VII Semiclassical Radiation Theory within a Quantummechanical Framework*

E. WOLF, PROGRESS IN OPTICS XVI @ NORTH-HOLLAND 1978

VII

SEMICLASSICAL RADIATION THEORY WITHIN A QUANTUMMECHANICAL FRAMEWORK* BY

I. R.SENITZKY Departmenr of Physics, Technion-Israel Institute of Technology, Haifa, Israel

* This work was supported in part by the US Army through its European Research Office.

CONTENTS PAGE

0 1. INTRODUCTION . . . . . . . . . . . . . . . . § 2 . DESCRIPTION OF SEMICLASSICAL THEORIES

415

. . . 416

0 3. SIMPLE EXAMPLE OF FIELD-ATOMS INTERACTION

419

.

426

OF QUANTUM-MECHANICAL RADIATION THEORY. . . . . . . . . . . . . .

434

P 4. DISCUSSION OF THE SEMICLASSICAL THEORIES . § 5 . CLASSICAL LIMIT

REFERENCES . . . . . . . . . . . .

. . . . . . 447

8 1. Introduction Much interest has been exhibited, recently, in the validity of semiclassical radiation theory (SCT). This interest has been partly generated by the widespread use of SCT in the analysis of many important phenomena, such as laser operation, atomic and nuclear resonance effects, and photoelectric detection. It has also been stimulated by the suggestion - on the one hand - that semiclassical radiation theory can be postulated (with, perhaps, some refinements) as a fundamental substitute for quantum electrodynamics (JAYNES[1973]), and - on the other hand - by the criticism of its use where it had been considered acceptable (GLAUBER [1963]). As an illustration of this interest, one may cite two recent review articles which place SCT in opposition to quantum electrodynamics, and discuss the arguments in favor of each theory as opposed to the other (MANDEL [1976], MILONNI [1976]). It is the purpose of the present article to look at SCT as part of quantum electrodynamics, and to review work that indicates the conditions under which SCT is valid - or may be considered a good approximation - from an orthodox quantummechanical viewpoint. The first question that should be asked is: “What is meant by SCT?” Not surprisingly, a number of differing descriptions, or definitions, are found (sometimes in implicit form) in the literature. In order to discuss SCT in a coherent manner, we need a precise formulation of the several descriptions. For convenience, the theories corresponding to the different descriptions will be labeled as SCT I, SCT 11, etc. After describing them, we will discuss the several theories from the present point of view. A remark about terminology is necessary. Radiation theory refers to the interaction of the electromagnetic field with matter, and the kind of matter mainly under consideration in the present discussion consists of microscopic systems of which only a few properties - say, dipole moment and energy spectrum - may be pertinent. These systems have been referred to in the literature by various terms, such as particles, atoms, molecules, atomic systems, and two-level systems (if only two levels are of 415

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[VII, 9: 2

interest). This practice will be continued in the present article, and, for linguistic convenience, no effort will be made to use the same term throughout. It should be borne in mind that use of one of these terms rather than another has no particular significance.

§

2. Description of Semiclassical Theories

2.1. SEMICLASSICAL THEORY I

Perhaps the most common description of SCT is one in which “we treat the electromagnetic field classically and the particles with which the field interacts by quantum mechanics” (SCHIFF [1949]). We will consider this statement to be the definition of SCT I.

2.2. SEMICLASSICAL THEORY I1

Another description of SCT has found wide applicability in the analysis of masers and lasers, self-induced transparency, spin echoes, photon echoes, and other coherent atomic phenomena. (For a number of references o n these subjects, see HAKEN[1970]. An illustration of the application of SCT I1 may be found in the analysis of a gaseous laser by LAMB[1964].) In SCT 11, the field is treated classically, obeying Maxwell’s equations in which the dynamical variables referring to the matter interacting with the field are replaced by their quantum-mechanical expectation values. Usually, the only matter variables involved are the electric or magnetic polarization; in order to avoid complications that are irrelevant to the arguments of the present article, we will consider these to be the only matter variables that enter into the field equations, as far as SCT I1 and the higher-numbered theories are concerned. The atomic systems constituting the matter are treated quantum-mechanically in SCT 11, obeying Schrodinger’s equation in which the field variables are classical quantities. The two sets of equations, those for the classical field and those for the quantum-mechanical matter are coupled - or made mutually dependent by using the latter to obtain the expectation values occurring in the former. One may describe the calculating procedure involved in this form of SCT as follows: Schrodinger’s equation is solved formally (in principle) in terms of the classical field variables, expectation values of the pertinent

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DESCRIYITON OF SEMICLASSICAL THEORIES

417

matter variables are then calculated, and these expectation values are substituted for the corresponding matter variables in Maxwell’s equations, the result being a set of equations for the classical field variables only. Frequently, phenomenological terms are inserted into Schrodinger’s equation (or an essentially equivalent atomic equation - that for the density matrix) in order to account for effects other than those due to coupling with the field described by the field equations. These effects are usually referred to as relaxation phenomena (LAMB[1964], HAKEN [1970]). For the sake of simplicity of argument, they will be ignored in SCT I1 and in the higher-numbered theories.

2.3. SEMICLASSICAL THEORY 111

A third description of SCT is more recent (JAYNES[1973]; a formally different but, from a physical viewpoint, essentially similar theory is given by EBERLY [1976]). It contains a classical description not only of the field but also of the microscopic systems that constitute the matter; it is also the latest version of what has been called the “neoclassical” theory (CRISP and JAYNES [1969], STROUDand JAYNES [1970], JAYNES [1973]). In SCT 111, the individual atomic system is described in terms of its natural frequencies and associated dipole moments. Let the natural frequencies of the atom be those that correspond to the set of the energy levels hw,, n = 1, 2, . . . . The Hamiltonian describing the free atom is then given by

where a, and uz are independent classical (complex) dynamical variables. For each natural atomic frequency there exists an associated dipole moment, the components of which are linear superpositions of the quantities

The coupling between the (classical) atom and the field is assumed to be

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P2

of the dipole-moment type described most generally by the interaction Hamiltonian

cc 3

H’= h

m = l i
d$F’F$;”’,

(2.3)

where F$“)is a linear superposition of electromagnetic field components that contains appropriate coupling constants. The total Hamiltonian for the atom and the field is then

H=H,+H’+H,,

(2.4)

H , being the (classical) Hamiltonian for the field only (see, for instance, HEITLER [1954]). The “canonical” equations of motion for the atomic variables are given by (JAYNES[1973]) iRa,

= dHJda:,

iha:

= -dH/da,,

(2.5)

and the equations of motion for the field are the canonical equations of the conventional Hamiltonian formalism for the classical electromagnetic field (Maxwell’s equations). The connection between S C T 111 and quantum mechanics is made in the choice of initial values for the atomic variables; these are determined by the requirement that the initial dipole moments coincide with the corresponding quantum-mechanical expectation values. 2.4. SEMICLASSICAL THEORY IV

Another form of SCT has been formulated recently (SENITZKY [1977]) that consists of the classicai limit of a quantum-mechanical theory in which the atomic systems are described by a boson-second-quantization formalism. Since the description of this theory is intimately connected with the classical-limit procedure discussed in § 5 : its description will be given there. 2.5. OTHER THEORIES

There exist other theories that could, perhaps, be classified as “semiclassical” (MARSHALL [1963, 19651, BOYER[1969, 19701). They have not been associated, however, with the types of phenomena for which SCT I-IV are intended, and they will not be discussed in the present article.

