expression for (p,) from formula (6.126) and allow for
the relation (6.122); this yields
? Гт tV
i rm{Yt)
1 V1 I"*m
Тг=г~^Т~7~
*m i лт r n
Sn^2m
Sm
(6.129)
Hence the result for the partial characteristics as well is easily obtained:
Pity) =
Гт Y^/jY)
к
{Pi} =
г\п!Гт
Ti + XlWWJ
к
(6.130)
Let us consider the results obtained. Under conditions of overlapping
Doppler lines of the isolated transitions, optical pumping does not lead to
a drastically decreased absorption probability and then drift velocity as
distinct from the above-discussed case of an isolated absorption line.
Formally this is manifested in the fact that in the expression for the
absorption probability (p) all the quantities (Y,) may be of the same
order, not differing greatly from their maximum value. Physically, this can
be easily accounted for, using velocity-changing collisions. During absorp-
tion, say, at the transition n,-m followed by spontaneous emission a
particle may come to the level nk Ф n, without changing its velocity. If the
Light-induced drift of gases
[Ch. 6
velocity of the particle was within the range of “resonant” velocities for
the level n(, this velocity is no longer resonant at level nk, i.e. the particle
no longer interacts with the radiation. Nevertheless, as a result of elastic
collisions the particle velocity changes and again is within the “resonance”
(already for level и*) interval after which the particle can again interact
with the radiation. This qualitative picture shows that, as the collision
frequencies v‘„ decrease, optical pumping must become more pronounced.
Indeed, as v'„ decreases, the parameter (see formula (6.125)) does not
reach an asymptotic value as in the two-level model but increases
unrestrictedly. This leads to an unrestricted decrease in the quantity (p)
and the drift velocity. Therefore, for a hyperfine structure of the ground
state, there is a maximum in the drift velocity as a function of the buffer
gas concentration in the range where v'„ ~ Гт and this decreases as the
concentration is either increased or reduced. If there are few hyperfine
components, the drift velocity near maximum is of the same order of
magnitude as in the two-level model.
If there are some components n, for which absorption is impossible
optical pumping is fully manifested, changing the quantity (p) to zero.
This situation formally reflects that for one or more components |G,|—>0
and, consequently, (p)—>0. In real systems (e.g. atoms of alkali metals)
such optical pumping appears in the radiation field of circular polariza-
tion. Optical pumping in such cases can be reduced only by taking special
measures to provide exchange between the levels n,.
The dependence of drift velocity on radiation frequency under hyper-
fine splitting becomes more complicated. For example, in the particular
case of two hyperfine components (r = 1,2) and similar collision fre-
quencies v*„ (vj, = v2„ = v„), for the drift velocity we obtain
к _ vn-Vm
к v„(rm 4- vm)
(6.131)
Under large Doppler broadening it follows from the above that
k_ v„-vm__________________[Г2-д(П-П)/Гт]/А:й________________
u kV v„(v„ 4- vm) t2 4- №1 exp{[(£? - d)/kv)]2} + a2 exp{[(i2 4- <5)/fcv]2}
Q = (Of+ £2)/2, <5 = (Q2 - Г20/2. (6.132)
§6.9]
Light-induced drift in physics, astrophysics and technology
373
Here 2d is the value of the hyperfine splitting and Q is the detuning of the
radiation frequency from the mean frequency of transitions гц-т, n2-m.
The drift velocity becomes zero at Q= д(Г^ — Г^)/Гт, which in the
general case does not coincide with the position of the absorption line
maximum. The Q dependence of the drift velocity is not antisymmetric: in
some direction (along к or opposite to k) the value of и is greater than in
the opposite case. With <хг = <x2 the absorption line becomes symmetric
about Q = 0 but in this case too the Q dependence of и remains
asymmetric.
Numerical calculations of the drift velocity for sodium atoms under
hyperfine splitting are given in refs [6,20] where all the above-discussed
qualitative relationships are treated. The role of hyperfine splitting under
partial collisional exchange between components has been analysed in ref.
[26].
6.9. Light-induced drift in physics, astrophysics
and technology
As originally intended in the present study, we have confined ourselves to
a unified mathematical description and physical consideration of kinetic
problems of non-linear spectroscopy and deliberately left out the accumu-
lated experimental material. According to our general approach we
performed no detailed analysis of the experimental data on the light-
induced phenomena of gas kinetics. However, since this field of research
is comparatively new we thought it necessary to give at least a brief
account of most important experimental results.
The LID effect was first experimentally observed in 1979 [27]. Sodium
vapour was excited by the radiation of a dye laser. The experimental
scheme of an optical valve was employed (see fig. 6.10): sodium vapour
Radiation
Na
Fig. 6.10. Schematic diagram of an experiment under the conditions of an optical valve.
374
Light-induced drift of gases
[Ch. 6
from the source entered the centre of a capillary through which radiation
was passed. Helium or neon was used as the buffer gas. Depending on
sign of the detuning Q, the LID effect caused sodium vapour to propagate
either to the left or to the right of the capillary. Owing to the fluorescence
of the vapour the effect was distinctly visible.
In the first experiments the physical adsorption of sodium vapour on the
walls of capillary masked the effect. The number of sodium atoms
accumulated on the capillary walls was several orders of magnitude
greater than that in the volume. Therefore, after the sign change of Q, a
long time (—15 min) was required for the sodium to move from one part
of the capillary to the other part.
Nowadays, on carrying out this experiment the capillary walls are
covered by a paraffin [7] or silane [28] film, which completely removes
physical adsorption. Under these conditions it took sodium vapour 10~2 s
to move from one part of the capillary to the other, i.e. the drift velocity
is equal to approximately 10 m s-1. In the experiment of ref. [7] changes in
the transport characteristics for the sodium atom were measured under its
excitation by resonant radiation. This change proved to be dependent on
the kind of buffer gas. The change is greatest (—40%) with xenon as the
buffer gas.
A wide range of experimental studies has been devoted to LID of
molecules under their rotational-vibrational excitation by CO2 laser
radiation. A typical experimental scheme for molecular gases is shown in
fig. 6.11. In some studies [5,25,29-31] isotopic modifications of molecules
were experimentally separated under conditions where one isotopic
component interacts with the radiation and the second component acts as
the buffer gas. Changes of isotopic concentration were recorded by a mass
spectrometer. Under optimum conditions the concentration of the rare
isotope over a cell length of =»1 m was changed by LID several times.
Complex studies of the LID dependence on the characteristics of the
medium and the radiation and on the kind of buffer gas [25,32,33] were
carried out. A dynamics of the LID process development was considered
Mass spectrometer
Fig. 6.11. Typical schematic diagram for the experimental study of molecular LID.
§6.9]
Light-induced drift in physics, astrophysics and technology
375
[34] which was shown to satisfy eq. (6.66). Experimental results showed, in
particular, that the model where transport frequencies of collisions do not
depend on velocity and rotational quantum number proves to be sufficient
for the molecular objects under study.. For each of these objects the
relative change of transport collisional frequency under vibrational
excitation has been found. This change turned out not to be large (~10-2)
and for molecule SF6 it was even less (~10 4).
LID has been used to separate nuclear spin modifications of heavy
molecules and to measure the characteristic conversion time [35,36]. The
conversion time of spin modification was found to depend on the isotopic
composition of the molecule.
A striking manifestation of LID under optical piston conditions was
demonstrated in ref. [37]. Further, the same research group studied in
detail the effect of the optical piston in rubidium and sodium vapours
[6,20,38,39]. In particular, rubidium isotopes have been separated [39].
In experimental studies [6,20,37-39] the adsorption of vapour on the cell
walls was quite pronounced, so the process was rather slow.
When investigators succeeded in avoiding physical adsorption [21, 40] it
became possible to observe the drift of a vapour cloud in a long capillary
[21,22]. In this way a considerable amount of data on sodium vapour LID
has been obtained; specifically, the essential role of the effect of optical
pumping has been established [20,22]. Experiments were usually per-
formed under the conditions of an optically thin cloud, but form
stabilization of an optically dense cloud has been also observed.
Studies of LID effect are at the initial stage as yet. However, even now
we may predict that it will be very significant for a wide range of research
and practical applications.
Nowadays the great importance of LID is acknowledged in the physics
of interatomic and intermolecular collisions. Previously there were no
methods reliable enough for measuring the transport characteristics of
excited states of atoms and molecules. However, it has already been
found that the LID effect is very sensitive to changes in those characteris-
tics under optical excitation. It has been experimentally shown that LID
has laid the basis for obtaining reliable data on the transport frequencies
of collisions of excited atoms and molecules. Therefore experiments on
LID, as it would be expected, stimulated theoretical studies on the
calculation of the transport characteristics of atoms and molecules in
excited states [14,15,40].
Light pressure is well known to play an important role in some
astrophysical phenomena. However, under certain conditions the LID
376
Light-induced drift of gases
[Ch. 6
effect is many times greater than the light pressure. It is natural to expect
that LID may prove very important for astrophysics. LID can be directly
connected with the phenomenon of so-called chemically peculiar stars.
Observed anomalies of the chemical composition of these stars have found
no adequate explanation until recently. As shown in ref. [41], under
conditions characteristic of such stars LID can be responsible for the
separation of chemical elements and observed anomalies. The thermal
radiation of the star acts as the exciting radiation and the spectral
non-homogeneity in the vicinity of the absorption line required for LID is
due to the Fraunhofer absorption lines of other elements and isotopes.
The latest data on the chemical and isotopic composition of planets of
the solar system brought about the hypothesis of the non-homogeneous
chemical and isotopic composition of the protoplanet material. If this is
true the LID effect may be regarded as a real physical mechanism for
separating chemical elements and isotopes in the protoplanet cloud. The
protosun radiation in the molecular emission lines is estimated [42] to be
able to cause the drift of molecular components (specifically, water
vapour) of the protoplanet material and their separation. The required
asymmetry of the spectrum is maintained by the red shift of the emission
lines of the protosun resulting from its gravitational field.
At present much attention is given to development of laser methods of
detecting and analysing small quantities of substances. The LID effect
makes it possible to improve the sensitivity of these methods significantly.
Laser methods (adsorption, fluorescence etc.) of impurity detection are
characterized by the fact that the volume of detection region is much less
than the remaining volume containing impurity. LID may help to set a
“trap” in the detection region where the impurity from the entire volume
can be collected. Thus the detection power of the corresponding method
of detection is improved by many orders of magnitude. The promise that
this method shows is illustrated experimentally in ref. [43] where the
example of sodium vapour is studied.
The possibility of controlling the concentration of admixtures by means
of the LID effect can in our opinion be successfully applied to micro- and
nanotechnology. LID can either prevent an admixture from entering the
region of production operation or, on the contrary, add to that region
measured proportions of admixtures.
The trapping of gas admixtures may prove useful in studying short-lived
particles (isotopes, nuclear isomers etc.) which emerge, for instance,
under nuclear reactions. Even if the lifetime of a particle is «40-28 the
concentration of studied particles can be increased by a factor of more
than 102, which has been demonstrated experimentally [43].
References
377
LID can be successfully employed as a selective “optical pump” in
production problems involving separation, enrichment and purification of
chemical and isotopic mixtures. Isolation of one particle of a mixture by
means of LID under optimum conditions requires ~10-102eV [9]. This
seems quite acceptable for many problems (e.g. purification of isotopes).
LID-based method of isotope separation may defy competition from other
laser methods.
The above examples of the application of the LID effect in various
research fields and practical problems by no means exhaust the pos-
sibilities of the effect. The scope of LID applications will surely increase
and quite unexpected possibilities could emerge as has already happened
with astrophysical applications.
References
[1] F.Kh. Gel’mukhanov and A.M. Shalagin, Pis’ma Zh. Eksp. Teor. Fiz. 29 (1979) 773 [JETP Lett.
29 (1979) 711].
[2] J.H. Ferziger and H.G. Kaper, Mathematical Theory of Transport Processes in Gases
(North-Holland, Amsterdam, London, 1972).
[3] C.V. Heer, Statistical Mechanics, Kinetic Theory and Stochastic Processes (Academic Press,
New York, London, 1972).
[4] F.Kh. Gel’mukhanov, Avtometrija (1) (1985) 49 (Opto-electron., Instrum. Data Process. (1)
(1985) 43). ‘
[5] P.L. Chapovsky and A.M. Shalagin, Kvantovaya Elektron. (Moscow) 12 (1986) 2497.
[6] H.G.C. Werij, J.F.M. Haverkort and J.P. Woerdman, Phys. Rev. A 33 (1986) 3270.
[7] S.N. Atutov, I.M. Ermolaev and A.M. Shalagin, Zh. Eksp. Teor. Fiz. 92 (1987) 1215 [Sov.
Phys. JETP 65 (1987) 679].
[8] V.N. Faddeyeva and N.M. Terent’ev, Tables of the Values of the Function w(z) = e“*2^l +
-?L [ e~'2d/) (Pergamon, London, New York, 1961).
V« Jo J
[9] V.R. Mironenko and A.M. Shalagin, Izv. Akad. Nauk SSSR, Ser. Fiz. 45 (1981) 995.
[10] F.Kh. Gel’mukhanov and A.M. Shalagin, Zh. Eksp. Teor. Fiz. 78 (1980) 1672 [Sov. Phys. JETP
51 (1980) 839].
[11] G. Ecker, Theory of Fully Ionized Plasma (Academic Press, New York, London, 1972).
[12] F.Kh. Gel’mukhanov and L.V. Il’ichov, Khim. Fiz. 3 (1984) 590.
[13] F.Kh. Gel’mukhanov, L.V. Il’ichov and A.M. Shalagin, Physica A137 (1986) 502.
[14] T.P. Red’ko, Opt. Spektrosk. 61 (1986) 946.
[15] W.A. Hamel, J.F.M. Haverkort, H.G.C. Werij and J.P. Woerdman, J. Phys. В 19 (1986) 4127.
[16] A.I. Burshtein, Kvantovaya Kinetika (University Press, Novosibirsk, 1968).
[17] T.A. Georges and P. Lambropoulos, Phys. Rev. A 20 (1979) 991.
