Table Of ContentPhoton-Atom
Interactions
Mitchel Weissbluth
Department of Applied Physics
Stanford University
Stanford, California
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Library of Congress Cataloging-in-Publication Data
Weissbluth, Mitchel.
Photon-atom interactions / Mitchel Weissbluth.
p. cm.
Bibliography: p.
Includes index.
ISBN0-12-743660-X
1. Photonuclear reactions. 2. Quantum theory. 3. Statistical
physics. I. Title.
QC794.8.P4W45 1988
530.Γ2—dc 19 88-12642
CIP
Printed in the United States of America
89 90 91 92 9 8 7 6 5 4 3 2 1
In Memory
of my Parents
Preface
A substantial part of the history of quantum mechanics is associated with
efforts directed toward an understanding of the interactions between light and
matter on the atomic and molecular level. For the first several decades of this
century, activity in this area remained at a high level, and the continuing
stream of advances in spectroscopy, both in theory and experiment, con
tributed enormously to fundamental physics and to various applied sciences.
By modern standards, the light sources available during that early period
produced light over a broad spectrum and at relatively low intensities, so that
on the whole, perturbative treatments to low orders were sufficient to deal with
experimental observations. A new era of light-matter interactions began in the
1960s with the invention of the laser. Its unique properties of high intensity,
monochromaticity, directionality, and coherence led to the disclosure of new
optical phenomena, the development of novel forms of high-resolution
spectroscopy, and the invention of numerous optical devices. As applications
proliferated, new subfields under various titles—Nonlinear Optics, Quantum
Electronics, Laser Physics, and Quantum Optics—came into existence.
Though each subfield is more or less unique in content, whether it be
spectroscopy, chemical analysis, medical application, communication, or any
of the myriad applications, there exists a significant body of theory shared by
all. This body of theory consists of a mixture of classical electromagnetism,
statistical physics, and quantum mechanics. These are, of course, well-
established branches of physics. Nevertheless, the manner in which they are
xi
XII Preface
applied and the particular combinations found to be appropriate in modern
optics are of more recent origin. The purpose of this book is to provide an
introduction to some of the new concepts and formulations with emphasis
on the quantum and statistical aspects.
The first chapter introduces the nomenclature, definitions, and certain basic
formulae associated with the mathematics of stochastic processes. Included is
a description of Brownian motion to illuminate the significance of the
Langevin and Fokker-Planck approaches. Chapter II is devoted to the
density matrix, evolution (time-development) operator, time-dependent per
turbation theory, correlation functions, Green's functions, and an intro
duction to two-sided Feynman diagrams.
Not infrequently in the history of physics, a new field, when examined more
closely, turns out to bear a close kinship to an older, well-established field.
Such is the case with modern optics in relation to magnetic resonance, best
exemplified by the close formal analogy between a spin-1/2 system in a time-
varying magnetic field and a two-level atom (or molecule) in a radiation field.
Indeed, evidence of this analogy is found in some of the optical terminology as
well as in the methods employed in certain types of experiments. Portions of
the theory of magnetic resonance are therefore included in Chapter III to serve
as a background for understanding these fruitful connections.
Quantization of the radiation field and the harmonic oscillator formalism
are treated in Chapter IV. Several types of states and their statistical properties
are discussed, including photon number, coherent, and squeezed states. In
Chapter V, the radiation field is coupled to an atomic system and the resulting
processes—absorption, emission, and scattering—are formulated in both the
semiclassical and quantized versions. Coherence functions in first and higher
orders are defined as well as the connection with light beams exhibiting
bunching, antibunching, and random statistics.
Chapter VI discusses damping and the master equation derived on the basis
of the interactions of a dynamical system with a reservoir (heat bath). Both
Langevin and density matrix methods are employed with applications to the
damped oscillator, the optical Bloch equations, photon echoes, spontaneous
emission from the standpoint of vacuum fluctuations, and several kinds of line
shapes. Finally, in Chapter VII, a number of nonlinear and multiphoton
processes—two-photon absorption and emission, stimulated Raman pro
cesses, three- and four-wave mixing, dressed states—are discussed. A
prominent role is assigned to the susceptibility function and its representation
in terms of two-sided Feynman diagrams.
The laws of nature are drawn from experience, but to express them one needs a
special language: for, ordinary language is too poor and too vague to express
relations so subtle, so rich, so precise. Here then is the first reason why a
physicist cannot dispense with mathematics: it provides him with the one
language he can speak ... Who has taught us the true analogies, the profound
analogies which the eyes do not see, but which reason can divine? It is the
mathematical mind, which scorns content and clings to pure form.
Henri Poincaré, Analysis and Physics
I Stochastic Processes
It is a matter of general experience that all physical measurements are sub
ject to fluctuations. Random perturbations, which may originate in molec
ular collisions, spontaneous emission, lattice vibrations, and various other
processes, manifest themselves in phenomena such as spectral line broaden
ing and relaxation effects. We find, for example, that light beams may have
different statistical properties depending on how they are generated and
that such differences have an important bearing on optical nonlinear inter
actions. Considerations of this sort are relevant in both classical and quantum
mechanical formulations; it will therefore be necessary to treat events that
can be described only in probabilistic language. The present chapter is de
voted to a summary of a number of properties of stochastic processes. For
readers interested in more extensive treatments, numerous sources exist,
some of which are listed in the general references at the end of this book.