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SIMPLE EXAMPLE OF FIELD-ATOMS INTERACTION

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Q 3. Simple Example of Field-Atoms Interaction Before an analysis of the various semiclassical theories is undertaken, it is instructive to investigate a simple - and somewhat idealized - problem involving the interaction between the field and a number of atomic systems (to which we refer as “molecules” in the present section). The results obtained will motivate and illustrate some of the arguments used in the analysis of the semiclassical theories. A pertinent feature of the investigation is the use of a formalism in which the dynamical variables referring to both the field and the molecules may be interpreted, independently, either as classical variables or as the corresponding quantum mechanical operators. It will therefore be possible to see, at a glance, the results of any combination of classical and quantum mechanical descriptions of the two interacting systems. The investigation follows the work of SENITZKY [1968a, 1968bl. Consider a number of identical molecules, located inside a cavity with perfectly conducting walls, and let the resonant frequencies of the cavity and the molecules be such that only one mode of the cavity field interacts significantly, and in an approximately resonant manner, with the molecules. The interaction is assumed to be of the dipole-moment type. It is also assumed that there exists no direct coupling between the molecules themselves. We can then write, for the Hamiltonian of the total system,

H = $ Z z o ( q 2 + p 2 ) + z H , ,+, h p x y , D , , m

(3.1)

m

where q and p are the (dimensionless) coordinate and momentum of the radiation oscillator associated with the mode, and Hm, D, and ym are the Hamiltonian, (dimensionless) dipole moment, and coupling constant, respectively, of the mth molecule. The coupling between the field and molecules is assumed to begin at t = O . (Note that the initial variables refer to the uncoupled systems.) The molecules are considered to be similarly and independently prepared systems, in the sense that the initial state of each molecule is the same, and can be described independently of that of the others. The dynamical variables of the molecules and the field may be regarded as either classical variables or as quantum mechanical (Heisenberg-picture) variables. The time derivative of any variable V( t) is given by (3.2)

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SEMICLASSICAL RADIATION THEORY

where the bracket [ ,Ip designates the commutator if V is a quantummechanical variable, or the Poisson bracket (in which the partial derivatives are taken with respect to the appropriate dimensionless coordinates and momenta) multiplied by i if V is a classical variable. The variables q and p satisfy the relationship

(3.3)

14, PIp = i.

We carry out a perturbation-theory calculation, up to second order in the coupling constants ym, of the energy absorbed by molecules in a resonant manner. The dynamical variables satisfy the equations of motion

The quantity we seek to calculate is

AH,,,~Hm(t)-Hrn(0) dtlp(t,)Drn(tl)

= -yrnh[

= - Ym N

F ( t)Dm ( t >- P (0)Dm (0)I + I’m fi

-yrnfio[

dtlq(tl)Drn(t,)

1‘ 0

d t1P (tl)orn([I (3.5)

where the last approximation may be regarded as a resonance approximation. This approximation is justified by our interest in the transfer of energy between the field and the molecules during a time long compared to a cycle but short compared to a molecular “lifetime” (the time during which the molecular energy undergoes a large change). Since D,(t) oscillates at approximately the same frequency as q ( t ) and p ( f ) , the integral will give a much larger contribution to AHm than the coupling energy evaluated at any time t, provided the coupling is sufficiently weak - a condition we assume throughout the present discussion. (The transfer of energy large compared to the coupling energy during a large number of cycles may be regarded as the essence of a resonance effect.)

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StMPLE EXAMF'LE OF F'IELD-ATOMS INTERACTION

42 1

Beyond the assumption of resonance between molecules and field, it is not necessary, for purposes of the present discussion, to choose a specific model for the molecules. The free molecule is described by the variables H$) and D$'(t), which are presumed to be known functions of the initial variables, for a given model. (We use a superscript in parenthesis to indicate the perturbation-theory order.) For the free -or zeroth order field, q ( t ) and p(t) may be written explicitly in terms of initial variables. In order to do this most simply, and also, for later use, it is convenient to introduce the complex variables a = (2)-1'2(q + ip),

a t = (2)--"'(q - ip),

(3.6)

which satisfy the relationship [a, atIp= 1. Quantum-mechanically, these are the well known annihilation and creation operators, respectively. We then have a"'(t) = a(O)e-'"',

a'"+(t)= at(0)eiw',

(3.7)

and can express q"'(t) and p"'(t) by

q W = CymJ ' dtlDm(t1) cos w ( t - ti), m

O

(3.10)

which lead, together with the above (approximate) expression (3.5) for Hm, to

(3.11)

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SEMICLASSICAL RADIATION THEORY

[VII. 5 3

In accordance with our resonance-type approximations, only the slowly varying terms in the integrand of HE' need be retained. Since [D',")(tl), D(mo'(t2)]pis an odd function of tl--t2, we neglect the even part of q(0)(t,)p(O)(tz),and retain the odd part, given by where the curly bracket is defined by { A ,B } = A B + BA. Approximation based on symmetry considerations can also be used in the m' = m term in the summation with respect to m'. Classically, Dg)(f2)D(mo)(tl) is symmetric with respect to f l - tZ. Quantum-mechanically, we can write D(,o)( t2)Dg'(tl) = f{D',")( tl), DE)(t2)}-tf[D',"'(tl), D',"'(t J ] .

(3.13)

Only the symmetric term need be retained, because of the factor cos o(t, - t2). One obtains, thus,

HE'

= f h w l O tdt,["'

dtz( yLID',")(fl), D',"'(t2)]p[2at(0)a(O)+A ] sin w(t,

-

tz)

where {a(O),a'(O)}has been rewritten as 2at(0)a(0)+A, with A = 1 if the field is treated quantum-mechanically, and A = 0 if the field is treated classically. The quantity aAho is the energy associated with the zero-point field. The total energy absorbed by the molecules is given by S,,,H$)+ X,,,H',2). If we sum the right side of the equations for H c ) and H:', we obtain summations of the form C m F t ' , where F stands for a molecular variable or the product of such variables. This is a summation over uncoupled, similarly and independently prepared (in the sense described earlier) molecules, since the variables are of zeroth order. For a large number of molecules, we can approximate a summation of this form by N(F"'), where N is the number of molecules, and the brackets ( ) indicate either classical average or quantum-mechanical expectation value, whichever is applicable. Both classically and quantum-mechanically, this approximation is based on the law of large numbers. Classically, this approximation is a familiar procedure. Quantum-mechanically, one may, at first glance, question the fact that an operator is replaced by a c-number. From a mathematical viewpoint, this poses no problem, since

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423

one can regard the c-number as being multiplied by the unit operator. From a physical viewpoint, the answer is contained in the statement that the numbers associated with - or the result of measurements of - the dynamical variable C,,,g;), are given by a probability distribution that becomes relatively sharp as N becomes large. Setting

(3.15) m

neglecting unity compared to N, and assuming ym and DE’(t) to be statistically independent, we can write & ( l= )

J

(y)tio

dt, q(0)(tl)(D(O)(tl)>,

(3.16)

0

(3.17)

x Jot dtlIot1dt2i([D‘”(t,), D‘O)(t,)],)sin o(t,- t,),

(3.18)

and

Since qndis the energy lost by the molecules due to the initial excitation of the field, we may regard it as induced emission, and since is the energy lost by the molecules independently of the initial excitation of the field, we may regard it as spontaneous emission. In order to simplify the discussion and concentrate only on the aspects essential for present purposes, we deal first with the special case for which or One can then write

(D“’(t))= 0,

(3.20a)

( Y ) = 0.

(3.20b)

x (Ai([D“)(t,), D(O)(f2)lP) sin w ( t , - fz) -({D(’)(t1),D(O)(tZ)}) cos o(t1-

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SEMICLASSICAL RADIATION THEORY

Consider a situation in which all the molecules are in the ground state. (For classical interpretation, this means that the molecular energy has its lowest value.) If the molecules respond to the field (or absorb energy from the field) in second order, which we assume, then &in,, must be negative. The expression for &ind leads, therefore, to the inequality

6'

dt,['

dtzi(0)[D'O'(tl), D'"(t2)],)0)sin w ( t l - t2)> 0,

(3.22)

where 10) is the ground state both quantum-mechanically and classically. It is clear that ([D'"(tl), D(o'(t2)]p) does not vanish in either formalism when the molecules are in the ground state. The result is different, however, with respect to the expression ({D(o'(tl),D(')(t2)}).Quantummechanically, this expression cannot vanish in the ground state, since D'O'( t ) does not commute with the molecular Hamiltonian (otherwise it would not oscillate). On the other hand, D'O'(t) and {D("'(tl), D'"'(t,)} must vanish classically, since internal motion ceases when a classical system is at its lowest energy. It is useful to have an explicit quantummechanical evaluation of the ground-state expectation values considered above in terms of the matrix elements of D(0). Noting that (3.23) ojE'(t)== D j k eXp (iWjkt), where D j k = D;E'(O),hwjk = Ej - &, with the Ei's being the molecular

energy levels, we obtain, with obvious labeling,

(ol(oco'(t,),D ' O ) ( t J ) l o ) q m

= 2 C IDok)' COS W k O ( t l k

t2).