[18] A.K. Popov, A.M. Shalagin, V.M. Shalaev and V.Z. Yakhnin, Zh. Eksp. Teor. Fiz. 80 (1981)
2175 [Sov. Phys. JETP 53 (1981) 1134].
[19] A.K. Popov, A.M. Shalagin, V.M. Shalaev and V.Z. Yakhnin, Appl. Phys. 25 (1981) 347.
378
Light-induced drift of gases
[Ch. 6
[20] H.G.C. Werij, Ph.D. Thesis, Leiden, 1988.
H.G.C. Werij and J.P. Woerdman, Phys. Rep. 169 (1988) 45.
[21] S.N. Atutov, St. Lesjak, S.P. Pod’jachev and A.M. Shalagin, Opt. Commun. 60 (1986) 41.
[22] H.G.C. Werij, J.F.M. Haverkort, P.C.M. Planken, E.R. Eliel, J.P. Woerdman, S.N. Atutov,
P.L. Chapovsky and F.Kh. Gel’mukhanov, Phys. Rev. Lett. 58 (1987) 2660.
[23] F.Kh. Gel’mukhanov and A.I. Parkhomenko, Zh. Eksp. Teor. Fiz. 92 (1987) 813 [Sov. Phys.
JETP 65 (1987) 458].
[24] A.M. Dykhne and A.N. Starostin, Zh. Eksp. Teor. Fiz. 79 (1980) 1211 [Sov. Phys. JETP 52
(1980) 612].
[25] V.N. Panfilov, V.P. Strunin and P.L. Chapovsky, Zh. Eksp. Teor. Fiz. 85 (1983) 881 [Sov. Phys.
JETP 58 (1983) 549].
[26] A.I. Parkhomenko and A.M. Shalagin, Physica C142 (1986) 120.
[27] V.D. Antzygin, S.N. Atutov, F.Kh. Gel’mukhanov, G.G. Telegin and A.M. Shalagin, Pis’ma
Zh. Eksp. Teor. Fiz. 30 (1979) 262 [JETP Lett. 30 (1979) 243].
[28] J.H. Xu, M. Allegrini, S. Cozzini, E. Mariotti and L. Moi, Opt. Commun. 63 (1987) 43.
[29] A.K. Folin and P.L. Chapovsky, Pis’ma Zh. Eksp. Teor. Fiz. 38 (1983) 452 [JETP Lett. 38
(1983) 549].
[30] A.E. Bakarev, S.M. Ishikaev and P.L. Chapovsky, Kvantovaya Elektron. 15 (1988) 1318.
[31] P.L. Chapovsky, Kvantovaya Elektron. 15 (1988) 738.
[32] A.E. Bakarev, A.L. Makas and P.L. Chapovsky, Kvantovaya Elektron. 14 (1987) 574.
[33] A.K. Folin and P.L. Chapovsky, Opt. Spektrosk. 62 (1987) 214.
[34] P.L. Chapovsky and A.M. Shalagin, Kvantovaya Elektron. 14 (1987) 574.
[35] P.L. Chapovsky, L.N. Krasnoperov, V.N. Panfilov and V.P. Strunin, Chem. Phys. 97 (1985)
449.
[36] A.E. Bakarev and P.L. Chapovsky, Pis’ma Zh. Eksp. Teor. Fiz. 44 (1986) 5 [JETP Lett. 44
(1986) 4].
[37] H.G.C. Werij, J.P. Woerdman, J.J.M. Beenakker and I. KuSSer, Phys. Rev. Lett. 52 (1984)
2237.
[38] W.A. Hamel, A.D. Streater and J.P. Woerdman, Opt. Commun. 63 (1987) 32.
[39] A.D. Streater, J. Mooibrock and J.P. Woerdman, oipt. Commun. 64 (1987) 137.
[40] M.L. Strekalov, Khim. Fiz. 5 (1986) 1555.
[41] S.N. Atutov and A.M. Shalagin, Pis’ma Astron. Zh. 14 (1988) 664.
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Siberian Branch, (1986).
[43] S.N. Atutov and A.M. Shalagin, Opt. Spektrosk. 64 (1988) 223.
7
Light-induced gas kinetics
7.1. Modifications of the light-induced drift effect
The discovery of the light-induced drift (LID) effect cast a new light on
kinetic processes in gas and gas-like systems exposed to radiation.
Radiation proved to have a specific influence on the velocity distribution
of gas particles, which leads to a series of new effects of gas kinetics. Thus
a new branch of gas kinetics arose: light-induced gas kinetics (LIGK),
whose effects will be briefly touched on in the present chapter.
The most interesting LIGK effect is still the LID effect. It is, therefore,
best studied both theoretically and experimentally. In particular, various
modifications of the effect have been revealed, the physical causes of
which are the same as that of “ordinary” LID but which are manifested in
a different way.
Suppose that ions act as absorbing particles and buffer particles are
neutral. Obviously, the transport characteristics of the ions (in particular,
they characterize the mobility of the ions) are determined by their internal
quantum state. Optical excitation selective in velocities causes the LID of
ions in buffer gas (by the ordinary LID mechanism). Ion drift implies the
presence of an electric current [1] which can be detected by an
appropriate device. In ref. [1] this effect is called a light-induced current.
ТЪе density of the light-induced current
J = qNu, (7.1)
(q is the ion charge) can easily be evaluated from the results of the
previous chapter. For и = 103 cm s-1 (values easily attainable for LID) and
N = 1012 cm-3 it follows that
J = IO-4 A cm-2; (7.2)
this value can easily be detected.
Ion drift goes on until a potential difference appears at the ends of the
absorbing cell which terminates the drift. If positively charged ions are
considered, because of gaseous discharge the stationary situation is
characterized by equal velocities of ion and electron drifts. The electron
379
380
Light-induced gas kinetics
[Ch. 7
drift velocity, in its turn, is connected with the potential difference by the
relation
uc = ii'£IL. (7.3)
where L is the length of the absorbing cell and ц is the electron mobility.
The relation between the established potential difference and the velocity
of light-induced ion drift follows from the condition и = ue:
^=иЦц. (7.4)
The quantity is proportional to the length of the absorbing cell. If
L = 10cm, then for u = 103cms-1 and /1 = 5 x 104 cm2 V1 s 1 (under a
buffer gas concentration Nb = 1017 cm-3 [2]) we have
£ = 0.1V. (7.5)
Current generation due to ion LID must take place in the atmospheric
conditions of chemically peculiar stars since heavy elements are fully
ionized there. Light-induced ion drift may account for the large-scale
magnetic fields inherent in some chemically peculiar stars [3].
The light-induced current effect is possible when a gaseous medium of
initially neutral particles [4,5] is exposed to light if ions are produced by
laser radiation (step ionization, associative ionization etc.). Common to
most mechanisms must be the fact that at one of the stages radiation-
induced excitation is selective in velocities [4]. Ions in this case have a
directed velocity. Electrons acquire the same velocity (in the ion rest
system electron emission may be considered isotropic). In the buffer
medium ions and electrons are differently decelerated (electrons lose their
directed momentum sooner), which results in the appearance of an
electric current and, as a consequence, charges are separated [4]. In the
absence of the buffer gas [5] electrons as light particles quickly reach the
walls, leaving in the volume of gas only ions in ordered motion. Since the
directed component of electron momentum is small compared with the
isotropic component, the current that arises is practically completely due
to the ion component.
It is known that electrons in the conduction band of semiconductors
have many properties typical of gases. The analogy proves to be so
profound that under some conditions an LID-like effect is possible in
semiconductors which gives rise to an electric current [6,7]*.
* Much later this effect was newly discovered in ref. [8] (in connection with this problem, see also ref.
[9])-
§7.1]
Modifications of the light-induced drift effect
381
Suppose that a radiative transition is allowed between two energy states
m and n of electrons characterized by energies Em and En dependent on
momentum p. The laws of conservation of energy and momentum under
photon absorption have the following form:
ho) + En(p) = Em(p + Др), Ар = йй. (7.6)
Normally the photon momentum is much less than the characteristic
electron momentum in a crystal. Therefore we may use an expansion of
Em(p + Ap), after which relations (7.6) take the form
3Em
(o-comn = k-v, v=-—,
dp
И(отп(р) = Ет(р)-Еп(р). (7.7)
In gases the quantity is not momentum dependent and then the first
of relations (7.7) reflects the Doppler effect, causing selectivity in
velocities for radiative transitions. If the dispersion laws Em(p) and E„(p)
differ significantly or, as it is said, zones are not parallel, radiative
transition is possible both at к • v > 0 and under к • v < 0. For excitation
to be selective and anisotropic the zones must be parallel. This condition
is fulfilled for transitions between Landau levels in a quantizing magnetic
field and also between levels due to space quantization in thin films or
transitional layers. Both of these cases are fully analogous with that of gas
media from the viewpoint of the LID effect [6,7]. Light-induced electron
drift takes place in the direction of free motion. Electron momentum
relaxation is due to the collision of electrons with phonons, impurities and
lattice defects. Relaxation velocities in the general case differ for different
quantum states of electrons, i.e. the second condition required for LID to
take place may be also fulfilled.
Reliable experimental evidence has been obtained for the existence of
the LID effect for conduction electrons in semiconductors [10].
Not only electrons but also other quasi-particles in solids can behave as
gas particles. If those quasi-particles can move freely and have quantized
values of their internal energy, this provides conditions for the LID effect
to be manifested. Wannier-Mott excitons have these properties and the
LID effect is predicted and estimated for them in ref. [11].
Let us return to gas systems and discuss one more interesting
modification of the LID effect, the so-called surface LID [12,13]. The
absorbing particle flows may relax not only as a result of their collisions
with the buffer gas but also because of their friction against the surface
382
Light-induced gas kinetics
[Ch. 7
(longitudinal momentum accommodation). If accommodation coefficients
are different for excited and unexcited particles, excitation is selective in
velocities and brings about drift of the absorbing particles. This type of
drift has been called the surface LID. The surface LID is due to particles
close to the surface in the layer whose thickness is of the order of the
mean free path. Obviously, for the effect to be most pronounced,
Knudsen conditions must be provided (the cell diameter must be much
less than the mean free path). Surface LID in this case will establish a
pressure gradient of the absorbing gas along the cell.
The simplest description of the surface LID effect can be given in the
following model. Let radiation pass through a long capillary tube
containing only the absorbing component of the gas. Volume collisions
will be neglected and collisions with the wall will be considered elastic.
The result of collisions with the wall will be characterized by momentum
accommodation coefficients am and an that give the proportion of particles
diffusely scattered by the wall (if a, = 1 the scattering is completely
diffuse; if a, = 0 we have specular reflection). The coefficients am and an
are analogues of transport cross-sections under collisions. Each particle in
a capillary experiences on the average about v/2R collisions with the wall
per unit time (R is the radius of the capillary). For density matrix
elements averaged over cross-section of the capillary and their transverse
velocities effective integrals of collisions with the wall can be introduced
which in the model of strong collisions assume the form (the z axis lies
along the capillary axis)
St(vz) = - v,p,,(Uz) + vtNtW (yz), i = m,n-,
vi = txlvl2R, (7.8)
where W(uz) is a one-dimensional Maxwellian distribution and the v, are
effective frequencies of strong collisions.
Under the above assumptions all the results referring to the volume
LID effect and obtained for the model of strong collisions (see ch. 6) hold
true also for surface LID if the parameters v, = &ivl2R are taken as strong
collision frequencies. In particular, by analogy with formula (6.17), we
have
(7-9)
<xn-<xm {p)
u =-----------------(p,
<Xn rm + vm
where the factor <p and the absorption probability (p) are given by the
formulae (6.19) and (6.40).
57.1]
Modifications of the light-induced drift effect
383
The results of section 6.7 oriented at the model of strong collisions are
valid for the description of changes in the absorbing particles concentra-
tion N. In particular, for an optically thin medium formula (6.35) is valid
where the diffusion coefficient is transformed in accordance with the new
meaning of the transport collision frequencies vm and v„. If the values of
the accommodation coefficients <xm and a„ are close to each other, then
It must be emphasized that in the case of the surface LID effect a
change in N is followed by a changed gas pressure, since the gas is a
one-component gas. This, however, does not imply that radiation trans-
fers momentum to the gas. The analogy with the Maxwell demon holds
here as well: the total momentum of the system of gas + absorbing cell
remains the same in the process of surface LID.
The concrete realization of the effect of surface LID considered here
can be compared with the classical problem of Knudsen flow in a thin
capillary. In this problem the pressure difference at the ends of the
capillary maintains the flow of gas particles along the capillary. The flow
velocity has a finite value owing to the collision of gas particles with the
walls. If there is no source maintaining the pressure difference, the
particle flow equalizes the pressures. Surface LID in the case described
above is the “reverse process”: a flow of particles is induced in an initially
equilibrium gas; this flow then leads to a differential pressure at the ends
of capillary (if at least one of its ends is closed). The pressure differential
increases until the flow terminates.
Experimentally the surface LID effect has been detected under
vibrational-rotational excitation of CH3F molecules [14]. The relative
change in concentration N owing to this effect proved to be rather small
(~10-3), which is due to an insignificant (~10-3) relative change in the
momentum accommodation coefficient when the molecule is vibrationally
excited. Under conditions when the volume LID effect is manifested, the
surface effect leads to vanishingly small corrections [15].
Elements of the volume and surface LID effects are combined in the
LID of aerosols predicted in ref. [16] and considered in more detail in ref.
[17]. On the one hand, aerosols may be treated as buffer particles and we
must consider the difference between transport cross-sections of excited
and unexcited absorbing particles under scattering by aerosols [16]. On
the other hand, aerosols can be treated as macroparticles on whose
384
Light-induced gas kinetics
[Ch. 7
surface momentum accommodation takes place; in this case surface LID
concepts can be used, as in ref. [17]. An experimental study of aerosol
LID would be interesting in particular because the motion of a single
particle could be observed.
7.2. Light pressure and light-induced drift
On the derivation in section 2.7 of the equations of gas kinetics in a
radiation field the light pressure effect was neglected. As a rule, this effect
is insignificant compared with other LIGK effects but under special
conditions it may easily be manifested. The light pressure effect is caused
by photon momentum transferred to an absorbing particle and, as a
consequence, may cause space non-homogeneity of the concentration of
the absorbing component. Light pressure was chronologically the first
effect of radiation on the motion of gas particles to be discovered. It was
first observed by P.N. Lebedev at the beginning of this century [18].