1.1 Discrete and Continuous
Random Variables
Consider a simple coin-tossing experiment. For each experiment there are
two possible outcomes: heads(H) or tails(T). If we let ξ represent the outcome,
1
2 I Stochastic Processes
then
£ = {"' (i.i)
In general it is preferable, for the purpose of further mathematical manipula
tion, to represent outcomes by numerical values. We therefore might construct
a function Χ(ξ) such that
The function Χ(ξ) is known as a random or stochastic variable, defined as
a variable whose value depends on the outcome of a random experiment.
The probability of an outcome ρ(ξ) must satisfy
0<ρ(ξ)<1 ΣΡ«) = 1· (1·3)
Beyond these statements, the assignment of numerical values to ρ(ξ) lies
outside the purview of the mathematical theory of probability which is
primarily concerned with the manipulation of probabilities and ultimately
rests on an axiomatic foundation. From a physical standpoint, one proceeds
by assigning values to ρ(ξ) based on a mixture of available knowledge (or lack
of it) concerning the system, physical reasoning, and possibly other con
siderations. In the final analysis, it is only through experiment that one can
determine whether the assignments are justifiable. If the coin is tossed N times
and n(H) is the number of times the outcome is H, it is reasonable to suppose
that the probability p(H) is given by the ratio w(H)/JV when N is large. This is
merely an assumption, however, since n(H)/N has no limit as JV approaches
infinity.
Χ(ξ) in the coin-tossing experiment is a discrete, random variable with
just two values: 1 and 0. In many experiments, the random variable X is
continuous, that is, the outcome of an experiment may lie anywhere on a
continuum. Furthermore, because much of our work will involve temporal
changes, the random variable will be regarded as a function of time and will be
written X(t).
To illustrate these ideas, consider the case of a fluctuating voltage. One may
obtain a continuous record of the voltage taken over a specified time interval.
A record, V(t\ is a curve of voltage vs. time and is regarded as the outcome of a
random experiment. If there are many replicas of the system that generate
the voltage, each replica would produce its own record V(t, ξ^ where £ is a
f
label to keep track of individual records. ν(ί,ξ) where ξ = ξ ξ ,..., rep
9 ί9 2
resents a family or ensemble of curves.
1.1 Discrete and Continuous Random Variables 3
x(U)
x + dx
x
t t
1 2
FIGURE 1.1 The set of curves Χ{ί,ξ) is a stochastic process; X(í,{¿) with ζ constant is a
ί
function of time; X(t £) with t¡ constant is a random variable. The quantity of interest is the
i9
probability that the random variable lies in the interval (x,x + dx) at various times t¡.
Now, in place of a voltage, we may generalize to a physical quantity X (e.g.,
position, velocity, phase) whose value x is subject to fluctuations. The
ensemble of records X(t ξ) (Fig 1.1) is known as a stochastic process. When ξ is
9
kept constant, say ξ = ξ the function X(t ξ^ is simply a function of time and
ί9 9
represents the outcome of an individual experiment (as in the case of a single
record of the voltage fluctuations). On the other hand, when the value of t is
fixed at t = t the function X(t ξ) is the random variable. If both t and ξ are
i9 i9
fixed at t = ti and ξ = ξ then X(t ξ) is simply a number (x). As in the coin-
ί i9 (
tossing experiment, we shall be interested in the probability of a particular
outcome, but since the variables are continuous the statements refer to a
probability that the random variable X(t ξ) has a value that lies between x
h
and x + dx at the time t. It is customary to suppress the dependence on ξ
{
unless it is explicitly required. The random variable Χ(ί ξ) is then written
ί9
X(t¡) but since t may be varied, t = t t ... (Fig 1.1), the random variable is
i9 29
usually written X(t).
There is a fundamental difference between a random variable, X(t)
9
associated with a stochastic process and a deterministic function, f(t). For the
latter, the value of / is completely specified at every value of the time t but for
the random variable there is no functional relation between the value of X and
the value of t. In fact, for any given t the value of X can be anything within its
range of variation and all we can say is that X has a certain probability of lying
in a particular interval.
4 I Stochastic Processes
1.2 Probability Densities
For the continuous random variable X(t) we define a function W^x, t) known
as the first-order probability density or probability distribution function such
that \ν(χ,ϊ)άχ is the probability that the value of X(t) lies in the interval
γ
(x,x + dx) at the time t (Fig. 1.2):
\ν{χ,ϊ)άχ = p{x < X(t) < x + dx}. (1.4)
γ
This definition is illustrated in Fig. 1.1 for two values of the time, i and t .
x 2
Under special circumstances, the probabilities at t t and other values of the
u 2
time may all be the same, but for a general definition such an assumption is not
required. Since probabilities are inherently positive,
^(x,i)>0. (1.5)
We may also include discrete random processes in which the random variable
X(t) is defined only for integral values s. For this case,
W (x,t)ô{x -s)dx = p{X(t) = s}= p(s). (1.6)
1
J - oo
When Wi(x,i) is integrated with respect to x over the interval (a f,t), we
obtain the probability that the random variable X(t) acquires values lying in
(a, b). Thus,
Cb
W(x,t)dx = p{a<X(t)<b}. (1.7)
i
Ja
and if the limits are extended to ± oo to encompass the full range of x, the
probability achieves its maximum value:
r oo
W (x t)dx = p{-oo< X(t) < oo} = 1. (1.8)
x 9
J —00
FIGURE 1.2 The curve W (x, i) as a function of x is the first-order probability density. The area
l
H^(x, i) dx is the probability that X{t) lies in the interval (x,x + dx) at the time t.