(3.25)

Classically, we merely write

~ i(ol[Dco'(tl), ~ ' o ' ( t ~ ) l , l O ~#c 0,

(3.26)

(Ol{D'"'(tl),D'"'(tz)llo>c,= 0.

(3.27)

Let us consider the expression €or spontaneous emission, with the molecules in the ground state, using all four possible types of description, as follows: (i) Both the field and the molecules are described quantummechanically. Utilizing the previously used approximation of retaining only resonant terms in the integrand, we obtain cancellation of the two

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SIhPLE EXAMPLE OF FIELD-ATOMS INTERACTION

425

terms in the integrand of (3.21), so that Espont

= 0.

(3.28)

(ii) Both the field and the molecules are described classically. In this case each term in the integrand of (3.21) vanishes, and Espont

= 0.

(3.29)

(iii) The field is described quantum-mechanically and the atoms are destribed classically. The first term of the integrand in (3.21) is now positive, according to the inequality (3.22), and the second term vanishes, so that (3.30) One notes that cSpont, in this instance, is exactly equal to &ind if we replace U’(O)U(O) by 1/2 in Bind. (iv) The field is described classically and the molecules are described quantum-mechanically. The first term of the integrand in (3.21) vanishes, and the second term produces a positive contribution to the right-hand side, so that Espont’o. (3.31) Since spontaneous emission is independent of the field energy, we apply these results to the case where the field is also in its ground state. In descriptions (i) and (ii), the results are entirely reasonable, since molecules in the ground state cannot emit energy. In descriptions (iii) and (iv), however, the conclusion contradicts the assumption; if the molecules and the field are both in their state of lowest energy, the molecules should not be able to emit or absorb energy. This internal inconsistency may be explained by the statement that, formally, the zero-point oscillation of the quantum-mechanical system is “seen” by the classical system as motion which can do work. Thus, in description (iii) the classical molecules absorb energy from the field as though it were induced absorption from a field with energy of quantum number 1/2. Likewise, in description (iv), the quantum-mechanical molecule appears to be doing work through its zero point oscillation on the classical field. ((Ol{D(”)( tl), D‘’’( t2)}10)may be regarded as a measure of the molecular zero-point oscillation.) Under certain conditions, the expression for cSpontis meaningful for all possible descriptions. From (3.19), these conditions are, clearly, those for which the term containing A is negligible compared to the other terms, in

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SEMICLASSICAL RADIATION THEORY

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D4

which case the formalism used for the field is irrelevant. For (D'"'(t ) ) = 0 or ( y) = 0, such a condition is given by where the subscript "max" indicates the maximum value during a cycle of tl-t2. This condition may be described as the existence of a large (essentially classical) molecular amplitude of oscillation. It is satisfied, for instance, by a harmonic oscillator in a high energy state (MESSIAH [1961]). If we extend our consideration to the case (D("(t))# 0, ( y ) # 0, the N 2 term will be dominant in E ~ for ~sufficiently ~ ~ large ~ N., Physically this case may be described as that of a large number of molecules oscillating in phase and producing a large (essentially classical) oscillating dipole moment. In other words, the method of description of the field in the expression for spontaneous emission is irrelevant when the oscillating dipole moment is essentially classical. We turn, next, to induced emission. In principle, both &(l) and &ind may be regarded as induced emission? since both quantities depend on the initial field; in usual practice, however, one has &ind in mind when referring to induced emission, and assumes that either ( y ) or (D'"'(t))is negligible. It is seen that neither qndnor E ( ' ) depends on the formalism used to describe the field, but only on the numbers that we associate with either at(0)a(O) or with q")(t).The result for qndwill be identical in both formalisms if, for instance, we say, quantum-mechanically, that the field is in an energy state corresponding to a given energy eigenvalue, or, classically, that the energy of the field has this same value. An identity of results also exists if the prescription of the field is such as to yield a statistical description of the energy, provided that this statistical description is the same in the classical and the quantum-mechanical prescriptions of the field. § 4.

Discussion of the Semiclassical Theories

We return to the consideration of the semiclassical theories described in 3 2 . 4.1. SEMICLASSICAL THEORY I

Classical treatment of the field and quantum-mechanical treatment of the atomic system is the proper analytical method of studying the atomic

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427

behavior, from a quantum-mechanical viewpoint, when the field is prescribed, that is, the behavior of the field is specified without reference to the atoms. If one deals with the case of a strong field where the reaction of the atoms is negligible, then a classical prescription of the field is clearly justified. The field assumes the role of a (time dependent) parameter in either Schrodinger’s or Heisenberg’s equations of motion, and the problem is a computational one only, soluble, in principle, to all orders of perturbation theory. If one deals with the case of a weak field, a classical prescription of the field is justified under certain conditions. Suppose that the question posed does not refer to the effect of the atoms on the field, and the expression for the answer depends on one field variable (the energy, say). Then as seen in § 3, an appropriate classical description of the field will yield the same result as a quantum-mechanical description. An illustration of the first case is the behavior of atoms in a strong laser field. An illustration of the second case is induced emission or absorption up to second order in perturbation theory. (Another illustration, photoelectric detection, will be discussed later.) Although the word “photon” may be used in connection with these phenomena, it should, from a strict point of view, be interpreted as referring to the atom rather than the field. Thus, when one speaks of the absorption of a photon-or a number of photons, in multiphoton phenomena - the statement should be understood as referring to the energy of the atom only, since prescription of the field (independently of the atomic behavior) is inconsistent with calculation - or consideration - of atomic effects on the field. Taking account of the reaction-or the action-of atoms on the field is, in general, logically impossible in SCT I, since quantummechanical atoms radiate a field that must be described, in general, quantum-mechanically. This follows from the fact that the inhomogeneous solution of Maxwell’s equations (which, in operator form, are the Heisenberg equations of motion for the electromagnetic field) is an expression containing the quantum-mechanical matter variables. Furthermore, as has been shown in § 3, a classical field “sees” the zero-point motion of quantum-mechanically described atoms as motion which can do work, a fact that leads to absurd results under certain conditions. Spontaneous emission is describable in SCT I only when the oscillating dipole moment is sufficiently large (in a sense described quantitatively in § 3 ) to be essentially classical. One may enquire if a prescription of the field (during the time of interaction with the atoms) can be given in quantum-mechanical terms.