It seems important to study the role of light pressure as compared with
LID under conditions when the latter effect is usually manifested. To
allow for the recoil effect causing the light pressure we shall proceed from
eq. (2.53) for the density matrix. Since the recoil veloctiy hk/m is usually
small compared not only with thermal velocity v but also with characteris-
tic scale of non-equilibrium structures in the velocity distributions of
populations the quasi-classical approximation may be employed, to which
corresponds eq. (2.55).
We shall confine ourselves to the model of two-level particles. In the
field of a travelling monochromatic wave the equations for density matrix
according to eq. (2.25) take the form
/3 \ d
(T,+ Гт + v ’ V+ NP(V) ~ % '
\dt / dv
/д \ * d
( + v . у jpnn(v) = S„(v) + Lmpmj(v) - Ap(v) - £N—p(v),
\dt / * dv
Г Э .1
— 4-V’V+y + v — i(£2 — к • v) p(v)
.at J
Э
= S(v) + iG[p„„(v) - - iG£ • — [p„„(v) + pmm(v)],
dv
57.2]
Light pressure and light-induced drift
385
2Vp(v) = -2Re[iG*p(v)], £ = йЛ/2т; (7.11)
[ <5[(«i - v)2 - (2§)2]pmm(v!) dvj.
ZJtnK J
(7-12)
The recoil effect has been manifested here as additional differential
dynamical terms and also in the fact that the spontaneous transition m -+ n
is described by the integral term Гтртт(г) due to the change of particle
velocity under the spontaneous emission of a photon. Expression (7.12)
for the term tmpmm(v) is obtained from formulae (2.83) and (2.84).
From eqs (7.11) macroscopic equations of gas kinetics may be derived,
the procedure being similar to that in section 2.7. Here we shall limit
ourselves to the analysis of the contribution of light pressure to particle
flows. Under stationary spatially homogeneous conditions we have
J=Yi^Jm + 2LN{p)y
J vl J Vt
(7-13)
On obtaining these equations we made use of the relation
\ L oV J/
(7-14)
the validity of which can be proved by integrating the left-hand side by
parts.
From eqs (7.13) we obtain the following expression for the flow of
absorbing particles j:
, vl-vT <vp(v)) £ Гт + (v? + v?)/2
J =--------~--------Ь 2/V------------------
vl Г + vT vl Г + vT
<p).
(7.15)
The flow j induced by the radiation consists of two parts: the flow caused
by the immediate action of the radiation force and directed along к (the
second term in expression (7.15)) and the flow due to the asymmetry of
the function p(v) and the difference in the transport characteristics for
levels m and n. The action of the force is a result of the absorption of a
radiation quantum with momentum ftk and subsequent isotropic spon-
taneous emission causing the particle to receive momentum kk.
Sometimes this radiation action is referred to as spontaneous light
pressure [19]. The mean force acting on a particle is, therefore, ftk(p'). If
v™ = v? only this force is responsible for the motion of gas particles.
386
Light-induced gas kinetics
[Ch. 7
The difference in v" and v™ causes the LID effect. It turns out,
nevertheless, that one more effect arises simultaneously with the LID due
to the recoil effect under absorption and induced emission [20]. Let us
consider this case in more detail.
Assume, as before, the phase memory to be absent under collisions.
Then for p(v) the following expression obtained from eqs (7.11) is valid:
A^p(v) = Npo(y) + N Ap(v),
2 |G|2 (y + v)
Wp°(v) = ^r + vy + ^Q_k.vy “ P™n(V)L
2 IGI2(v + vl d
N = (y+v)2?(Lj.v)2 * • Tv[P^ + (7-16)
The main term Np0(v) in this expression is caused by induced optical
transitions selective in velocities and leading, in particular, to the LID
effect. The second term N Ap(v) is due to the recoil effect. It is
proportional to the photon momentum and its velocity dependence is
specifically asymmetric. The non-equilibrium parts in the velocity dis-
tributions at levels m and n due to the term Ap(v) prove not to be
equalized at v"^vf, i.e. they are transferred to the total velocity
distribution function of the absorbing particles as in the case of LID.
Part of the total flow (7.15) connected with Ap(v) is proportional to the
quantity (v Ap(v)). If the total velocity distribution pmm(v) + pnn(v)
does not differ much from the Maxwellian distribution, it follows from
expression (7.16) that
(v Ap(v)) = #v4=^
(y + v)u2exp(-u2/i)2)
(y +v)2+(£2-£uJ2
duz.
(7-17)
Note that this expression remains unaltered when the sign of Q is
changed, i.e. the part of the flow (7.15) due to the recoil effect under
induced optical transitions is symmetrically dependent on Q in the same
way as the flow due to spontaneous light pressure. The new effect,
therefore, has both the features of the LID effect (it is non-zero at
v^^v?) and those of the usual light pressure effect (symmetrical
dependence on the detuning Q).
Take Q = 0 so that the LID effect should be absent and consider the
case of homogeneous broadening for which the solution of eqs (7.11) is
the simplest. The velocity distributions of the populations will be
presented as in the relation (6.25), after which eq. (7.13) for Jm, if we take
§7.2]
Light pressure and light-induced drift
387
into account relation (7.17), is transformed to
1 Г t
1 + к, L 2 7
<p)
г + v?
Lm ' rl
+ y)^
K1 = 4|G|2/(y + v)(rm + vr).
(7-18)
On substituting this expression into the equation for j we finally obtain
. (v" — vT <p) rrm + (v7+vT)/2 11
1 + к I v? + v? v" L rm + vf J J
_ 2 |G|2 v?+v7 . Гтк/2 __ 4|G|2
K Гт + vf v?(y + v) ’ P 1 + к ’ K rm(y + vY
(7-19)
For the quantity (p) we used expression (6.32), discarding corrections
due to the recoil effect as unimportant in the approximation employed.
The first term in braces of the expression for j is due to the recoil effect
under induced optical transitions. The remaining part of the flow j is due
to spontaneous light pressure.
Interestingly, sometimes conditions are possible under which the effect
caused by induced transitions prevails. At Ki » 1 when the rate of induced
transitions is greater than that of spontaneous decay, formula (7.19) takes
a particularly simple form:
(7.20)
If in this case |v7 — vT\»2rm, the second term in formula (7.20) can be
neglected and the recoil effect under induced transitions becomes practi-
cally completely responsible for the flow. The direction of the flow,
depending on the sign of the difference v? - v”, may be either along the
wavevector k or opposite to k.
In the above-discussed limiting case the physical interpretation of the
effect is the simplest (see ref. [20] where the effect is called “negative”
light pressure). Figure 7.1 illustrates how the flow j can arise only from
induced processes. Induced processes equalize the populations of the
levels m and n, but the velocity distributions are displaced from each
other at a distance equal to the recoil velocity 2§ = hklm (in fig. 7.1 the
recoil velocity is arbitrarily taken to be large for easier interpretation).
Therefore they cannot simultaneously be at equilibrium. Equilibrium
distributions can be established by collisions. However, when collision
frequencies differ (v“ Ф v") the equilibrium components of distributions
388
Light-induced, gas kinetics
[Ch. 7
Fig. 7.1. Drift initiation due to the recoil effect under induced transitions. The model of strong
collisions is assumed and saturation is taken to be strong.
also become different. This results in stationary velocity distributions
similar to those shown in fig. 7.1 (yT> v"). The total distribution in this
case becomes asymmetric, which implies the presence of a particle flow. If
the usual procedure is used, the flow due to this effect may be shown to
lead only to a change in the partial pressures of the mixture components
and not to alter the total pressure. The gas total pressure is changed only
under the action of the forces of spontaneous light pressure.
If Q Ф 0, under a wide range of conditions the LID effect is essentially
greater than those due to the photon momentum. Consider the ratio of
flows due to the LID effect (jLID) and to spontaneous light pressure (ji.P).
From formulae (7.15) and (6.17) we have
/lid = mv Vj-vT
7lp hk rm + (V[+vT)/2'
The factor mv/hk in the above expression is equal to the ratio of the
thermal momentum of a particle to the photon momentum and its value is
large. For atoms and molecules for which the molecular weights equal 10
and for an optical spectrum (Л = 2n/k = 10-4cm), at room temperature
(u = 5 x 104 cm s-1) we have mv/hk ~ 104. Under conditions optimal for
LID the remaining multiplier in relation (7.21) may be of the order of
unity and then the LID effect is greater by a factor of 104 than the
spontaneous light pressure. Both effects are comparable in three cases:
when the difference between collision frequencies is anomalously small
(|v"— v7|/(v7+ v?)~ 10-4), under a very low pressure of the buffer gas
when у\/Гт ~ 10-4 and when the value of the factor <p is very small (for
<p < 10-4 under all values of the radiation frequency the buffer gas
§7.3]
Light-induced pulling and pushing
389
pressure or radiation intensity must be sufficiently large that kv/rs~
IO"4).
The contribution of the recoil effect to the total particle flow under
induced transitions increases as the radiation intensity and buffer gas
pressure become larger, so in order to obtain its maximum influence we
may use the approximation (7.20). The ratio of /nn to Д/ (the part of the
flow due to the recoil effect under induced transitions) is, therefore, given
by the formula
Д/ hk rm + (v? + v?)/2 V k 7
Under conditions optimal for LID we have у1Ш/Лу ~ 104. The effects of
the contributions are comparable under a very high pressure of the buffer
gas or under very intense radiation. Under these conditions when the
detuning Q is optimal for LID we have (see expression (6.32) for the
factor <p)
/lid. mv 4Гт kv
Д/ hkvl + vT Г/ 1 ‘ }
The value of this ratio becomes of the order of unity at rmkv/(y" +
vr)i;~io-4.
The above analysis shows, in particular, that under not too exotic
conditions the recoil effect under photon absorption and emission may be
neglected when describing the LID effect.
7.3. Light-induced pulling and pushing
When the spatially non-homogeneous equations (6.63) and (6.64) for
particle concentration were analysed in section 6.6, the effect due to the
spatial inhomogeneity of radiation [21] was neglected. Now it will be
treated separately. The essence of the effect consists in the fact that
absorbing particles may be either pulled into the region of high radiation
intensity or pushed out of it.
By way of an example of the physical basis of the pulling-pushing effect
consider the spatially inhomogeneous radiation intensity in the cross-
section of the light beam. Figure 7.2a shows the distribution of radiation
intensity |E|2 along one of the coordinate axes (y) in the cross-section.
390
Light-induced gas kinetics
[Ch. 7
Fig. 7.2. (a) Distribution of radiation intensity and concentration of (b) excited and (c) unexcited
particles over a cross-section of the light beam. Radiation brings about concentration gradients of
excited and unexcited particles equal in magnitude but opposite in sign.
The radiation resonantly excites level m. Since the lifetime of the excited
state is finite (sufficiently short), excited particles are present only inside
the light beam. Consequently, the concentration distribution of excited
particles proves to be spatially non-homogeneous. The same is true for
unexcited particles: their concentration outside the light beam is greater
than that inside. The distribution of concentrations N„(y) and Nm(y) is
shown qualitatively in figs 7.2b,c. It is known that when there is a
concentration gradient of some component of a gas mixture there is
diffusion flow of this component proportional to the concentration
gradient. Assume that as well as the absorbing gas there is a buffer gas of
sufficiently high concentration. The resulting partial diffusion flows Jm and
J„ are connected with the corresponding concentration gradients in the
following way:
Jm = -DmVNm, — (7.24)
where Dm and D„ are diffusion coefficients. According to figs 7.2b,c the
flows jm and j„ are directed out of and into the light beam.
Optical excitation in itself does not cause a particle to travel, i.e. does
not change the spatial homogeneity of the concentration. This implies
§7.3]
Light-induced pulling and pushing
391
that, at every point, under the action of the radiation, gradients of
concentrations of the same value and of opposite sign are created
(V7Vm = —V7V„). If the diffusion coefficients Dm and D„ are, however,
different, the flows Jm and J„ do not compensate each other. There arises,
therefore, a flow of the absorbing component as a whole directed either
inside or outside the light beam. It can easily be seen that if Dm < D„
particles will be pulled into the light beam and under Dm > D„ they will be
pushed out of it.
As an LID effect, the pulling in-pushing out effect is possible only
when there is a buffer component, the concentration change of the
components taking place under unchanged total pressure of the mixture.
Component separation also may take place without energy (or momen-
tum) loss by the radiation. This is justified by the general analysis (see
section 6.3) of eqs (2.335)-(2.339) containing also the effect of pulling
in-pushing out.
This effect will be quantitatively studied under conditions of homoge-
neous broadening. The results of eq. (6.69) can be used immediately.
Assume that the decay time of the excited state 1/Гт is much shorter than
the time of particle diffusion across the light beam, i.e.
rm>>v2/a2vT (7.25)
where a is the mean radius of the beam. Then the populations Nm and N„ of
levels m and n are determined by the local characteristics of the radiation
and expressions obtained from formulae (6.70) and (6.32) hold true:
„ - V S „ _ K,rm+g 2\G\2 Г
m rm + 2g’ " rm + 2g’ 8 r2 + Q2
The simplest expression for diffusion flow jd from eq. (6.69) is obtained
in the approximation
(7.26)
vT,vr»rm-l-2g, (7.27)
which means that the transport frequencies of collisions greatly exceed the
velocities of spontaneous and induced transitions. Then the diffusion
coefficients Dn and Dm have the usual form
= v2l2v\, i = m, n,
and diffusion flow jd may be represented in the following way:
DM =
v2 Гт+g(vi +v”)/v”
2v? Гт 4- 2g
(7.28)
(7.29)
392 Light-induced gas kinetics [Ch. 7
The effective diffusion coefficient Def(g) introduced here depends on the
radiation intensity and, as a consequence, on the coordinates. This very
circumstance is formally responsible for the effect of pulling in-pushing
out.