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Why cannot the field be prescribed “quantum mechanically” with the statement that it is initially in a given quantum state and its subsequent behavior is unaffected by the atoms? Such a prescription implies that any (ideal) measurement performed on the field -which, presumably, could be carried out by means of the atoms - will also leave it undisturbed. This implication is inconsistent with the uncertainty principle. Thus, if the field is prescribed independently of the atoms during the interaction process, the prescription must be considered to be essentially classical. It need not necessarily be deterministic, however; the prescription may be given in a statistical form. 4.1.1. Photoelectric detection The use of SCT I in photelectric detection has been the object of some criticism which is based on the point of view that one is dealing with a “photon field”, and “there is ultimately no substitute for the quantum [19631). It is instructive to discuss theory in describing quanta” (GLAUBER this subject in some detail. Although there exist several ways of looking at the phenomenon of photoelectric emission (KELLEY and KLEINER [1964], GLAUBER [1964], MANDEL, SUDARSHAN and WOLF[1964], MANDEL and WOLF[1965,1966], SENITZKY [1968b], LAMBand SCULLY [1969]), it is hardly necessary to point out that what one observes experimentally are electrons, and not photons. The fundamental expressions in photoelectric detection should, therefore, refer to the excitation (or ionization) of the atoms rather than to the disappearance of photons. We can use our simple model of § 3 for this purpose, letting the atoms in the cavity be the “detector”. If we consider the emission of a photoelectron as the excitation of an atom to an energy in the neighborhood of Eo + hw, where E, is the ground state energy, then the photoelectric current is given, from (3.18), by

I=

1 d hw d t

Eind

J

= ~ ( y ’ > a ~ ( o ) a ( o dt,i([D“’(t), ) 0

~ ‘ “ ’ ( t ~ ) sin ] , ) w ( t - r,).

(4.1)

Assuming a spread in the excited atomic energy levels with a level density r ( E , + h w ) , we obtain, approximately, I = .rrNhy?ai(0)a(O)r(E,,+hw)lD12,

(4.2)

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where 10)2 is the average of I D O k ) * for Ek near E , + h o . This is the photoelectric current during the time t for which second-order perturbation theory is valid. Setting w = (I/N)t,

(4.3)

and noting that the expectation value in (4.1) came from the averaging procedure implicit in the law of large numbers, we must interpret w as the probability per atom for the emission of a photoelectron during the time 1.

The above theory describes the effect of the field on atoms in a lossless cavity. This theory has been developed further by taking into account cavity losses and the presence of a source that drives the field (SENITZKY [1967, 1968133). The source is assumed to be negligibly affected by the detector, and the spectral width of the field is assumed to be less than that of the atoms. The result for the photoelectric current is formally similar to (4.1), except that the operator at(0)a(O)is now replaced by the c-number a * ( t ) a ( t )which describes the energy of the field generated by the source. (The insensitivity of the source to the detector makes the source appear [1965, 1966, 1967]).) 'We can ''classical'' to the detector (SENITZKY therefore write dw/dt = const. x a*(t)a(t).

(4.4)

In other words, the emission probability per atom per unit time at time t is proportional to the field energy at time t, described classically. One can proceed, using this expression together with the assumption that the atoms emit independently of each other [which is implicit in the factor N of (4.1), and is the result of the use of lowsst order perturbation theory] to calculate the various statistical properties of the photocurrent as a function of the statistical properties of the field, now viewed as a (classical) random process (MANDEL[1958, 19591, MANDELand WOLF [19651). Thus, if the reaction of the photoelectrons on the field is ignored, SCT I is valid, from a quantum mechanical viewpoint, in the analyses of photoelectric phenomena. It should be emphasized that this result follows naturally from the fact that we address ourselves to the question concernand WOLF ing the emission of photoelectrons (MANDEL,SUDARSHAN [1964], SENITZKY [1968b], LAMBand SCULLY[1969]) rather than to the

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question concerning the disappearance of photons (GLAUBER[ 1963, 19641). As is well known, the form of the question, or the choice of variable with respect to which the question is posed, plays a significant role in quantum theory. 4.2. SEMICLASSICAL THEORY I1

In contrast to SCTI, where the mutual interaction between the field and the molecules cannot be described generally in a self-consistent manner, S CT I I provides a prescription for doing so. As mentioned earlier, SCT I1 calls for replacement of the matter variables in Maxwell’s equations by expectation values derived from Schrodinger’s equation. It is useful to outline the method of derivation of these expectation values for the case of an atomic system coupled to a classical field through its dipole moment. Consider, for the sake of simplicity, only a finite number of discrete levels of the atomic system to be of relevance in the interaction. We describe the state of the atom in the Schrodinger picture by (4.5)

Iq(t)) =Ctct (t)Iqt)>

where the lq,)’s satisfy the relationship

HJl%)= ELbA

(4.6)

H , being the Hamiltonian of the free atom. The dipole moment operator associated with the ij pair of levels can be expressed as a linear superposition of the operators

9:; =4(1cp,)(cp,l ’

’:I’

= -ii(l

+lcpCX@,l) ‘ P I >(qZ

I- 1

q I

9:;’= i(lqj)(qj I - Iqz ) ( q t

zf

J.

1)

>(qJ

I),

(4.7)

The total Hamiltonian for the atom in SCT I1 is Ha,= Ho + H’,

(4.8)

with

where F:j’’’) is, most generally, a linear superposition of (classical) elec-

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DISCUSSION OF TH!3 SEMICLASSICAL ‘THEORIES

431

tromagnetic field components that contains appropriate coupling constants. Schrodinger’s equation of motion for the atom can be written (SCHIFF [1949])

with the summation being taken over j only. The expectation value of the dipole moment is given by (9:;))

=+(CTCi

+ C*Cj)

(gf)) = -+i(cTci - ci* ci)

(4.10)

($3:;)) = &CTCj - C * C i ) .

According to SCTII, the dipole moment of the atom that yields directly the polarization vector in Maxwell’s equations (polarization being dipole moment per unit volume) is to be considered a linear superposition of these quantities. Before we discuss the degree of validity of SCTII from a quantum mechanical viewpoint, certain features of SCT I1 that may be regarded as unsatisfactory from a conceptual viewpoint should be pointed out. Firstly, the connection between the classical field equations and the quantummechanical matter equations is made through an ad hoc prescription, namely, “let the matter variables in Maxwell’s equations be replaced by their expectation values”. This prescription does not arise from a formal theory based on conventional laws of physics. Secondly, one can regard this prescription as containing a degree of arbitrariness. Why should the matter variables in Maxwell’s equations be replaced by their expectation values before the equations are solved rather than after the equations are solved? It is clear that the results will, in general, be different. For instance, the solution for radiated field energy from a dipole oscillator into free space depends on the square of the dipole moment (see, for instance, SCHIFF[1949]), while Maxwell’s equations contain the first power of the dipole moment, and as is well known, the expectation value of the square is not necessarily equal to the square of the expectation value. More generally, if one wants to replace the operator 0 by a number, any one of the (generally different) possibilities (On)’”’, n = 1, 2, . . . , is available, and there exists no a priori reason for choosing one rather than another.

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SEMICLASSICAL RADIATION THEORY

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SCT I1 ignores the statistics inherent in a purely quantum-mechanical description. Quantum mechanics is applicable, in principle, not only to microscopic systems, where the macroscopic experimental result is obtained by an averaging process, but also to macroscopic systems. Consider, for instance, the radiation by an excited macroscopic harmonic oscillator. Now, there is nothing to prevent us, in principle, from describing the quantum-mechanical state of this harmonic oscillator (if it is suitably prepared) as an energy state, no matter how high its energy may be. In this case, the expectation value of the matter variables in Maxwell’s equations are zero (since the variables are linear in the harmonic oscillator coordinates) and the radiated field (or energy) obtained is zero! The reason for this incorrect result is evident. An energy state describes a harmonic oscillator only statistically, and, in the limit of high quantum numbers, the energy state describes an ensemble of classical oscillators with precise amplitude and random phase (MESSIAH [1961]). An ensemble average over the coordinates of the oscillator before one calculates the radiated field (or its energy) will therefore yield nothing meaningful. When does SCT I1 provide a useful formalism for the analysis of the interaction between matter and the field? The above discussion serves to motivate part of the answer. If the matter variables occurring in Maxwell’s equations are macroscopic, and have a relatively small quantummechanical dispersion (so that the effect of the purely quantummechanical statistics is insignificant), then the replacement of the matter variables by their expectation values may be regarded as a good approximation. It is instructive to analyze a simple, idealized, example in order to illustrate this point. Consider N identical, but distinguishable, two-level systems with parallel dipole moments. Let the energy states of the ith two-level system be labeled by 11,) and 12,), and the respective energy levels by E l and E,. The total dipole moment along the direction of polarization is given by

2 di, N

D

=

(4.11)

i = l

where di is the dipole-moment operator of the ith two level system. We consider, for simplicity, the case in which di has only off-diagonal matrix elements. Using dimensionless units, we can then set d = 9 y d for each two-level system, so that

di =4(11i)(2iI+ l2i)(1il).

(4.12)

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DISCUSSION OF THE SEMICLASSICAL THEORIES

433

Let the total system be in the state (of the Schliidingerpicture)

n N

PI)=

[c,(t)lll)+ C2(Wl)1?

(4.13)

1=1

where (c112+lc212=1, and t is short compared to the time during which coupling produces significant changes in the atomic state. (This type of state has been referred to as a coherent state, or a Bloch state, in the literature (SENITZKY [1958], ARECCHI,COURTENS, GILMOREand THOMAS [1972]).) Then the expectation value of the total dipole moment is given by ( D )=+N(kTk,e-'"'+ k:kle""'), (4.14) where w = ( E 2 - E l ) / h and k, = c,(O). We also obtain, for this state (SENITZKY [1977])

( D 2 )= $ N ( N - l)(kyk$e-2'"'

+ k;2k:e21u1t

+21k1121k2)2)+$N.(4.15)

One notes that the term gf order N 2 in the expression for ( D 2 ) is precisely (D)', and that ( ( D 2 ) - ( 0 ) 2 ) / ( D 2N-l. )~ (4.16) Thus, for a large number of two-level systems described by the state /TI), the statistical spread in the dipole moment introduced by the quantum mechanics is negligible, and replacement of the dipole moment (or polarization) by its expectation value in Maxwell's equations is a good approximation. Consider now the situation in which the two-level systems are described by the state (4.17) where P is the operator that permutes the atomic indices so as to produce different product states, and the summation is to be taken over all such permutations, of which there are N ! [ N - r)!r!]-'. One may describe as an energy state that is symmetric with respect to the interchange of any pair of atoms. (In the literature it has been described as a symmetric Dicke state (SENITZKY [1958], ARECCHI, COURTENS, GILMORE and THOMAS [1972]).) Its energy eigenvalue is rE,+(N- r)E2.For this state, we have (SENITZKY [1977]) PIIIDlTd= 0, (4.18) (~IllD21'P~J = $ [ r ( N - r) +*N].

(4.19)

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SEMICLASSICAL RADIATION THEORY

[VII. 5 5

One sees that the (quantum-mechanical) statistics associated with are such that the dispersion of the total dipole moment is of the same order of magnitude as the dipole moment itself. The replacement of the dipole moment of two-level systems in the state I*I ) by its expectation value in Maxwell’s equation is, therefore, unjustified, and will yield qualitatively incorrect results. It is interesting to note that for r = Nlk,12 the term of order N 2 in the time average of ( ~ I ~ D 2 ~ is *equal I ) to the term of order N 2 in (UI,(D’l*II), so that under certain conditions, substitution for the classical 0’ of the quantum-mechanical (0’)is a more meaningful procedure than substitution of (0)’.

4.3. SEMICLASSICAL THEORY I11

SCT 111 is closely related to SCT 11. In fact, it is a formulation of SCT I1 that constitutes a self-consistent classical dynamical theory without any ad hoc prescriptions. SCT I11 achieves the same result as SCT I1 by describing the field classically and by introducing a classical atomic model. An appropriate definition of dipole moment makes it equivalent to the corresponding quantum-mechanical expectation value. In order to demonstrate how SCT I11 produces this result, we refer to the “canonical” equations of motion (2.5) for the atomic variables a, and a: introduced in the definition of SCT 111. Explicit differentiation of the Hamiltonian yields, as the atomic equations of motion, equations for the a,’s that are identical to those for the c,’s, eqs. (4.9), in SCTII. Furthermore, the expresion for the dipole moment in terms of the d!;”’s of SCTIII is exactly the same as the expression for the expectation value of the dipole moment in terms of the (9;;”’)’s of SCT 11. Thus, the “dipole moment” of SCTIII is identical to the “expectation value of the dipole moment” of SCT 11, since, as pointed out in 0 2.3, the corresponding initial values are the same. From a conceptual viewpoint, SCTIII has the advantage of avoiding the ad hoc prescription connecting the classical description of the field with the quantum-mechanical description of the atoms.

9 5. Classical Limit of Quantum-Mechanical Radiation Theory If SCT is viewed as an approximation based on quantum theory, it should be possible to derive its formulation directly from quantum theory

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CLASSICAL LIMIT OF QUANTUM-MECHANICAL RADIATION THEORY

435

for those conditions for which it is a valid approximation. We do so in the present section, following SENITZKY [ 1974, 19771. In addition to furnishing insight into the validity of SCT, this derivation yields a theory that contains new features.

5.1. BOSON-SECOND-QUANTIZATION FORMALISM

We begin by considering a number of identical atoms that are characterized by their spectrum and associated dipole-moment strengths. (The dipole moments may be either electric or magnetic.) Let the relevant spectrum consist of energy levels hw,, i = 1 , 2 , . . . n. Although the atoms are, in principle, distinguishable, we assume that they couple to the electromagnetic field identically, so that, as far as their effect on the field is concerned, they are indistinguishable. A boson second quantization (BSQ) formalism will be introduced to describe the state of these atoms. The space in which these states are described is spanned by the orthonorma1 basis vectors Ir, . . . r , . . . rn),

(5.1)

where the ri’s are non-negative integers. The fundamental operators, in terms of which all dynamical variables are constructed, are ui(t)and a t ( t ) (we use the Heisenberg picture, unless stated otherwise), such that

ai(0)lrl . . . ri . . . r,,) = r!’*Irl . . . ri .- 1 . . . rn), u!(0)lrl

. . . ri . . . r,) = (ri + I)”*Irl . . . ri + 1 . . . r,).

(5.2)

(5.3)

The operator commutation relationships are

(5.4) with all other equal-time commutators vanishing. The Hamiltonian which describes the collection of atoms is given by

One sees immediately that ai and a f are the usual annihilation and creation operators associated with a harmonic oscillator of frequency mi. The dipole moment of the entire collection is a linear superposition of the

436

SEMICLASSICAL RADIATION THEORY

[VII, 8 5

operators

For brevity, the d$,!’”’s, rn = 1 , 2 , 3 , will be referred to as the dipole moment components associated with the frequency lwi - mil. The coupling between the atoms and the electromagnetic field - assumed to be described quantum-mechanically, at this point - is given most generally by the interaction Hamiltonian (5.7) where the fiy)’~are linear superpositions of the components of the electromagnetic field multiplied by appropriate coupling constants. For the total Hamiltonian, we have

H

= Ho

+ H’+ Hf,

(5.8)

Hf being the field Hamiltonian (which we do not need in explicit form). For later discussion, it is convenient to introduce the “reduced” variables A,(t), Ar(t), defined by the relationships a j ( t )= Ai(t)epiwl‘,

a:(t) = A;(t)elw1’.

(5.9)

One verifies easily that the reduced variables are constants in the absence of coupling to the field, and vary slowly (compared to the natural atomic oscillations) for sufficiently weak coupling, the only kind of coupling to be considered. The basis vectors l r l . . . r,) are, clearly, eigenvectors of the occupation number operators ni(0),defined by

n. = ata. = A . 1

t

(5.10)

and satisfy the eigenvalue equation, ni(0)lvi... r n ) = r i l r i . . . r,).