In directions orthogonal to the direction of radiation propagation the
LID effect is not pronounced and only diffusion flow jd is possible, which
redistributes the concentration (pulling in-pushing out). Under stationary
conditions jd becomes zero at every point so that from eq. (7.29) we
obtain
Def(g)N = constant. (7.30)
This indicates that not only Def(g) but also the concentration N is
coordinate dependent. If we denote the concentration outside the beam
by No, from the formula (7.30) we find
N-No = g(vT-v4)/vT
No rm+g(vn1 + vT)/vT }
The result of the above implies that the concentration of absorbing
particles at a given point is determined by the radiation characteristics at
the same point, which is a consequence of the approximation (7.25).
According to the above qualitative considerations, the relative con-
centration change given by formula (7.31) is proportional to the difference
vf — v" of transport frequencies of collisions. At v? > v? particles are
pulled into the area of increased radiation intensity and vice versa. As the
radiation intensity increases (with increase in g), the relative concentra-
tion change becomes greater until it reaches an asymptotic value:
N — No _ v” — v"
No v” 4- v"
If v” » v" the concentration in the beam becomes twice as high. This can
easily be interpreted in the following way. In the intense field the
populations of levels are equalized. The condition v™» v" at the same
time means that excited particles practically do not diffuse, whereas
unexcited particles because of the comparatively fast diffusion “manage”
to remove the inhomogeneity of their concentration. As a result particles
in the ground state are uniformly distributed in space. Consequently,
because of the equality Nm = N„ = No, inside the beam the general
concentration is twice as great as No. In the opposite case v” « v", almost
all particles are pushed out of the beam. This situation may have the
following interpretation. The radiation field excites particles and excited
(7-32)
§7.4]
Anisotropy of the pressure tensor
393
particles rapidly diffuse from the beam. The reverse flow of particles is
strongly hindered since particles outside the beam are in the ground state
and their diffusion coefficient is small.
Note that condition (7.27) from the viewpoint of the validity of the
relation (7.31) is actually not necessary; it will be enough to satisfy the
much weaker condition
vr,vT»Tm, (7.33)
which can be proved by more rigorous calculations. However, this is quite
evident also from qualitative considerations. If in the central part of the
beam g » Гт, the populations and total concentration there are practically
independent of the coordinates, the quantity N reaching its asymptotic
value (7.32). Therefore there is no significant contribution from this area
to the concentration redistribution.
Intensity inhomogeneity along the direction of radiation propagation
evidently also produces a pulling in-pushing out effect. If Q = 0 so that
the LID effect is absent then according to eqs (6.68) and (6.69) the only
cause of the spatial inhomogeneity of the concentration is the effect of
pulling in-pushing out. The concentration change in all directions is given
in this case by formula (7.31). When the change in concentration N
along the direction of radiation propagation is mainly due to the LID
effect. The effect of pulling in-pushing out produces additional deforma-
tion of the concentration’s spatial distribution (see, for example, ref. [21])
but this is as a rule not large.
Formula (7.32) predicts that the concentration in the light beam will
become at most twice as great, which is specific for the considered
two-level model. In systems with more levels the concentration increase in
the beam may become significantly greater [22].
The effect of pulling in-pushing out was observed and experimentally
studied in sodium vapour with noble gases [23,24]. This effect was used to
measure the relative change in diffusion coefficient of sodium atoms under
their optical excitation with high accuracy.
7.4. Anisotropy of the pressure tensor: “cooling”
and “heating” of gas components
All the light-induced phenomena of gas kinetics arise because, owing to
the velocity-selective influence of radiation on gas particles and owing to
394
Light-induced gas kinetics
[Ch. 7
the difference in transport characteristics of particles in various quantum
states, a non-equilibrium function of the velocity distribution of the
particles is formed. The most important phenomena are connected with
the creation of the first moment of the distribution function, i.e. the
particle flow, and they have already been considered. In the present
section we shall draw your attention to the second moment of the
distribution function and the related macroscopic characteristics: the
pressure tensor and energy of translational motion. It follows from eqs
(2.337) that the pressure of the absorbing gas component acquires a tensor
structure, and its translational energy becomes different from the equi-
librium energy [25].
According to the structure of the tensor (?b (see relation (2.351))
describing the collisional relaxation of the pressure tensor, this relaxation
may be accompanied by the transfer of the non-equilibrium part of the
distribution from one translational degree of freedom to others. This
process is reflected by the transport collision frequencies v3. Indeed,
TrC/b includes no frequencies v‘3, i.e. the terms of expression (2.351)
containing v3 are responsible only for the anisotropy of the tensors
causing no energy changes of the ith mixture component. The remaining
terms in formula (2.351) evidently describe the independent relaxation of
the tensor component P“p during collisions with the buffer gas. Usually
the transfer of non-equilibrium parts from one degree of freedom to
others is insignificant and may be neglected if there is another (more
important) cause of the anisotropy of the pressure. We neglected this
transfer in order to obtain the simplest formulae*.
In addition, we shall proceed from the model of velocity-independent
transport frequencies. Under spatially homogeneous conditions eqs
(2.337) for pressure tensors take the following form:
/ d \ N
(-Z + rm+ 2v”= Wodap + NQ?-,
\dt / N
/ d\
— = —2v"Pap + 2(v" - vr)P^ +
\dt/
~ [(vT- v?)^+ vj]w0^; Wo = lmv2N. (7.34)
It must be recalled that in the model of velocity-independent transport
* In the case of a one-component gas, exchange of the non-equilibrium structure between
translational degrees of freedom is fundamental and will be accurately allowed for (see section 7.5).
§7.4]
Anisotropy of the pressure tensor
395
frequencies of collisions the required stability of the equilibrium
distribution leads to the equality v\ = v'2 (see formula (2.360)).
Under stationary conditions, from eqs (7.34) we obtain the following
expression for the pressure tensor of the absorbing component as a whole:
" + + 2УГ) " 2 6^vVN- <7'35*
The last equality is obtained by allowing for the relation rmNm = N{p),
which follows from eq. (2.335) under stationary, spatially homogeneous
conditions. When the collision frequencies are the same (v“ = v”) the
pressure, as expected, becomes a scalar corresponding to equilibrium
conditions. This is the same result that we have in the case when the
function p(y) is not selective in velocities, i.e. when it is proportional
to a Maxwellian distribution. Therefore the pressure becomes a tensor
when v” differs from vt and when the function p(y) is not at equilibrium.
When these conditions are satisfied, the mean energy of translational
motion of the absorbing component also becomes different from the
equilibrium value, i.e. the absorbing component is selectively “cooled” or
“heated” owing to the buffer component (recall that the energy of
translational motion of the gas as a whole is not changed). According to
the definition of the energy density W, it follows from relation (7.35) that
(vt - vf}N
W = j Tr P = Wo + {m(v* 2 - lv2)p(v)), (7.36)
"lUm ' ZV1 )
which in the general case is different from the equilibrium value of the
translational energy density Wo.
In the system of coordinates with the azis z directed along the
wavevector к of the travelling wave of the radiation, the pressure tensor P
is diagonal. For p(v) similar to the expression (6.40) the pressure tensor
components are
P- = pyy = 2W(b
Vt~ Vt
1
2pr) —1 y(v)dv.
\ V/ J
(7-37)
396
Light-induced gas kinetics
[Ch. 7
Under the imposed restrictions the components and Pyy are not
influenced by radiation.
It is easily confirmed that the factor <p2 is symmetrically dependent on
the detuning Q of the monochromatic radiation. Under large Doppler
broadening (kv » rs) we have
/fl\2
ф2 = 2 — -1. (7.38)
\kv/
When Q is altered the factor <p2 changes sign at the points Q= ±.kvl\fL
These points are characterized by a kinetic energy mv2J2 of the
radiation-excited particles equal to the mean kinetic energy of motion
along the z axis. If, for instance, v? > v? then under |£?| > kvly/2 particles
with translational energy exceeding the mean energy are excited and its
faster relaxation implies that the absorbing component is “cooled”. At
|£?| <kv/y/2 the energy of excited particles is less than the mean energy
and during collisions they acquire additional energy from the buffer gas.
As seen from formula (7.38), the value of the factor cp2 may be of the order
of unity. As already shown when the LID effect was estimated, the
factor | v" — v?| (p)/v"(rm 4- 2 V'") under optimum conditions may also reach
a value of the order of unity. Therefore the tensor component Pzz may be
changed by a value of the order of the equilibrium value.
The change in Pzz can be particularly large in a radiation field with a
specially fitted spectrum [26]. It is shown in ref. [26] that in principle
Pzz«1% can be obtained.
7.5. Kinetic effects in a one-component gas
If a gas consists of a single component resonantly interacting with
radiation, the usual (bulk) LID effect is impossible in it owing to the law
of momentum conservation. Collisions of excited and unexcited particles
combined with velocity-selective optical excitation, nevertheless, lead to a
peculiar deformation of the velocity distribution function of absorbing
particles and have a certain influence on the macroscopic properties of the
gas. In particular, anisotropy of the pressure tensor [27,28], heat flow [29,
30] and light-induced viscous flow of particles [31] can appear in a
one-component gas.
§7.5]
Kinetic effects in a one-component gas
397
Solving specific problems of a two-component mixture, we assumed the
buffer component concentration to be comparatively high, which enabled
us to consider the buffer gas state to be given. This made the collision
integral linear, which simplified the problem. In a one-component gas
both colliding particles belong to the same component of an unknown
state. This leads to a complex problem when the collision integral is
non-linear.
To facilitate the analysis of the main light-induced kinetic effects in a
one-component gas the so-called linearization of the collision integral will
be employed; this method is rather popular in classical gas kinetics. We
assume that only a small portion of the particles is excited by radiation, so
that collisions between excited particles can be neglected. The velocity
distribution of unexcited particles pn„(v) will be represented as a sum of
an equilibrium term p(^(v)x W(v) and a non-equilibrium component
Др„„(и). Under such conditions we may confine ourselves only to
collisions of particles disturbed from equilibrium with unexcited equi-
librium particles. Consequently, the approach of section 2.7 which is
based on the introduction of transport collision frequencies can be
employed.
The general equations (2.335)-(2.339) for the density matrix moments
hold with the exception of the equations for the flow J and energy density
W which, by the laws of conservation of momentum and energy, in a
one-component gas take the form
m|y + V-P = 0, (7.39)
+ div 9 = 0. (7.40)
Let us now direct our attention to the pressure anisotropy in a
one-component gas, i.e. to the analysis of the pressure tensor in the field
of resonance radiation. In eqs (2.337) for Pm and P the relaxation
tensors and C appear, expressions for which must now be analysed.
According to the above assumptions and the definition (2.325) of the
398
Light-induced gas kinetics
[Ch. 7
tensor Qj, we have
'-'m ^mn
= m J ьаур5тп(у) du
= m J du dut du dut (v“vf - и^^ЭДи, - и - i(Ui - u)] x
d(u2 - ufio^fu, | u)p<°>(v - u)pmm(v). (7.41)
fa# — fap i fap i fap
''mn ' vwn ' ''nn)
C% + C“m = m J du dut du dui (v“v? - v“vp) x
d[ui - и - |(ui - u)] x d(u2 - u2) x
[cu.(«i I u)p<°„>(u - u)pmm (u) +
®nm(®l I ®)Рш?(®0Рmm (v - u)],
C“® = m J du dut du dui (v“vf - v“up) x
<5[u! - и - i(u( - u)]<5(u? - u2)o„„(ui | u) x
[p£?(v - ») Apnn(v) + p^(v) Apnn(v - и)]. (7.42)
The tensor C“® can be seen to be quite analogous to the tensor (at
i = m, b = zi, p = m/2) introduced in section 2.7. In particular, the
expressions (2.351) are applicable to the tensor C“p as well. In the model
of velocity-independent transport frequencies of collisions we have
= -2(v? - v?)P^ + U^NmdaP - iv?(Tr Pm)dap,
1 /• 2
£ = -j-, Ry" [Pmm(v) + p„„(v)] du = jmv2. (7.43)
/V J 2
It has been taken into account that in the assumed model v” = v“.
We perform all the possible integrations in the expressions for the
tensors and using the spherical symmetry of the interaction
potential, which leads to the relations
^„„(ui | u) = | u) = onm(-ui | -u) = a„„(u, 0),
COS 0 = U • Uj/uUp
(7-44)
§7.5]
Kinetic effects in a one-component gas
399
On integration (allowing for the isotropy of the function p£}(v)) those
tensors may be represented as
C% = C% = -2 J v3m(v)m(vavp - $v2daP)pmm(v) dv,
= -2 J v"(v)m(vavp - $v2dap) Ap„„(v) dv. (7.45)
Here the transport frequencies of collisions vT(v) and v"(y) have been
introduced which can be expressed in terms of the transport cross-sections
(compare with formula (2.355)) as follows:
. . 3 f 3(v-u)2-v2u2
v3(v) = 77 <^(“) —-------4------pS(v ~ du,
10 J V
3 Г 3(v • и)2 — v2u2
v"(v) = - <e>(u) --------L4-----p<°„)(v - u)u du. (7.46)
10 J V
The quantities tr^(u) and a<2)(u) are given by formula (2.358).
It can readily be seen that the trace of each tensor (7.45) becomes zero,
which reflects energy conservation during collisions.
In the model of velocity-independent collision frequencies we obtain
from the expressions (7.45)
C* = 2(v; - Tr Pm) - 2vs(p“p - у Ndap). (7.47)
In this model eqs (2.336) for the pressure tensors Pm and P have the
following form:
P^ + divq^
= lsvTNmdap - 23v?daP Tr a + mNQf,
(^ + 2v^PaP + dwqap
= lvn3eNdaP + 2(v? - v?)(P^ - ^aP Tr Fm). (7.48)
First let us consider the solution of these equations under stationary
spatially homogeneous conditions. In this case from the second equation
we obtain
PaP = iENdaP + (P* - jdap Tr a). (7.49)
V3
d
- + rm + 2(vr-vT)
400
Light-induced gas kinetics
[Ch. 7
If the collision frequencies v” and v" are equal, the overall pressure
tensor P is isotropic and in the usual way is connected with the density of
energy of the gas. Under v" =# v™ anisotropy of the pressure tensor P can
arise as a result of the anisotropy of Pm. Note that the anisotropy of Fean
appear only as a result of collisional transfer of a non-equilibrium state
from one translational degree of freedom to other since the trace of the
tensor F is a motion integral. The collision frequencies v" and v? are,
consequently, responsible for such a transfer.
From the first of the equations (7.48) we obtain
P% = (mNQ?p + ^Т^тдаР) +
2v™mN z л o л 4
(Гт + 2vT)(rm + 2vr - 2v?) ~3 “p 2>
(7.50)
The expression for Р“д naturally breaks into two parts with different
physical meanings. The first part characterizes the deformation of the
distribution function for those velocity components where radiation
creates non-equilibrium structure. Indeed, if p(y) = W(vx)W(vy)W(vz)
then, for instance,
V2 2 £
Q? = <(vx)2p(v)) = — (p ) = — rmNm =-—rmNm,
ill £ J
(7.51)
In this case if in eq. (7.50) there had been no second term, then
P% = &Nm, (7.52)
which corresponds to an equilibrium distribution in the velocity projection
vx. On the contrary, the second part of expression (7.50) describes only
the exchange of non-equilibrium structures between orthogonal directions
involving no kinetic energy change of the excited particles. The latter is
proved by the fact that the trace of this part is zero. Since the second part
of Р“д is proportional to the collision frequency v™, this very quantity
provides for the exchange of non-equilibrium structures between or-
thogonal directions. The analogy of frequencies v™, v™ and v” is now
evident.
On substituting expression (7.50) into eq. (7.49) we have
y** — у™ WlKI
Pap = liNdap + —---------------------(Q“p - ldap Tr Q2)
P vn3 Гт + 2(уТ-Уз)
v? — v?* tnN
- ieN6ap + ((v^ - &pV2)p(v)).
У3 1m T Уз J
(7-53)
§7.5]
Kinetic effects in a one-component gas
401
We can, therefore, see that in a one-component gas under velocity-
selective excitation pressure anisotropy is present. The trace of the
pressure tensor, evidently characterizing the translational energy of the
gas particles, is not changed.
If the function p(y) is non-equilibrium only in one velocity projection,
then
P^ = pyy = IeN -1ДР,
Pzz = jsN + i ДР,
Д/»_У^-Уз' mN
v” rm + 2(vT-v^)
/Г -> v2l \
( (H~y p(y)Y
\ L Z J 7
(7-54)
The expression for Pzz formally coincides with the corresponding expres-
sion from formula (7.35); therefore all results obtained for Pzz in section
7.4 may be applied to the case under consideration*.
In ref. [31] an interesting phenomenon was experimentally discovered
and theoretically explained: the differential of gas pressure arose in a
narrow cell when radiation passed through it. The effect is due to bulk
collisions and was named light-induced viscous flow. Now the approach
developed in the present book will be applied to this effect.
The z axis will be directed along the wavevector к of radiation
propagating through a cylindrical cell. The problem, therefore, becomes
very symmetrical. Proceeding from the general equations (7.48), we set up
stationary equations for the tensor components P" and Pzr, where r is a
coordinate directed along the radius from the centre of the cell. In the
chosen coordinate system the tensor Q2 is diagonal. The terms div qzr have
the form
div qzr = m I vzvrv • Vp„(v)dv.
(7.55)
Radiation produces a non-equilibrium distribution in projections of
velocity vz. The distribution p„(v) in orthogonal velocity projections is
practically isotropic and approaches the Maxwellian distribution.
Therefore
д C mv2 d
div qfr = m — (vr)2vzp,,(v) dv = ——j,,
dr J 2 dr
(7.56)
* The results obtained here are in agreement (if the parameters are adequately redefined) with those
of ref. [28] based on the model of strong collisions with arbitrary parameters and on the model of
similar cross-sections which, in particular, justifies the use of models in ref. [28] and attaches some
physical meaning to their parameters.
402
Light-induced gas kinetics
[Ch. 7
where /, is the particle flow of component i collinear with the z axis.
Therefore eqs (7.48) under stationary conditions take the form
/и 5
[rm + 2(v? - VT)]P" + —jm = 0,
Z Cfr
л „ mv2 d . л/
2v"Pzr +
2 dr
(1.51)
The tensor component Pzr has the meaning of longitudinal momentum
flow radially directed through a unit area. Under v"' = v" the second of
the equations (7.57) becomes an ordinary viscosity equation:
ди P
pZr = ~^ P = iTrF, (7.58)
CrK £ rj
where и is the velocity of the gas directed along the cell axis and r] the
viscosity coefficient explicitly related to the collision frequency v" and
pressure P. Under v? =# v" a non-zero source appears in the second of
equations (7.57) which, as can be seen from this equation, induces a
longitudinal momentum flow to a side wall of the cell. Consequently a
momentum flow of the opposite sign must be initiated in the bulk since it
is already known that radiation itself does not change the medium’s
momentum. Let us substitute the quantity P£ from the first of equations
(7.57) into the second. Then
mv2 d Г 2(v" - v”)
4^d^ lrm + 2(vT-v?)lm ~J
(7-59)
When both ends of the cell are open the longitudinal momentum flow
directed to a wall is stopped, i.e. Pzr becomes zero. Assuming that flows
jm and j„ are zero on the wall of the cell we have
2(^~уП .
Гт + 2(vT-vT)}mf
(7.60)
i.e. the induced particle flow of the gas as a whole is proportional to the
partial flow of excited particles jm and has the same spatial dependence.
Now imagine that the cell is closed. This brings about a pressure
gradient along the z axis, causing the flow averaged over the cross-section
of the cell to stop, i.e.
JrR
I j(r) 2nr dr = 0, (7.61)
0
§7.5]
Kinetic effects in a one-component gas
403
where R is the radius of the cell. Assuming that the pressure P over the
cross-section of the cell is practically constant, we write down the
condition for equilibrium of forces for a gas layer with thickness dz and
radius r:
Ttr2 dP = 2лгР" dz. (7.62)
On substituting the expression for P" which follows from this condition
into eq. (7.59) we obtain
2(vg-vT) . ,
dz 2v$ rdr\.rm + 2(vT-v?)Jm 7J‘ 7
The left-hand side of this equation is independent of r. If flows on the
walls become zero, from eq. (7.63) and the condition (7.61) we obtain
dP _ 4mv2 V3 — V3 T
a? - " v?/?2rm + 2(vr-vT)7m’
1 CR
/m = ~~D2 x 2jtr dr’
ЛА Jq
2( v? — vT) Г / r2 \ - 1
7(r) = rm + 2(vr-vr) L7m(r)" 2(X “^)7m]- (7,64)
The pressure gradient is proportional to the flow of excited particles
averaged over the cell cross-section. The particle flow of the gas as a
whole is represented by superpositioned light-induced flow and opposing
viscous flow characteristic of Poiseuille flow.
The flow of excited particles jm can be expressed in the usual way in
terms of the absorption probability (p). According to formulae (6.3),
(6.17) and (6.18) we have
<7-65»
lm “Г Vi
For the pressure gradient and velocity of gas flow u(r) = j(r)/N we Anally
obtain
1 dP _ 8v V3 — V3 {р)ф
P~di~~ vn3R2rm + 2(vT-vT)rm + vT’
2v(v?-v^) 1 Г /
“W =Г. + 2(гГ-г?)Г. + гГ Л1
_T_ rR
3tR2(p}<p= I (p)<p Х2лг dr. (7.66)
404
Light-induced gas kinetics
[Ch. 7
We have therefore obtained relations describing the effect of light-
induced viscous flow. Note that the pressure differential over a given cell
length decreases as the particle concentration increases (recall that
collision frequencies are proportional to the concentration, and the factor
{р)(р/(Гт + Vi) cannot be much more than unity). On the contrary,
under very low concentrations when the mean free path is significantly
greater than the cell radius the effect is also bound to vanish. Indeed, in
this case collisions in the volume are so rare that it is impossible for them
to affect the distribution function properly. Certainly, the transport
equations employed are no longer valid. Therefore the effect of light-
induced viscous flow is most pronounced under conditions intermediate
between hydrodynamic flow and the flow of free molecules.
Within the scope of the above-taken assumptions consider the initiation
of light-induced heat flow. Moments of the collision integrals Rmn and R
from eqs (2.339) are expressed in terms of differential cross-sections as
follows:
Rmn = ? I dv dv, du dui (v,v, - v2v) x
£* J
6[v, - v - i(ui - u)]<5(u? - u2)amn(ui | u)p£>(v - u)pmm(v),
R = Rmn + Rnm + Rnn,
Rm„+Rnm=™ j dv d^ du du! (vfa - v2v) x
- v - |(ui - u)]d(ui - u2) x
| u)p^(v - u)p
mm (v) +
a„m(ui | u)p£>(y)pmm (v -«)],
Rnn = T I dv dv, du dui (vfo - v2v) x
Xr J
<5[v, - v - i(ui - u)]<5(u? - u2)a„„(u! | и) x
[p^(v - u) Ap„„(v) + p£>(v) Ap„„(v - u)]. (7.67)
The moment Rma is described by the expression (2.352) with collision
frequencies (2.353) and (2.356). All possible integrations are performed in
the expressions for the moments R„„ + Rnm and Rnn, and use is made of
the symmetry properties of the differential cross-section and the function
§7.5]
Kinetic effects in a one-component gas
405
Pnn(v)> tbis yields
8 f , . mv2 , . ,
Rmn + Rnm = “Г *>"(«) VPmm(v) d” +
J J £*
£ У4т(и)«Ртт(«)<1«,
8 f . . mv2 , . . ,
Rm = -z *T(v) ~z~ v &pnn(v) dv +
J J Л*
v4"(v)v Ap„„(v) dv.
(7.68)
The transport collision frequencies v" and v™ introduced above are
defined by formulae (7.46) and the newly introduced quantities v4 and v4'
are given by the expressions
v“(v) = J a® (u) p<°)(v - »)u3 dw,
v3(v) = 72 [ a<2„>(u) p£>(v - u)u3 du.
4c J v
(7-69)
In the approximation of velocity-independent transport frequencies eqs
(2.339) for heat flows take the form (under stationary, spatially homoge-
neous conditions)
(Гт + 3v™)qm = vTejm + — NQ3,
v" — V3 V4 — v“ _.
9 — Ят £Jm-
(7.70)
Deriving the second of the relations (7.70) we allowed for the fact that no
particle flow is initiated in a one-component gas (Jm = —j„). This relation
shows that the heat flow q is due both to partial heat flow qm and to the
flow of excited particles. Each of them differs from zero under asymmetric
velocity-selective particle excitation caused by radiation, i.e. when X2=#0.
The part of the flow q due to the quantity qm is proportional to the
relative difference between the transport collision frequencies v” and V3
responsible for the exchange of non-equilibrium structures between
translational degrees of freedom. The same parameter evidently brings
about pressure anisotropy. The difference in transport frequencies of
406
Light-induced gas kinetics
[Ch. 7
collisions leads to the appearance of a total heat flow due to the flow Jm of
particles in the excited state even in the absence of qm.
Let us take the expression for qm from the first of the equations (7.70)
and the quantity jm from formula (7.65). Allowing also for the definition
(2.335) for Q3 we obtain
q = jev{p)^
V" - Уз фз
V? гт + 3v“
V4 V3 — v"' V4 — v“\ ф
rm + 3v“ V? rm +v?J’
(7.71)
If the function p(v) has the form (6.40) then the factor tp3 is fully
determined:
<n
vz
— V(v)dv,
v
(7.72)
where the z axis is directed along the wavevector k. As a function of the
detuning Q, the factor tp3 as well as the factor tp are antisymmetric. In the
limiting case of large Doppler broadening we have
Я Г /Й\21
фз = тт 1 + I тт I ,
kv \kv/ J
|£2| <^kv.
(7.73)
If there is an obstacle, the heat flow must bring about a temperature
gradient followed by a gradient .of the gas particle concentration.
However, direct experimental observation is difficult on account of the
following. First, the walls of cells have a much higher thermal conductivity
than the gas so temperatures will be equalized by heat flow through the
walls of the cell. Secondly, a much stronger effect is produced by the flow
of excitation energy [32] which may be released at the obstacle leading to
a much greater temperature change than that produced by heat flows.
Experimentally, we may only hope to find indirect manifestations of
light-induced heat flow.
The analysis of light-induced phenomena of gas kinetics presented in this
monograph is not claimed to give a complete picture of this rapidly
advancing field. Our task was to formulate a general theoretical approach
to the solution of such problems, providing a basis for the simplest
References
407
description of some specific effects most of which have already been
experimentally recorded. Several effects predicted and then more or less
studied theoretically but not yet experimentally realized have not been
considered. Examples of these are the effect of sound wave generation
[33,34] and the change in the dispersion law of sound waves for a gas in a
radiation field [35,36]. A number of new effects are due to anisotropy of
the interaction potential of colliding particles. This anisotropy leads to
violation of the simple (as in eq. (2.349)) connection between the friction
force and distribution function and brings about various vortex motions of
the gas components [37,38]. An anisotropic interaction potential also
makes spatial and magnetic orientation of molecules possible [39-42].
The theory of light-induced phenomena of gas kinetics is being
continuously developed. Experimental needs require the consideration of
more and more complex models of objects, making it necessary to allow
for real experimental conditions and for the effect of additional factors.
In this situation we have to confine ourselves to simple models of
collisions and to sacrifice rigour from the viewpoint of gas kinetics (see,
for example, refs [43-45]). Fundamental problems of the maximally
correct solution of the kinetic equations for a gas in a radiation field are
still attracting much attention. Various approaches and approximations
are being developed in this field, e.g. those based on Grad’s method
[38,46] or on other approximate methods [47,48].
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Appendix
Some properties of Clebsch-Gordan
coefficients and of 6j symbols
{jimlj2m21 /3/П3) = (-iy,+'2 - m2j2 - m21 j3 - m3) (AI.l)
= I/3W3)
V2/3 +1 .
, л <Л"Мз - rn31j2- m2)
Z/2 T 1
V2/3 + 1 .