(5.11)

They are also eigenstates of Ho, and are, therefore, energy states of the free (uncoupled) collection of atoms. It can be shown (SENITZKY [1974]) that, with the above interaction Hamiltonian, the operator Zini is a

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CLASSICAL LIMIT OF QUANTUM-MECHANICAL RADIATION THEORY

437

constant of motion. All the energy states are eigenstates of this operator,

(c 1

ni Ir, . . . r,,) = Nlr, . . . rn),

where

N

=

(5.12)

Cri, I

and those corresponding to a given N form a subspace which is invariant under the transformation of the total Hamiltonian. We describe the collection of atoms under consideration by a state (the initial state, if the Schrodinger picture is used) which lies in the subspace for which N is the total number of atoms. We obtain, thus, a BSQ formalism in which the atoms may be considered to be the bosom. From a first-quantization viewpoint, this means that the states used are fully symmetric with respect to the atoms, and from a physical viewpoint, this means that a situation of maximum atomic cooperation (or complete similarity of atomic behavior) is being considered. The operator n, now corresponds to the number of atoms (regardless of their identity) found in the ith (one-atom) energy state by an experiment designed to yield such a result, and the fact that Zini is a constant of motion merely means that the total number of atoms is invariant under interaction with the field. In addition to the set of energy states labeled by the integers rl, . . . r,,, it is useful to define a set of “coherent” states labeled by the complex constants c l r .. . c, (which we normalise by 1 1ciI2= 1 and designate collectively by {c}) defined as follows:

. . c : [ r l . . . r,,),

(5.13)

where the parenthetical superscript ( N ) in the summation indicates that the summation is to be taken over all values of rl, r,, . . . r,, for which

C ri = N. I

A t present, we merely note the properties (5.14) Certain other properties and the significance of the coherent states will be

438

SEMICLASSICAL RADIATION THEORY

[VII, P 5

discussed later. (Coherent states have been discussed in various contexts by SENITZKY [1958], ARECCHI,COURTENS, GILMOREand THOMAS[1972], GILMORE [1972], ARECCHI, GILMORE and KIM [1973], GILMORE,BOWDEN and NARDUCCI [1975], SENITZKY [1977].) 5.2. CLASSICAL LIMIT (SEMICLASSICAL THEORY IV) AND RELATIONSHIP TO SEMICLASSICAL THEORIES I1 AND I11

As is well known, a boson formalism is particularly suitable for passage to the classical limit. (One might remark that this is the reason, from a quantum-mechanical viewpoint, for the success of the classical theory of the electromagnetic field in contrast with the classical theory of the atom.) It should be noted that our collection of atoms has been described in the BSQ formalism by y1 harmonic oscillators, the operators a, and a: being the complex amplitudes

a, = 2-l”(qJ

+ ip,),

a: = 2-”2 (qJ-ipJ)?

(5.15)

where q, and p, are the dimensionless coordinate and momentum of the jth oscillator, satisfying the commutation relationship [q,, Pa1 = i,

(5.16)

and in terms of which the Hamiltonian of the jth oscillator is expressed by H, =+hw,(q;+p;)= hw,(n,+i).

(5.17)

Now, the conditions under which a harmonic oscillator may be regarded as classical for all purposes are well known. The traditional description of these conditions is the Correspondence Principle, which presents them as t h e limit of high quantum numbers. We obtain these conditions in quantitative form for the present case in the following argument. The essential difference between the quantum-mechanical description and the classical description of the harmonic oscillator is the fact that a, and a! are (noncommuting) q-numbers rather than (commuting) cnumbers. If, however, the numbers associated with a,a: and a:a, are approximately the same, that is, if [a,, a:] is relatively negligible, then a, and a: may be treated as c-numbers and the description becomes classical. This is indeed the case when the numbers associated with the operator n, are large compared to unity. What are these numbers? If our system is in an energy state, then, clearly, the only number associated with n, is its eigenvalue r,. If the system is not in an eigenstate of n,,then there appears to be more than one number that is associated with n,.For

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CLASSICAL LIMITOF QUANTUM-MECHANICAL RADIATION THEORY

439

instance, a coherent state consists of terms which belong to eigenvalues of ni that range from 0 to N. These terms are weighted by their coefficients, however, and a reasonable estimate of the significance of the various eigenvalues of ni is provided by the quantity (q). We will use this quantity as a measure of the numbers associated with ni.Our criterion for treating the oscillator classically is, therefore, the following: if ( n , )>> 1, a, and a: may be treated as c-numbers, and the ith harmonic oscillator may be regarded as classical. In order that the entire system may be regarded as classical, this criterion must be satisfied by all the oscillators:

( n , ) >> 1,

i = 1, . . . a.

(5.18)

It follows immediately, of course, that the condition N >. 1,

(5.19)

must also be satisfied. For clarity, the symbol tilde will be used henceforth over a variable where it is necessary to indicate that it is being treated classically (that is, as a c-number). When the inequalities (5.18) are satisfied, Ho can be regarded as the classical Hamiltonian of the free (uncoupled) collection of atoms, expressed either in terms of the complex variables ii,, 6: or in terms of the (dimensionless) coordinates and momenta iji and pi,and is the same as the Hamiltonian of n classical harmonic oscillators. (It should be noted that h appears in Ho only for dimensional reasons.) The time variation of any variable is obtained from the Hamiltonian by the usual Poisson-bracket relationship. The Poisson bracket expressed in terms of partial derivatives of ij, and pi must be written so as to take into account the fact that these are dimensionless coordinates. Thus, for any variable X ( t ) , in the present notation, the equation of motion is (5.20) if X is considered to be a function of the Q’s and

ax aH

aH

ax

AS,or (5.21)

if X is considered to be a function of the Gi’sand 5:’s. The equations for iij and ii? become (5.22)

440

SEMICLASSICAL RADIATION THEORY

[VII, 5 5

which, for the free collection of atoms, lead to %+ = /pelO,f, 6J = AJ e-’“,‘, J J

(5.23) where AJ and are now classical constants, obeying the inequality lAJl*>> 1. The dipole-moment components are, in this case, classical quantities, two of which, d!:) and d::), oscillate with the frequency Iw, - wJI. It will be shown later that the electromagnetic field interacting with a classical atomic system may be treated classically, so that the fI;”s in the interaction Hamiltonian (5.7), as well as the variables of the field Hamiltonian, may now be regarded as classical variables. The classical limit of the BSQ formalism gives us, therefore, a self-consistent classical radiation theory for a number of n-level atoms interacting with the electromagnetic field. We refer to this theory as SCTIV. Comparison of SCTIV with SCTIII shows that the equations of motion for the dynamical variables (GI,67, and the field variables) are identical in both theories. It might therefore appear, at first glance, that we have derived the equivalent of SCT I11 (and also SCT 11, for which SCTIII constitutes a Hamiltonian formalism) by means of the above limiting procedure, provided, of course, that the conditions for this procedure are satisfied by SCT 111. Further inspection reveals, however, that SCT IV is more general than SCT 111, the reason being the specification of initial conditions. The initial conditions in SCT I11 are specified by identification of the initial values of the variables with the corresponding initial quantum-mechanical expectation values. In SCT IV, however, the initial values are automatically determined by the limiting procedure in a manner to be described below, since passage to the classical limit, which converts q-numbers into c-numbers, does not allow an arbitrary prescription of initial values. These must come from the quantum-mechanical specification of initial values, that is, from the quantum state of the system. Now, description of a system by means of quantum states is statistical. It is important to note that some statistical properties do not disappear, generally, even in the classical limit. Consider, for instance, the energy state (rl . . . r,,). According to the previous discussion, the system described by this state may be treated classically (at least initially, if it is interacting with other systems, and for all time if it is free) provided

r,;.>l,

t = 1 , . . . n,

(5.24)

a condition we assume. The relationships

. . . r,,lai(rl. . . r,)= 0 , t ( I I . . . rnlaiailrl.. . r,)= r,, (rl

(5.25)