, л </3 - m3j2m21Л -m,) .
4/1+1
0iW1/2/n2|/3/n3) = (-iy*->2+m’V2/3 + 1 (71 72 7з V (AI.2)
\/И1 m2 —m3)
^mx j2m2 I 00) = (AI.3)
Oi/njOO I/3W3) = бЖ1ЖДл. (AI.4)
S (ШгМг I/3W3) ^хт^2т2 \jm) = (AI.5)
mim2
2 {}imxj2m2 I jm)\jm) = (Al.6)
ym
2 I jm ) = (2/ + l)d70. (AI.7)
S (^зМз^гМг I jimi} (^зМз/г^г | X
М1М2МЭ
mjm2
(/ig!/2g21 j3m3) (jimij2m21 jm)
= 6mm3d>/3(-l)Z1+Z2+»+«V2/1 + 1 V2/\ + 1 f71 72 7з]. (Al.8)
Ul l2 I3J
409
(sriv) rE/ 7 Y] f£/ Zz ’/i |^£y Zf lyJ [Ey zf iy.j(74-7+'/)'3-(l ) fzZ 7 Tip 7 7-»p 7 7i ’ [£f я E;jlv я Z/Jl1/' Я +
(wiv) и £f V] f-* 7 Tip* 7 7i * (/ 7 7J= 11 ч 7/(7 7 + *z)/+.'+*(l~) Z
(tl'IV) 9 = b 7 7JI.Z 7 xJ(T + ^Z^ + /z)
(znv) I + z./gA_i + zzgA /T . r7 7 71 I lEz Zz oJ
(niv) ,r7 Zz £-r 7] f7 V 7'> ,сЧ ч у-» lEz 7 l[)\zi Ez TzJfl£z li< zzJ = lEz Zz TzJ
(onv) [z 4 v) (Ш' । Zu/VlM/l.O {ш[ 1 zrizlliill) X1 + \/гд 1 + TZZA 5 tf+u+w(i-) = %. | zt«7rZz) (lM/71 zrizjri]y
(6IV) rEz zl j//- z} tJ<E»"7lzt"7li"7) x i + 7^a i + TzzA^»r+^+.,(i -) = (Eu/71 Wt'rlif) {W] | zwzf£ri£i){4u41 zriz]£ri£])
Olfr
I xrpuaddy
Appendix
II
Wigner D matrices
P,y) = e-MadJMM.(P)e~iMX
dJMM(P) = d%M.(P) = (-1)"-" dJM.^P)
P, y) = p, y)
= DJJM (-a,p, -y).
^MM'(O) = *
(AII.l)
(All.2)
(AII.3)
(AII.4)
2 DJMM\a, P, y)Dmm-(<x, P,y) = ^ D^a, P, Y)DJ^M(a, P, y)
M M
= <W- (AH.5)
f daf dpf dy sin p DJMM.(a, P,y) = (AII.6)
Jo Jo Jo
( daf dpf dYsinpD^M(a,P,Y)DJ^(a,P,Y)
Jo Jo Jo
Sit2 „ . . Z A TT -,4
~ 2J + 1 ^771®А/А/1®Л/'Л/'' (AII.7)
£>мм,(а, P, Y)DJm ^{<x, P, y)
= S S (AII.8)
j=\J-J’\ mmi
ip(JM | ity) = 2 D7MM.(a, P, Y^iJMr | (AII.9)
Ml
P, У) = {JMJ'M' I Kq)DJMMi(a, p, У) x
DJM ^a, P, y^JM.J'M'. | кЧ1). (AII.10)
411
Problems
Problem (1) Express the matrix element for the interaction of an atom
with an electric field in terms of the oscillator strength and radiation flow.
Hint. For monochromatic resonance radiation linearly or circularly polar-
ized, according to eqs (2.75) and (2.76),
V(nMn'M', f) = —G(nMn'M') exp(-i£2t),
I n||d||n')||(JMJ'— M'\lo)\.
Use the relations
|(n||<l||n')|2 = --------
2tt>„„jn
Solution
Ve2
VAPf„,„(2J' + l)|(JAfJ' - M' | la)|
zmc n
J
—M o)
= 0.633 x 109VAPf„.„(2J' + 1)
= 1.001 x lCWPfn.n(2J' + 1)
= 0.334 x 10-2VAPf„„(2J' + 1)
—M
M'
—M
MHz
—M
cm \
where Л (gm) is the wavelength, P (W cm 2) is the flow and fnn is the
oscillator strength for the transition n'—»n.
413
414
Problems
Problem (2) Derive the equation for the density matrix (2.67) in the
polarization moments representation.'
Solution. Proceed from the equations for the density matrix in the JM
representation
/ 3 \
( — + v • V]p(nMn'M') = R(nMn'M') + S(nMn'M') —
\at /
i 2 [V(nMnxM1)p(n1M1n'M') —
niMi
pinMniM^V(niMpt' AT)].
Using relations (2.61), (2.62) and (2.70), transform these equations into
the polarization moments representation:
/ d \
+ v ’ v)pnn'^Kq^ = + Snn (Kq) -
i S S -M'\Kq)x
{(-1У’-*(/Ж - Mx I Ла)УЯЯ1(Ла) x
(-ly'-^XAM.J' - M' I Kiqi)pntn,(Kiqi) -
(-1)7,-M,(JA/Ji - Mx | Krq^p^K^) X
| Ла)УЯ1Я,(Ла)}.
Sum over the moments projections using relations (9) of Appendix I; this
yields
/ d \
I + v . V p„n(xq) = R„„(rq) + S„n.(Kq) -
\at / ______
i 2 S (-1)»+k>-7'V2k + 1 V2k, + 1 x
nuri^i Ла
z r , , , v fA Ki k)
(-1) 4'{KqK1 — qx | Ла) x
V. J J JIJ
К1Я«(Ла)рЛИ1(к1д1) ~
(-1)-7'-’(k - q^qt | Ла) x
J JIJ
Introduce columns p„„-, R„„- and S„„. with components p„„.(icq), Rnn.(Kq)
and Snn (xq) and matrices U^, and U„„, with elements 1/ЯЯ](к<7 | Kxqx) and
Problems
415
Un„„,(Kq I Ki<]i) specified by relations (2.69) and (2.72). As a result we
come to eqs (2.67) written down in a matrix form.
Problem (3) Express the collision integral 5 in terms of the Moller
operator Q and the T matrix. Take into account the interaction with an
external field.
Solution. A two-component gas of particles interacting with each other
and with an external field is described by the multiparticle density matrix
p depending on internal and external variables of the numbers Na and Nb
of particles of respectively types a and b. The matrix p obeys the equation
ift|p = [^+V> + ^,p], (1)
at
where Й is the Hamiltonian of isolated particles a and b, and W and V are
Hamiltonians of the interaction of particles with each other and with the
external field.
On evaluating the trace of both parts of eq. (1) over the variables of all
the particles but one of the a type we obtain the equation for the
one-particle matrix:
Pa = [Ла + Pa] + Trb[lVab, pab] + Tra,[lVaa,, paai], (2)
at
Here pab is a two-particle density matrix, Йа + P"a is the Hamiltonian of
particle a in the external field, V^ab and V^aai are Hamiltonians of the
interaction of particle a with b and av Omit the last term in eq. (2),
assuming that the concentration Nb of particles b substantially exceeds Na.
Calculation of the trace of eq. (1) over variables of all particles except a
and b yields an equation for pab containing a three-particle matrix
(Bogolubov’s chain of equations). If the concentration of particles Nb is
small enough, i.e.
pX«i, (3)
the three-particle matrix can be neglected and then
T- Pab = [Ла + Йь + Va + Vb + V^ab, pab], (4)
at
the initial conditions providing the transition to the equilibrium state
416
Problems
(Bogolubov’s initial condition) being
lim pab(r) = pa(t) x pb(t), (5)
00
where pa(f) x pb(f) is the direct product of the matrices pa(f) and pb(t).
The purpose of the next stage of the derivation is to express the solution
of eq. (4) under the initial conditions (5) in terms of standard operators of
the theory of binary collisions of a and b particles. Introduce the Green’s
functions (propagators) of free motion of the particles, i.e.
ift 4 G(t, t') = (Йа + Ра + Йь + Vb)G(t, t') + iftld(t - r'), (6)
at
and of motion with interaction:
ift 4 П = (Я + + Я + К + Wab)G(t, f) + - f). (7)
at
The matrices pab at times t and t' are connected by-the relation
Pab(0 = G(t,f)Pab(f)G4t,f) (8)
The operators G(t, t') are not defined for t’ —» — oo. In order to make use
of the initial conditions (5) we can introduce the operator
t') = G(t, t')G\t, V),
which has a finite limit at t' —» —
Я(0= lim G(M')G4M'), (9)
—oo
since for t'—» —oo the motion of the particles is free. Taking into account
formula (8) and the relation
Gf(t, t')G(t, t') = 1,
we come to a chain of equalities:
Pab(0 = G(t,f)Pab(f)G4t,f)
= G(t, t')G\t, t')G(t, f)Pab(f )G\t, t’)G(t, t')G\t, t')
= Sl(t, t')G(t, i')pab(i')Gt(i, f)£T(r, t').
According to the initial conditions (5)
lim pab(f) = pa(f) x pb(f).
f'—►— 00
Problems
417
Then
G(t, f )[pa(f) x pb(f )]G*(r, t') = Pa(O x рь(Г)
and, finally, by relation (9)
рл(0 = й(0[р.(0хрь(0]йХ0. (10)
From eqs (6) and (7) we obtain an equation for Я(Г, t'):
'Л si(t, t') = [йа + % + йь + ?b, ii(t, e)] + r) + iftid(t - e)
at
(11)
or its equivalent
Й(/,Г') = 1Ц (12)
1ft Л'
Going over to the limit t'—» — °° we obtain
Я(0 = 1 + К(0,
K(0 = i f G(t, f)^ab«(r1)GXr, Л) dG. (13)
\Tl J—QO
The matrix T(t) is introduced by the relation
T(r) = WabQ(t), (14)
so that the operators K(t) and T(t) are related by
iftK(t) = Г G(t, QT^G^t, drv (15)
J — QO
If H does not depend on time then
G(t, n = G(t - e)
iftK = [ G(T)TGf(T) dr, Я(0 = Я (16)
Jo
where Я is the Moller operator.
Summing the results (10) and (14), we obtain for the collision integral
5 = ^Trb[U;b, pab]
In
= Trb[T(0(pa X рь)ЯХ0 - Я(0(р. X рь)Г(0]. (17)
1ft
418
Problems
Problem (4) Derive the formula
Т(раръ \ра1ры) = д(Р-Рг)Т(р I Pl).
Hint. Allow for the fact that Wab depends on ra — rb.
Problem (5) Prove the validity of eqs (2.173) and (2.178) using the law
of conservation of total momentum of a system of colliding particles.
Hint. Make use of the model of isotropic collisions and expand the
scattering amplitudes into spherical functions, introduce the total momen-
tum representation
f(JMJbMb,u\JlMlJblMbl,ul)= 2 J x
fQJhJ, l,j, U I JiJbiJi, 11, ], U1) X
2 {JMJbMb I JM) (JMlm \js)x
{JiMiJbiMbi | JiMx) (JiMilimi\js).
The law of conservation of total momentum consists in the fact that the
scattering amplitude in the JJbJjs representation is diagonal in j,s and is
independent of s.
Problem (6) Prove the validity of relations (2.182), (2.184) and (2.187).
Hints. Proceed from the expression
у(пп'к\П1П[к)= 2 (-1)7_м+л-лл x
2К + 1 дММ'ММ
{JMJ' — M' | Kq) (JiMiJ'i — M'i\ Kq) X
«f(JMj81 AMjSOf *(J'M'P I J'lM'M. (1)
(1) For the derivation of formula (2.182) consider the expression which
is the complex conjugate of the right-hand side of eq. (1) and perform the
substitutions M^M’i, use the hermiticity of the scattering
amplitude and the properties of Clebsch-Gordan coefficients (see Appen-
dix I).
(2) At n' = n, n‘i = n, K = q = 0, expression (1) can be reduced to a
sum of the squares of scattering amplitude moduli with positive
coefficients.
Problems
419
(3) Consider the expression
2
а | -M'\Kq}~
M
о M\
Evaluate the square modulus choosing the coefficient a so that the sum of
sum squares takes its minimum value. The cross-term coincides with the
left-hand side in relation (2.187).
Problem (7) Derive equalities (2.243), (2.245) and (2.246) for in-
frequencies and kernels of a collision integral expressing scattering
symmetry about time inversion.
Solution. Use the identity (2.245). Assume the perturbing particles b to
be in a state of thermodynamic equilibrium. Taking into account the law
of conservation of energy we obtain
W(v1)A(aa'v | = W(y) x 2 2 I du duiX
/3/31 J
Г 2 1
d u2-u21 + -(Ea-Eai + Ep-Efil) x
L fl J
d V-Vr-— (u-uj exp[-(Ea-Ea,)/T] x
m J
pb(fi, v-u)f(afru | a1j81«1)/*(a'j8« | a^Uj).
(1)
Reverse the signs of the velocities и and и, in the integrand and apply the
reciprocal theorem (2.238). As a result we obtain
W(vr)A(aa'v | aiafa) = W(v) exp[(Ea,
du du, x
2
6 u2-ul + -(Ea- Ea, + Ep — Ep,)] x
fi
Г и 1
5 V — Vl-(u-Ui) Pb(fi, V + u) X
m J
(-lyytatfitu! | a*j8*u) x
/’(arjSrui | a'*/?*u), (2)
o — J — M+J'—M' + Ji — Mi + J'i — M\.
420
Problems
The quantum numbers ft* and ft* take the same values as and /Jp
Therefore, summation over and ftA in eq. (2) may be replaced by
summation over /?* and Taking into consideration that under
thermodynamic equilibrium pb(ft) = Pb(ft*) and using the definition
(2.147) for a kernel we find from eq. (2) a relation coinciding with eq.
(2.245). Integrating both parts of this formula with respect to v and taking
into account that with v « vb the relation
W(v)A(aa'v! | a^v) = W(vt)A(aa'v | (Xioftvi)
holds true, we obtain the formula (2.243).
Problem (8) Prove the formula (2.273) and derive a formula for the
amplitude of scattering through classical angles by an isotropic potential.
Consider the “glory” phenomenon.
Hints. Proceed from expression (2.271). For large scattering angles and
high values of d(p) use the asymptotic expansion
7/ 4 1 Г \(
+ exp
./ л\
“VP-j
I Р| I
J’ 9 =
and find the point of stationary value of the phase function in the
integrand in relation (2.271):
1 6
l<5'(Ps)l=9 = *sin-.
At this point expand <3(p) up to the quadratic term.
Solution.
<5'(Ps)>0:
<3'(Ps)<0:
/(9) = 4 +;
/(e)=4b%j]
For small (diffraction) scattering angles the contribution to /(0) from the
range in the vicinity of the extremum of the function <3(p) is
i / 2л \1/2
- ~ I ~ ) PoJoiqpo) exp[-i<5(p0)], <3'(p0) = 0.
л \ о /
Owing to the condition |<5(Po)l »1 the glory contribution to forward
scattering is small.
Problems
421
Problem (9) Evaluate the correlation function and line shape under
phase modulation in the impact approximation.
Hint. For successive collisions accompanied by a phase shift (p1 we have
ФМ = (e-^)) = 2 Рт(т)е-^т,
m=0
ml
where Pm(r) is the probability of m collisions’ occurring over time т and Vj
is the mean frequency of collisions. Summation over m yields
0i(t) = exp[-VjtCI - e-*”)].
Averaging over collisions with independent phase shifts, we obtain
Ф(т) = П <Mr) = ехр[-(Г + iA)r],
r + iA = <v/(l-e-i”0>/
Problem (10) Evaluate the radiation spectrum for the case (3.61) and
(3.66).
Solution.
I(Q} = R 2Fl + ^12 + ^21 ~ ~ kV^ + ~ + ^12 ~ ^21 +
2n[rt + at — i(42 — A:v)][74 — a — i(42 — fcv)]
Ц = у + i(vi + v2 - vn - V22),
r2=v1- vn-(v2- V22), Av = v1-v2,
v = i(V] + v2), a2 = v12v21 + l(T2 + ik Av)2.
Problem (11) Derive and solve the system of equations for the quantities
г,, rmn, rmn given by the formula (3.221).
Hint. Introduce expressions (3.221) in the system (3.220) and separate the
parts with different time dependences.
422
Problems
Solution.
(Гт - ie)rm + iG*rmn - iGf™ = iG^,
(T„ - ie)r„ - Amnrm - iG*rmn 4- iGr*„ = -'1С„рпт,
[Г - i(e + Q - + iG(rm - r„) = -iG„(p^ -
[Г - i(e - Q + 4)]^ - iG*(rm - r„) = 0.
From the solution of the system (1) we obtain
_ i^i2 г .n „0 J
g-i£
Gu, .
rm - rn = i ~ (g* - if){(g - i£)[(rm - i£)p„m + (r„ - Am„ - if)p„m] 4-
iG*[Fm + Гп-Атп- 2i£](pL. - p2„)}
g = r-i(Q-A),
D = (g — ie)(g* ~ ™)(Гт ~ ie)(TH ~ ie) +
2(Г - i£)(FM + Гп - Am„ - 2i£) |G|2.
Problem (12) Derive eq. (3.235)
a(m, 0; t) 4- ymla(m, 0; t) = 0
for the case of decay of the lower state /.
Hint. Consider a spontaneous cascade decay in a three-level system m, I, j
(Em >E,> Ej), the level j being stable:
d(m, 0; t) = H'{m, 0 11, lA)e“iOua(/, 1A; t),
a(l, 1A; t) = — — 1A | m,G)Q,Ol‘a(m, 0; t) —
ft
H'{1,1л, ом I/, 1л, 1м)е-^а(/, 1л, 1„; t),
d(i, 1л, 1„;0 = 1л, 1„ | /, 1л,0M)e‘^a(Z, 1A;t).
n
Formally integrating the latter equation, introducing the result into the
equation for a (1,1A; t) and summing over p (analogous to the procedure of
Problems
423
section 3.4), we find
a(l, 1Л; Г) = у/уа(/, 1л; Г) = - |н'(Л 1A | т, 0A)eiOi,a(m, 0; t).
ft
Now the obtained equation must be formally integrated and the result is
to be introduced into the equation for a(m,0;t). On summation over Л
we arrive at eq. (2.235).
The case of interaction with an external field is treated in a similar way.
Formal integration is performed by making use of the corresponding
evolution operator.
Problem (13) Estimate the spectral density of spontaneous emission for
the transition m —> n.
Hint. Employ the solution of problem (11), allowing for difference
between the right-hand sides in the systems of equations (3.220) and
(3.254) and truncate the population p°nn and the term (Tn - Am„ — is)pnm in
the expressions for rm„.
Solution.
-PM = 2^|GJ2Re^3—,
x = (g* - i£)[(g ~ i£)(rm - +
1в*(Гт + Гп-Атп- 2iE)p^].
Problem (14) Find an expression for the radiative in-term of eqs (2.42)
and (2.43):
R™(nMnM') = An„ 2 (JMlo\JlM1){JM,lo\JlM{)p(nlMlnlM{).
oM\M\
Hint. Proceed from the equation
/d \
|-r + Yn)a(nM, iA;0
\dz /
= -^H\nM, 1A | П1Мг,0A; t). (1)
ft
424
Problems
Set up an equation for
p(nM, nM') = 2 a(nM, 1A; t)a*(nM', 1A; t)
л
which has the meaning of an atomic density matrix element (the
amplitudes a(nM, 1A; t) describe the states of the “atom + field” system;
the summation over Л separates states of an atom). This equation has the
form
+ 2y„\p(nM, nM') — — i 2 H'(nM, 1A | nxMx, 0A) x
\dt / лм,
e’^'a^iAf], 0A; t)a*(nM', 1A; t) —
еГ,а^'а*(п\М\, 0A;t)a(nM, 1A; t),
V2nft
—-eA-d(nM,n1M1),
<oAV
da(nM, n.M.) = <n|| d - Mx | la). (2)
The amplitudes a(nM, lA;t) and a*(nM', 1A; t) in eq. (2) must be
substituted by the formal solution of eq. (1), and summation over modes Л
must be broken into integration with respect to &A and angles and
summation over polarizations. As a result of these calculations the
right-hand side of the equation (2) is reduced to R(2)(nM, nM').
Problem (15) Express the work done by the field (4.20) in terms of a
probability integral of complex argument (compare with relation (3.32)).
Solution.
p=2“uvr^f“(z)(N"-N")’
z = [Ts — i(Q — A)]/kv, Г, = П/1 + к, u(z) = Re w(z),
2 (г
w(z) = ez2[l — 0(z)], ф(г) = —/= I e'2df.
Vjt 'o
Problem (16) Find density matrix elements for moving atoms interacting
with a standing monochromatic wave under conditions of exact resonance
Problems
425
(<d = шт„) and equal relaxation constants (Гт = Гп = Г, Amn = 0). Perform
averaging over coordinates.
Hint. Make use of the evolution operator (3.112).
Solution.
Pa(r, (б™ + Qn ± (Qn - Qm)r( e rTcos[/(r, v, r)] dr|,
Zi I Jo >
Pmn(r,v)= W(v)(Qm - Q„) ( e^sin[/(r, v, r)]dr,
-------- W(v)
Pii(r, v) = {Qm + Qn ± (Qn - Qm)r x
Problem (17) Study the problem of the asymmetric frequency depend-
ence of the work done by the field P(Q) (standing wave) for the model of
strong collisions with a complex in-frequency of the non-diagonal collision
integral.
Hint. Proceed from the expression (3.52). Determine the first moment
and the position of the maximum Q2 of the function P(Q) at |v|« kv.
Expand P(Q) into a series in the vicinity of the point Q = v".
Solution.
= v", Q2 = v" - 2v".
Problem (18) Calculate the matrix reciprocal to К given by the formula
(4.99).
Solution.
P = K~\
DPu ~ + 2т1лт1т |G|2y ,
DPi2 = 2TlnTlm |G|2y
426
Problems
DP13=—iTimG*y = DP*4,
DP2l = т1тт1л(2 |G|2y' 4-Лтл),
DP22 = т1л + 2т1тт1л |G|2 у',
DP23 = irln(l - rlm4m„)G*y = DP*4,
DP31 ~ ~irimG;y(l — AmnTln)DP2i,
D P32 = it} „Gy = DP*2,
DP33 = y + (т1т + т1л - т1л,т1лЛтл) | G |2 I у |2 = DPt,
DP34 = (Tim + ~ TlmTinAmn)G2 I у |2 = DP*3,
Ty^lYj + Vj, у-1 = у + v -i(£2- к • v),
D-1 = 1 + 2(т1т + т1л - т1л,т1лЛтл) |G|2у'.
Problem (19) Obtain the solution of the equations for the density matrix
in the model of strong collisions with changing velocity, rotational
quantum number J and projection M.
Hint. Neglect spontaneous transitions m—>n and “phase memory” for
non-diagonal elements pm„. Consider the case of a linearly polarized
standing wave inducing transitions mJ0 — nJ0 (choose the quantization axis
along an electric vector of the field).
Solution.
\ \ KI , |G| Nnm Tlm У m “I" ^2m Y m
= W(JMv) N„ + ww>l) 1 + 2|c|,l2rJ
2|G|24,
*iny'„+ -hnY'n '
pnn<JMv)=W(IMv) N„-
L tvi/oAf) l+Z|Gj T2r J
yW(v) 1
p.(Mv) + p.(Mv) = iGNm ,
f2(M)x(v) .
,=rnm7WW' xW = RexW, y=Rey,
x(v) = l/[y + v — i(42 — к • v)] 4- l/[y 4- v — i(42 4- к • «)],
Ут=У'5/Л, y'n=y'^rre,
Y; = S f duy'W(JMu),
JM J
M
f(M) = (-1Лм(Л)Л//о - M' 110),
Problems
427
_ ВД)
^2 ~ <"% r 14 ^2m ' л Tf 4 ^2л>
N N=0,
2Jo + 1 2J0 + 1 2y, + Vj - vy
tv = l/(2y, + v,), t2;- = Tvv;./(2yy + Vj - vy),
T1 = Tlm + T1„, W(JMv) = W(v).
Problem (20) Derive a formula for the velocity distribution at the lower
level of a transition resonant with a strong monochromatic field assuming
the model of the difference kernel of the collision integral.
Hint. Use Green’s functions (4.119) and (4.122) and recurrence relations
(4.65).
Solution.
pnn(v) = W(v)(n„ + 2т1Л |G|2y^-" X
( Г (y + v'Y
(1 - TlmAm„) 2 + n„Z„(x)
I L(y + v) +xz
x = Q—v" — kv, sn= sn\/l + n„ .
Problem (21) Calculate the velocity distribution of populations and the
work done by a standing wave field in the approximation of first
non-linear corrections, making use of a two-component kernel (4.155) of
the collision integral.
Hint. In a non-diagonal kernel A(y | v,) neglect the term v„„W(v). For
the description of selective scattering use one-dimensional difference
kernels At/(v - vA).
428
Problems
Solution.
Pj](y) = W(y)
2\G\2(Nm — N„)
1 + 2|С|2(т£> + т£>)У
[t^Y +
Fh— kv — krj) + B(Q + kv
2VS Г G2 1 ,2. v;(2)
kv exp[ (£v)2J’ ОД+у(2>)’
The functions F^ri) and B(z) are defined by formulae (4.122), (4.138)
and (4.117) but in expressions (4.117) and (4.122) the following substitu-
tion is required:
Г^Г+ v<2>, vmn v<!>, nt = у,!Г^ + v)2>).
The expression for the work done by the field has the form
Z’(fi) = 4ft<u |G|2 (N„ - N^nl^Q) x
f1 “l + 2|G|2(r^+r^)Y+ T^Y + ’
in which Z(£?) and I2(Q) are described by formula (4.160), where the
above-mentioned substitution is to be performed.
Problem (22) Evaluate the work done by the probe field for a three-level
system with k^<k in the approximation of first non-linear
corrections and Ц « kv.
Hint. Proceed from expression (5.4).
Solution.
P„ 2»o>B|G„|2^.exp * "(W)i * x{M n~ +
2 I<3|= ()V„ - N.) % Re!~~~~ + -p—- + 7_ % Jl,
к LZ] 142J I2 (Z2 1^2) J-*
C2 = 1/[Гт1 - r„i + Гтп + i(zA„; - A„i - zlm„)],
k-k^^-Nt
к Nm-Nn ’
Problems
429
к. ku к — ки
r1 = rmt+-£rmH, r2=^rnl + —j^rml,
= ~ Aml 7” — Amn),
К
i2M2 = &ц ~ Ami 7" (^2 + A„i — Aml)-
К
Problem (23) Calculate the two-photon absorption of one of the
travelling components of a standing wave under the following conditions:
the wave frequency <u satisfies the condition of a two-photon resonance
with respect to the forbidden transition n-l and is far from one-photon
resonance with the allowed transitions m-n and m-l.
Hint. As a consequence of the assumed condition |42/m|, |42m„|»fcv, in
the equations for ptm and pm„ relaxation constants and terms containing
v • V can be omitted. Evaluating the work done by the field, one must
substitute one of the travelling components of the wave in the general
relation and perform averaging over coordinates and velocities.
Answer.
P = kaiin
\dm„dlm\2
I ^lm I
412?й4
| «|4 N„ Rel 1,_) + —* exp
[Tln - iQ2 4kv
Q1 = (d- w„m, Q2 = 2(o- (Oi„ - Ai„.
Problem (24) Calculate the work done by the probe field in a three-level
system on condition that |G| «kv, k^>k (formula (5.11)).
Hint. Separate the non-saturated part of p„ — pmm in expression (5.2).
Apply the residue theorem, taking into account the fact that the zeros of
the denominator as a function of к • v in eq. (5.2) are in the same
half-plane, if k^ > k; make use of the poles of expressions (5.3).