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441

express the fact that the average of 6 is zero, and the average of (ii(' is ri. The description must be interpreted as being a (classical) statistical description of 6 ; in other words, ii must be regarded as a random variable. Thus, in the classical limit, the quantum-mechanical variables become classical random variables, with their initial values specified by probability distributions - or some other method of statistical description - determined by the quantum state which describes the system (the initial quantum state, if the Schrodinger picture is used). The connection between the statistical description and the quantum states will be discussed later. Presently, we merely note that this is not the conventional SCT approach. As mentioned previously, in both SCTII and SCT 111, the classical variables are equated to the respective quantummechanical expectation values. In those cases where the quantum state is reasonably deterministic (that is, the statistical spread of the dipole moment is relatively negligible) SCT I1 and SCT I11 are essentially equivalent to SCTIV, if the classical-limit conditions are met. However, in those cases where the quantum state is not sufficiently deterministic, SCT I1 and SCT 111- as pointed out in 4 4 - cannot be used, while SCT IV remains applicable. In order to complete the formulation of SCTIV, we must set up a method of prescribing the initial classicaI variables iii(0) and 6T(O) (from which all other initial variables can be constructed) from information furnished by the (initial) quantum state. A random variable may be described completely by a probability distribution; it may also be described completely by an appropriate infinite set of moments. The two methods of description are related, of course (KATZ[1967]). Our method of description will consist of specifying a finite number of moments, which makes the description somewhat incomplete. For sufficiently large values of the (ni)'s, however, the moments that can be specified are sufficient to make this limitation insignificant. The variables Gj(O) and ZT(0) are identical with /$(O) and Ay(O), and if the collection of atoms is free (uncoupled to the electromagnetic field), the time argument can be omitted in the reduced variables, since they are constant. For simplicity of notation, we consider such a free collection. Let it be described quantum mechanically by the state (9). A statistical description of the variables Aj is furnished by the specification of the moments (5.26)

for all integral vi and wi, where the average is understood to be taken

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SEMICLASSICAL RADIATION THEORY

[VII, 0 5

over an ensemble of atomic collections (that is, an ensemble of N-atom systems) associated with the statistical description of the collection. Such an ensemble is also involved in the quantum-mechanical description of the collection of atoms, or in the interpretation of the quantum state of the system. It is most reasonable, therefore, to set each moment equal to the corresponding quantum-mechanical moment with respect to the state under consideration. In general, there are many “corresponding” quantum-mechanical moments, since all ordering arrangements of the classical variables constitute the same moment, while different ordering arrangements of the quantum-mechanical variables may yield different values. In the present instance of the classical limit, however, we are dealing with a situation in which the commutators are considered relatively negligible, so that all ordering arrangements of the quantummechanical variables yield approximately equivalent results. Implicit in this approximation is the assumption u j + wj<<(nj),

j = 1,.. . n,

(5.27)

since, otherwise, the contribution of the commutators may become too large, and violate the above consideration. We limit the order of the moments by this inequality (thereby limiting the number of moments) and prescribe the moments by

(5.28) where 0 indicates a particular ordering arrangement of the operators in the curly brackets that is chosen for convenience of evaluation. This is the SCTIV specification of the initial values of the random variables 6, and 67.

For purposes of physical insight, it is useful to express a statistical description not only in terms of moments but also in terms of probability distributions. It is, therefore, worthwhile to choose a probability distribution consistent with the above moments. This probability distribution will not be unique, of course, on account of the limited number of moments, but one may pick distributions in a way most suitable to a lack of knowledge of higher moments (KATZ [1967]). It is instructive to illustrate the conversion, in the classical limit, from a quantum-state description to a random-variable description for both the energy states and the coherent states. For an energy state, we can write,

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443

according to the above prescription

(nmii:.)

'I"

=(rl.. .r " p =

n

%sw.o,

/ I I . . . r")

(5.29)

rVw,u,,

I

where the ordering in the operators in terms of nl's has been chosen for ease of evaluation, and use has been made of (5.11). For a coherent state, we have

where normal ordering of the operators was chosen for convenience of evaluation, and use has been made of (5.14) and the inequality (5.27). An application of these results to the dipole moment yields, for an energy state, ( d y ( t ) ) a v =0,

rn

(5.31)

= 1,2,

(di;3'(t>>av = i(rj - TI),

(5.32)

and for a coherent state, (J!,l)(t))av

where

= ~ I c ~ ccos i I [(wj - wz)t+

@,,I

(5.33)

(d!;)(t))av= ~lcjczlsin [(wj - wi)t+ 0jiI

(5.34)

(hl:)(t)>av=iN(I~j1~Ict12),

(5.35)

@,,is defined by c:cl

= Ic,~,le'~J~.

Simple probability distributions for the random variables 6, and 6: can be found that are consistent with the above moments. For an energy state

444

[VII, D 5

SEMICLASSICAL W I A T I O N THEORY

. r,), the probability distribution for Aj may be given by

)II..

Aj = r;/’ei’>,

(5.36)

where 6, has a uniform probability distribution P ( e j )between 0 and 27r,

(5.37)

p(eJ = ( 2 7 ~ - l ,

and all the Ai’s are independent random variables, the joint probability, P(&, 62,. . 0,) being

.

p(el, e, . . . en) = p(e,)p(e2) . . . p(e,). For a coherent state the probability distribution for Aj = N1”cieie,

Aj may

(5.38) be given by

(5.39)

where 8 has a uniform probability distribution between 0 and 27~,

p(e)=(27r)-1,

(5.40)

but in contrast to the case of the energy state, 8 is the same for all Aj’s. The A,’s are thus dependent random variables; if the phase is specified for any Aj, it is determined for all Aj’s. From the above distributions, we can obtain distributions for all dynamical variables. It is easy to see why (d:;))av and (d$f))av vanish for energy states and do not vanish for coherent states. The phase of the dipole moment (at any one time) is uniformly distributed over the ensemble of N-atom systems associated with the statistical description of the energy state, while it is identical for all members of the ensemble associated with the coherent state. If we take rj = NIcj(’,

(5.41)

the amplitude of the oscillating components of the dipole moment associated with any one frequency has the same magnitude in both ensembles. It is seen that SCT I 1 and SCT I11 are restricted forms of a classical limit theory that are applicable to those quantum-mechanical descriptions that become entirely deterministic in the classical limit, but inapplicable to those quantum-mechanical descriptions that become statistical (in the sense of non-deterministic) in the classical limit. The coherent states are an example of the former, and the energy states are an example of the latter. In the discussion of the classical limit of the BSQ formalism, we began with a quantum-mechanical description of both atoms and field, and

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445

examined the conditions under which the description of the atoms becomes essentially classical. What can be said about the field under the same conditions? The solution of the equations of motion for the fieldMaxwell’s equations - indicates that classical sources produce a classical field, given by the inhomogeneous part of the solution, and containing source variables only. Thus, in a quantum-mechanical description of the field, only the homogeneous solution can remain quantum-mechanical when the sources become classical. If the homogeneous solution contains a part that is due to external sources, this part is prescribed, and is, therefore, also classical, in accordance with our analysis of prescribed behavior in the discussion of SCTI. The only remaining part of the solution is the zero-point field. We have seen in the example of 0 3 that in order to have a meaningful radiation theory involving the interaction between the field and classical systems, we must assume that the classical systems do not “see” the zero-point field, since, to such systems, the zero-point oscillation appears as oscillation which can do work. Thus, if our interest lies in the atoms and in the field with which they interact or which they generate, then a classical description of the atoms validates a classical description of the field. 5.3. ADDITIONAL REMARKS

It should be recognized that the condition for the validity of SCTIV, namely, that all the (nj)’s be sufficiently large, involves quantities that change with time, since the (nj)’s are time dependent when the atoms interact with the field. It may, therefore, happen that the condition will be satisfied at certain times and not satisfied at other times, for a given system of atoms and field. Such cases are well known, and have been investigated by a combination of quantum theory and SCT, the former being used during the time when the latter is inapplicable (SENITZKY [1972]). On the other hand, if the time during which SCT is invalid is sufficiently short compared to the time during which the system undergoes a significant change according to quantum theory, then one may reasonably expect SCT to be a good approximation throughout. The time during which the system undergoes a significant change depends on the state of the system, of course. If the atoms are in the ground state, that is (al)= N, (n,)= 0 , its 1, and the field is in the ground state, then the system undergoes no change at all. This is obviously a trivial case, and its method of description is inconsequential. One could