Problem (25) Estimate the work done by the probe field in a three-level
system when the condition |G| »kv is satisfied.
Hint. The roots of the denominator of the expression (5.2) on condition
that |G| »kv are equal to
2*!,2 = rml + F„i + i(fi + Ami + Ani) ± Vi22 + 4|G|2 - 2iM1>2A • v.
430
Problems
Take partial fractions of the expression (5.2). Allow for the v dependence
only in resonance denominators containing <r12 and in W(v). As a result
we arrive at formulae (5.13) and (5.14).
Answer.
P- ll.ro l2VJtjtpC^l) + C2^2)
P. - -2»ш, |G,| —Re—
•21,2 = [fm/ + rnl — 2i(£2M — £21i2)]/2|Am — Afj>2A| v,
2Й1>2 = Q + Aml + A„,± VX22 + 4|G|2,
C.,2 = ±,, -%-Г, [№ - Л4)(Й1.2 - О) +
- Af1>2A| L
IGI2 ( Гт
The function w(z) is given by the formula (3.32).
Problem (26) Calculate the work performed by the probe field at an
adjacent transition in the model of strong collisions.
Hint. Neglect the “phase memory” at transitions m-n, m-l and n-l.
Separate pmm(v) into parts that are at equilibrium and selective in
velocities; formula (5.2) must be transformed as described in problem
(24).
Answer.
P, = PM1 + ХРЦ2, X = 1/[1 + 2 |G|2 r2( Y'(v))]
yjt
PMi = 2ftct>M |GJ2—exp
Af
кцЬ/
x
[A, - Nm + r^(Am - Nn)X |G|2 (F(v))].
The function P„2 is described by formulae (5.11) and (5.12) with the
substitution (5.15). The values of r^, r2 and Y(v) are specified
respectively by formulae (4.71), (4.88) and (4.104).
Problem (27) Calculate the work done by the probe field in a two-level
system on condition that Гт = Г„ = Г, |G|» kv.
Solution. Assuming in the solution of problem (11) that the relaxation
constants are equal and |G|» Г, reduce the expression for the work done
Problems
431
by the field to the form
/ 1 ie'
= -2йюм |G|2 ((pmm - pnn) Re r.n, r . , x
\ Г + iQ Г —is
Й'2- Q's' + 2|G|2 \
[r-i(£' + ОД[Г-1(е' - ОД/ ’
= Vi2'2 + 4|G|2, Q' = Q-kv,
s' = Q^- Q-(k^-k) • v.
On taking partial fractions of the obtained expressions and allowing for
the condition |G| »kv, the velocity dependence must be retained only in
the resonance denominators Г + iQ', Г—is' and r — i(s’±Q0). The
value of Qo must be replaced by an approximate expression:
Ql> = Q0~kv, i20 = VX22 + 4|G|2.
"0
This results in formula (5.23).
Problem (28) Assuming in formula (5.23) that Q = 0, perform averaging
over velocities at кц = —k.
Answer.
(Nn - Nm) Гг^- exp
kv Lii2M \2kv/
Q |G2|G| {1-exp
— 2|G| I
|G| L
Ц, +2 |G| I
(£2M -2 |G|)2]
(M2
(SM + 2|G|)2-
(kvy J
Г « kv.
Here ф(г) is the probability integral determined by the formula (3.32).
Problem (29) Derive the expression (5.38) for the spectral density of
spontaneous emission intensity in a two-level system provided that
Гт = Гп = Г, \G\»kv.
432
Problems
Solution. It follows from the general formula of problem (13) when
Гт = Г„ = Fand |G| »kv that
I W(v)
Рц = -ba), |GJ2 Re^(Nm + Nn) x
2\G\2+Q'e'-e'2 \
[Г -i(£' + £>')][Г - i(E' - Л6)]/ ’
i2o = Vi2'2 + 4 |G|2,
Q' = Q-k-v, e' = - Q - (км - к) • v.
Expression (5.38) is obtained by a procedure similar to that given in
problem (27).
Problem (30) Calculate the spectrum of spontaneous emission at the
transition m-n for atoms moving in the field of a standing wave under the
conditions Гт = Гп = Гтп = Г, oj = a)mn, A = 0.
Solution. A term describing emission (C+) must be taken from expression
(3.177) and the formula (3.143) must be used
=-2ftct>M |GM|2ReJ drJ d^ exp[-i(£2M - кц • v)(t - ti)]<p(t, Л),
<?(/, Q = Snn(t, Л)[Ь01)ЬХЛ)8ЧЛ
b(/j) = S(ti, Го)Ь(^о)-
The evolution operator S(t, t0) is defined by the formula (3.122), where
the following should be taken:
f(t) = J G cos{* • [r - v(ti - r0)]} d/i
2G . [k-v 1 Г Г v JI
= -----sin —— (t -10) cosj к • г - - (t -10) 1.
Л • v L 2 J I L 2 JJ
For a single emission act, the spectral and angular density of spontaneous
emission is (normalization of |GM|2 by formula (3.247) should be
employed)
d4 d'' e“n{l + cos 2[f(t) -f(t’)] ±
lOJt Jo J/
cos 2f(t) ± cos 2f(t')} cos[(S2M - k^ • v)(t - f)]-
The signs + and — correspond to upper and lower level excitation.
Problems
433
Averaging of over r yields
А Г°° Сж
^=77^1 dr I dr'e-ncos[(DM-AM-v)(r-r')]x
lOJt Jq Jt
±
Expansion of the Bessel functions into a Fourier series makes it possible
to perform integration over t,t' explicitly. To calculate must be
multiplied by Qm = rNm (or Qn = Z7V„):
„ Й<ИМ a
Р*~
Re (Nm + Nn) 2
<5,о + /?(г)
Г — i(DM — • v — Ik • v)
//(*)____________1 + <5/0_______
Г + \lk-v Г — i(DM — кц • v — Ik • v)
z = 2G/k • v.
Problem (31) Prove the identity
Г 1 1 к" 12
.iw
Kq '•«'I J2
“(-ir'-sl,1 J Л1,1 , <*>
Kq '•*3 J3 '•*'3 J3 J2'
Solution. By definition (5.50) we have
I^Kq)r{Kq)= 2 (-l)-c,-‘73(lorl-ori|^) X
<7<71<72<73
(la2l-a3|^)GSGS,*G:2Ga3.
Using formula (10) of Appendix I, change the scheme of addition of
momenta and go over from 4(/c<?)Z*(/c<?) to |J(/c<?)|2:
2 = 2 (- 1)K+K,(2x- + I)!1 J K‘} IJOcitfOl2.
q Kiqi Ч J
434
Problems
S(-l)‘(2«: + l){ J *
к U I Ki
Using formula (15) of Appendix I, sum the products of 6/ symbols over к:
1 1 К1Г1 1 к
J3 J3 Jl' 1<7з J3 «/2
v 7 U л j3J
Then the right-hand side of relation (1) proves to be equal to its left-hand
side.
Problem (32) Obtain relations between the components of the tensor cafi
and transport cross-sections. Consider the case of elastic scattering by a
spherically symmetric potential.
Solution. Let the distribution of buffer particles be isotropic (in the
laboratory coordinate system). Then the tensor cat> is axially symmetric,
the axis of symmetry being collinear to the velocity v before collision. Let
с,, and cx designate the principal values of the tensor cafi corresponding to
the axis of symmetry and to an orthogonal direction. Obviously
Tr с = с,, + 2cj. = - J (v - V!)2A(v! I v) dvi,
(1)
cn = ' (v “ I v) dvP
J
(2)
Integration with respect to Vi after substituting the expression (2.155) for
A(Vj I v) yields
/ u\2 f
C|| + 2cx = ( — I I (и — «1)2о(и I И!)д(и2 — U2) X
\mJ J
pb(|v -BiDded»!,
C" = J [v • (u - u,)]2o(u | »i)d(u2 - u?) x
Pb(|v -»i|)d»d«i.
Integration with respect to и (it is convenient to choose the polar axis
along »i) yields
/ u\2 f
cll + 2c± = l —) Otr(«i)Pb(|v-«iDufd»!,
Problems
435
/и\ Г
Сц = 1 — I [cos2datr(u1) + i(l-3cos2d)a2(u1)]pb(|v-«1|)uid«1,
\m/ J
where the following designations are introduced:
otr(ui) = J (1 — cos 0)о(и | «j) dO,
o2(«i) = J (1 - cos2 0)o(u | »i) dO,
cos & = v • uJvUi, cos в = и • ujuux.
Subject index
absorption coefficient 2,174,183
- of probe field 183
absorption probability 328
accommodation coefficient 382
adjacent transition 154
alignment, tensor of 25
anisotropic collisions, model of 61
anistoropy of medium induced by field 307
areas theorem 224
band of homogeneous saturation 228,259
Bennett hole and peak 199,259
Bennett structure 201
Boltzmann distribution 69,235,361
Boltzmann equation 8
Born approximation for kernel 72
broadening of spectral lines 110
- Doppler 111
-field 153
classical description of translational motion 22
Clebsch-Gordan coefficient 24
collapse of spectral structure 128
collision frequency, dependence on velocity
of 88,90
collision hole (dip) 224,241,259
collision integral 18,31
- in-frequency of 45
- in-term of 33
- kernel of 44
- moment of 98
- out-frequency of 37
- out-term of 33
collision integral in a model of
- isotropic collisions 51,55
- non-degenerate states 45
- relaxation constants 44
- strong collisions 87
- structureless particles 45
combination scattering 159,180
- reverse 181
continuity equation 101
correlation factor 162
correlation function 110,172
crossed polarization tensor 295
Dicke effect 124
difference kernel 76
diffusion approximation for collision
integral 58
diffusion coefficient 58,345,346,391
diffusion flow 347, 391
diffusion halfwidth 123
dipole interaction 26
dipole moment 15
- matrix element of 15
Doppler broadening of spectral lines 111
drift velocity 323, 333
eikonal approximation 79
Einstein coefficient for spontaneous
emission 28
energy of translational motion 97
equation
- of continuity 101
- of viscosity 402
exponential difference kernel 84
field broadening of line profile 153
field splitting of levels 143
flow
- of momentum 97
- of particles 97
- of translational energy 97
Fokker-Planck equation 58
force of internal friction 91,101
frequency modulation of atomic
oscillator 111,119
Grad’s method 336
Green’s function 113,220
impact approximation 31
in-frequency of collision integral 45
- one-dimensional 83, 89
- v dependence of 90
interference collisional resonance 283,285
437
438
Subject index
interference at non-linear resonance 272
interference of real and virtual states 160
interference of sublevels 159
isotropic collisions, model of 49
iteration method 220
kernel of collision integral 35
- difference 76
- exponential difference 84
- in a Lorentz model 61
- in the Bom approximation 72
- in the model of impenetrable spheres 72,85
- in the model of isotropic collisions 54
- in the model of strong collisions 87
- in the model of structureless particles 45
-Keilson-Storer 73
-one-dimensional 83
kinetic factor of kernel 73
Knudsen conditions 382
Lamb dip 215
light-induced current 379
light-induced drift 319
light-induced flow 321,345,347
light-induced heat flow 404
light-induced viscous flow 396,401,404
light pressure 384
-negative 387
-spontaneous 385
macroscopic equations 95,385
matrix
- of spontaneous decay 14
- of spontaneous transitions 27
Maxwell’s demon 331
memory factor 162
method of probe field 154
model
- Lorentz 61
- of constant transport frequencies 105
- of difference kernel 76
- of isotropic collisions 49
- of non-degenerate states 24,45
- of relaxation constants 44
- of strong collisions 87
- of strong collisions, generalized 235
moment
-dipole 15
- of collision integral 98
- of density matrix 97
- of kernel 54,58
-polarization 24
momentum accommodation 382
non-linear interference effect 173
ndn-linear resonance
-collisional 285
- interference 272
- of field work 207,248
- of probe field 264,266
normal wave 307
number of collisions 223
optical piston 349,352
optical pump 377
optical pumping 364
orientation, vector of 25
out-frequency of collision integral 37,63
out-term of collision integral 33
persistence of velocities 56
polarization moment 24
polarization tensor 205
- crossed 295
pressure anisotropy 323,393,401
pressure tensor 97
probe field 155
pulling-pushing effect 389,391
quantum mechanical factor of kernel 73
quantum kinetic equation 17
recoil effect 21,311,315,385
representation
- coordinate 17,19
-energy 18
-JM 24
-momentum 19
- of irreducible tensor operator 24
- of polarization moments 24
-Wigner 20
- Kq 24
resonance approximation 136
resonance scattering 160,178
saturation 138
- band of homogeneous 228,259
-parameter 201
scattering amplitude 42
splitting of levels 143,149
statistical factor of kernel 73
stepwise processes 158
surface light-induced drift 382
Subject index
439
transport cross-section 56,92
transport frequency of collisions 56,90,92
- v dependence of 92
triplet of spontaneous emission 192,289
two-photon absorption 181,268
two-photon fluorescence 181,268
two-photon processes 159
velocity distribution of population 199
Voigt contour 117
Weisskopf radius 80
wind effect 60,135
work done by the external field 15
The advent of non-linear laser spectroscopy of gases dates back to the early
sixties In the three decades since then, a host of new and quite peculiar
phenomena have been discovered and studied for wnich a simple and
transparent physical interpretation I las been worked out
This Ьоок is devoted to theoretical non-linear spectroscopy of atomic and
molecular gases. A special emphasis is placed on the analysis of dynamic
and stochastic kinetic processes. The entire presentation is based on the
quantum kinetic equation for a one-particle density matrix with the generalized
collision integral of the Boltzmann type Consistent sequential analysis of
spectroscopic, optical and gas kinetics phenomena is generally believed to be
possible only on the basis of this equation The tool of the quantum kinetic
equation has been significantly developed over the last two decades in
connection with problems of non-linear spectroscopy and gas kinetics in the
field of laser radiation
Engineers and research workers, students and post-graduates in the fields of
optics, spectroscopy, quantum electronics and physical kinetics will welcome
this book
Difference between the scattering of an excited (upper trajectory) and an
unexcited (lower trajectory) atom by a scattering centre (solid circle)
ISBN: 0 444 88357 6