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claim, in this instance, that the system can be described classically, for N >> 1, since all levels other than the first may be considered irrelevant, and the “relevant spectrum” consists of one level for which the requirements for a classical description are formally met. Consider, now, the case in which one of the higher atomic levels is populated, the lower ones are empty, and the field is again in the ground state. In this case, the lower levels are relevant, since they participate in spontaneous emission (assuming that the transitions are allowed), and the requirement for the validity of SCTIV is not met. If, however, with an upper atomic level occupied and the lower ones empty, the field is not in the ground state, and is sufficiently strong so that its action on the atoms will fill up the lower levels much faster than spontaneous emission, then the time during which SCT is invalid is sufficiently short to be ignored. This example also serves to illustrate the fact that the “relevant spectrum” used to define the classical atomic model depends not only on the real atom but also on the interaction under consideration, and must be chosen accordingly. Since SCT I1 and SCT 111 are special cases of SCT IV, the conditions for the validity of SCT IV also apply to SCT I1 and SCT III. However, in addition to the requirement that all the (n,)’s be large compared to unity, the requirement that the collection of atoms under consideration be in a fully symmetric state (in a first quantization formalism), implicit in the fact that SCTIV was derived from a BSQ formalism, must be stated explicitly as a requirement for the validity of SCTII and SCTIII. The physical significance of the symmetrization requirement may be understood, intuitively, as a requirement that all the atoms behave similarly (SENITZKY [1974, 19771). One may, of course, divide a number of atoms into separate collections, and treat each collection as bosons of a different kind. For instance, in the semiclassical laser analysis by LAMB[1964], a number of atoms distributed among two levels is divided into two collections, each one of which consists of atoms that are in the same one-atom energy state. SCTII and SCTIII must also satisfy the additional requirements - which is responsible for the fact that they are special cases of SCT IV - that the quantum-mechanical description of the dipole moment be deterministic (within the limits of quantum-mechanics), that is, that the quantum-mechanical dispersion in the dipole moment must be relatively small. This requirement does not apply to SCT IV. Needless to say, the validity conditions do not permit the application of SCT 11-IV to a single atom, an application that has been suggested in the case of

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REFERENCES

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SCTIII (JAYNES [1973]) as part of a proposal to make it a fundamental microscopic theory. The above discussion concerning the requirement for the validity of SCT I1 and SCT 111 needs an additional comment for completeness. Logically, it has been shown only that these “requirements” are sufficient conditions, from a quantum-mechanical viewpoint. In other words, we found conditions under which the theories are valid. Are these, however, necessary conditions? In order to answer this question, we consider SCTIII only, since this may be taken to be the formal dynamical theory that yields SCT 11. Let us begin with SCT 111 in its Hamiltonian form as a classical theory for describing atoms interacting with the electromagnetic field, and convert it into a quantum-mechanical theory. Using the conventional methods of passage from a classical Hamiltonian description to a quantum-mechanical Hamiltonian description, we obtain precisely the present BSQ formalism. One sees that this formalism is the quantummechanical version of SCTIII. Thus, there can exist no other quantummechanical formalism the classical version of which is SCT 111. In other words, if SCTIII is to be derived as the classical limit of a quantummechanical description, this description must be that of the BSQ formalism. The quantum-mechanical requirements for the validity of SCT 111 are, therefore, necessary and sufficient. Strictly speaking SCT 111 and SCT IV are not semiclassical theories at all, but completely classical theories. However, the atomic model in these theories is of a schematic nature, and is derived as the classical limit of a quantum-mechanical description. (Note that, in a sense, this is the reverse of the usual procedure followed in obtaining a quantum-mechanical model.) This classical model is, therefore, not divorced from its quantummechanical origin, and may be considered to be only “semiclassical”. References ARECCHI, F. T., E. COURTENS, R. GILMORE and H. THOMAS, 1972, Phys. Rev. A6, 2211. ARECCHI, F. T., R. GILMORE and D. M. KIM, 1973, Lett. Nuovo Cim. 6, 219. BOYER,J. H., 1969, Phys. Rev. 182, 1318, 1374. BOYER,J. H., 1970, Phys. Rev. D1, 1526, 2257. CRISP,M. D. and E. T. JAYNES, 1969, Phys. Rev. 179, 1253. EBERLY,J . H., 1976, in: Physics of Quantum Electronics, Vol. IV, eds. S. Jacobs, M. Sargeant and M. 0. Scully (Addison-Wesley, Reading, Mass.) p. 421. GILMORE, R., 1972, Ann. Phys. (N.Y.) 74, 391. GILMORE, R., C. M. BOWDEN and L. M. NARDUCCI,1975, Phys. Rev. A 12, 1019.

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GLAUBER, R. J., 1963, Phys. Rev. Letters 10, 84. GLAUBER, R. J., 1964, in: Quantum Optics and Electronics, eds. C. de Witt, A. Blandin and C. Cohen-Tannoudji (Gordon and Breach, New York) p. 65. HAKEN,H., 1970, Handbuch der Physik, Vol. XXV/2c (Springer-Verlag, Berlin). HEITLER,W., 1954, The Quantum Theory of Radiation (3rd ed., Oxford University Press, London) ch. I. JAYNS, E. T., 1973, in: Coherence and Quantum Optics, eds. L. Mandel and E. Wolf (Plenum Press, New York) p. 35. KATZ,A., 1967, Principles of Statistical Mechanics (W. H. Freeman and Co., San Francisco and London) ch. 4. KELLEY,P. L. and W. H. KLEINER,1964, Phys. Rev. 136, A 316. LAMBJr., W. E., 1964, Phys. Rev. 134, A 1429. LAMBJr., W. E. and M. 0. SCULLY,1969, in: Polarization: Matiere et Rayonnement (Presses Universitaires de France, Paris) p. 363. MANDEL,L., 1958, Proc. Phys. SOC.(London) 72, 1037. MANDEL,L., 1959, Proc. Phys. SOC.(London) 74, 233. MANDEL,L., 1976, in: Progress in Optics, Vol. XIII, ed. E. Wolf (North-Holland Publishing Co., Amsterdam) p. 27. MANDEL,L., E. C. G. SUDARSHAN and E. WOLF,1964, Proc. Phys. SOC.(London) 84,435. MANDEL,L. and E. WOLF, 1965, Rev. Mod. Phys. 37, 231. MANDEL,L. and E. WOLF, 1966, Phys. Rev. 149, 1033. MARSHALL,J. W., 1963, Proc. Phys. SOC.(London) A276, 475. MARSHALL, J. W., 1965, I1 Nuovo Cim. 38, 206. MESSIAH,Albert, 1961, Quantum Mechanics (North-Holland Publishing Co., Amsterdam) ch. XII, Ex. 4. MILONNI,P. W., 1976, Phys. Reports 25, 1. ScHiw, L. I., 1949, Quantum Mechanics (McGraw-Hill Book Co., New York) p. 240, p. 190. SENITZKY, I. R., 1958, Phys. Rev. 111, 3. SENITZKY, I. R., 1965, Phys. Rev. Letters 15, 233. SENITZKY, I. R., 1966, Phys. Rev. Letters 16, 619. SENITZKY, I. R., 1967, Phys. Rev. 155, 1387. SENITZKY, I. R., 1968a, Phys. Rev. Letters 20, 1062. SENITZKY, I. R., 1968b, Phys. Rev. 174, 1588. SENITZKY, I. R., 1972, Phys. Rev. A 6, 1175. SENITZKY,I. R., 1974, Phys. Rev. A 10, 1868. SENITZKY, I. R., 1977, Phys. Rev. A 15, 284. STROUDJr., C. R. and E. T. JAYNES, 1970, Phys. Rev. A 1, 106.