Stationarity and the random phase condition in the statistical theory of the electromagnetic field

ANNALS

OF PHYSICS:

30,

Stationarity Statistical

127-137

(1964)

and the Random Phase Condition in the Theory of the Electromagnetic Field* Y. KANO~

Department

of Physics

and

Astronomy,

University

of

Rochester,

Rochester,

New

York

The quantum mechanical analogue of the classical definition of stationarity of an electromagnetic radiation field is discussed. It is shown to imply that for an ensemble radiation field corresponding to a single momentum-spin state the phase space distribution function of the complex eigenvalues of the photon annihilation operators does not depend on the phases of the eigenvalues, i.e., the stationarity condition implies a uniform distribution of t’he phase angle of the eigenvalues. However, for a radiation field with a finite or countable infinite number of states, the stationarity condition is not a sufficient condition for the phase independence of the distribution function; it is shown that, in addition, the commutability of the density matrix and the number operator for each mode is required. The physical meaning of this commutability is investigated and it is shown to imply homogeneity of the radiation field. INTRODUCTION

Since the development of lasers, quantum theoretical descriptions of optical phenomena have become of increasing importance. The electromagnetic field can be considered as an ensemble of photons, so that the density matrix plays naturally an important role for the quantum mechanical analysis of such phenomena. Using the eigenstates of the photon annihilation operator as basis, Sudarshan (1,2) has introduced a representation of the density matrix which is particularly suitable for describing coherence properties of light.’ With the aid of this representation equivalence between the quantum theoretical and the semi-classical descriptions of statistical light beams has been demonstrated (4). In some investigations of interference and coherence properties of stationary light beams, it has been tacitly assumed, without any justification, that the phase space distribution function of the field is independent of the phases of the eigenvalues of the photon annihilation operator. It is the main purpose of this paper to investigate the stationarity condition from a quantum-theoretical standpoint and to analyze its implication on the question of phase-independence of the distribution function. * This research was supported by the Air t Present address: Department of Physics, 13. c. 1 Some special cases of this representation

Force Office University have 127

been

of Scientific Research. of British Columbia, noted

by Glauber

(3).

S’ancouver,

128

KANO

In the first section of this paper the correspondence between the quantum mechanical and the semi-classical description of stationarity of electromagnetic radiation is discussed. In the second section it is shown that for a stationary radiation field of a single momentum-spin state, the phase space distribution function is indeed independent of the phase angles of the complex eigenvalues of the photon annihilation operator. However, it is found that for a stationary radiation field consisting of a finite or countable infinite momentum-spin states, the distribution function is generally phase dependent, unless the density matrix and the number operator for each mode commute. This latter condition implies that the phase angles of the eigenvalues for each mode are distributed uniformly. The physical meaning of this commutability is investigated in Section III. In particular it is shown that under the condition of this commutability, the ensemble average of the photon number in a local volume depends neither on time nor on position. Accordingly this condition may be interpreted as the “homogeneity” condition of the radiation field. I. THE

CLASSICAL

AND

THE

QUANTUM CONDITION

MECHANICAL

STATIONARITY

Let us consider n time-dependent operators in a finite or countable infinite dimensional space. The time development of these operators can be expressed in the form o,(t)

=

e-Oj(())ptlh,

(j = 1, 2, *** ) 72))

(1.1)

where H is the Hamiltonian describing the total energy of the system. The quantum mechanical condition for a stationary ensemble is expressed by the operator equation

[H, PI = 0, where p is the density matrix characterizing density matrix is a constant of motion, i.e.

(1.2)

the ensemble. In other words

dp/dt = 0.

the (1.3)

As is well known, the ensemble average of any operator can be calculated by taking the trace of the product of the density matrix and the operator. We denote the ensemble average of the product of the above n operators by b :

Assuming the stationarity Thus Eq. (1.4) becomes

condition

(1.2))

we can interchange

p and eiHtl”.

STATIONARITY

AND

THE

RANDOM

PHASE

129

CONDITION

since any cyclic interchange of the operators does not affect the value of trace. Next we insert the unit operator iut,/r --iHtllh = 1 e .e (1.5) between pression

every pair of the exponential

factors,

. . .

= Tr(p&(O)

e-iHtj’heiHtk’h and obtain

e

iH(tn-tl)/h

the ex-

iHWl)/h}

(1.6)

%(O)e-

%(tz - tl> &(t~ - tr) . * . O,(t, - tl)).

The expression (1.6) depends only on the differences of the two time instants, tj - tl , so that it is invariant under the translation of the time and resembles the definition of stationarity used in the classical theory of stochastic processes (5). In the preceding discussion we did not use the explicit form of the Hamiltonian. Accordingly, even in the presence of an interaction, so long as the density matrix commutes with the Hamiltonian, the ensemble average (1.4) is invariant under the translation of the time. Let us next consider the converse. We assume the invariance of the ensemble average of the product of these operators under time translation and will show that this implies that equation (1.2) holds. Let us suppose that tl , b, . * . are increased by T. Then Eq. (1.4) becomes 0 = Tr(p(t)eiH(t’+‘)/L~l(O)e-iH(tl+r)/heiH(t2+?)/kOZ(O)e-iH(t2+r)lh . . . eia(t,+r)lLOn(0)e--;H(t.+r)lh} = Tr(eiHr’hp(t)eiHr’~Ol(t1)02(t2) The first three factors -iHr/hp(

e 011

. . . &(t,)].

in the bracket

can be written

t)e”Hdh

substituting

=

P(t)

+

(1-J)

;

b(t)

+

;

ii(t)

+

as follows: . . .

+

pyr)

+

. . .

.

from this equation into (1.7), we obtain the equation b = Tr(p(t>ol(t,)e,(tz>o,O

. - - O,(L)

+ fi Trlp(t>ol(tl>oz(tz>c730

}

- . . %(t,)

1

(1.8) + .. .

+ 2 Tr{ p’“‘(t>o,(t,>o,(t,>o,(t,> + *se.

. . . az(tn> 1

130

KANO

Now in view of our assumption about the invariance of CI with respect to time translation all the derivatives with respect to T vanish: dd -=g= dr

. . . =!!I$

. ..=

0.

Differentiating both sides of (1.8) with respect to r and then setting T = 0 we obtain the equation

0

dd = Tr(b(t)&(tJ&(tz) XT 74

Differentiating finds that

..a 0,(k)}

= 0.

(1.8) twice with respect to r and setting T = 0 afterwards, one

On repeating this procedure, we finally arrive at the equation

d% = drm 7=0 (4

Tr~p’“‘(t>o,(t,>o,(t,)

. . . 8,(&J}

= 0,

(m = 1,2, . . *).

(1.9)

In the above equations the operators, &(tl), &(tz), . * . , are arbitrary Heisenberg operators in the Hilbert space. Therefore, we obtain the simultaneous differential equations : p(t) = p(t) = p(t) zzz. . . = /p(t)

= . . . = 0.

The solution of the above equations is a time-independent obviously commutes with the Hamiltonian. II.

THE

PHASE

INDEPENDENCE

OF

Let us introduce the eignfunctions operators for a single mode* radiation alz>

= zlz>,

THE

<++

operator p(O) which

DISTRIBUTION

of the photon field (6) :

(1.10)

FUNCTION

annihilation

and creation

= z*
These eigenfunctions can be expressed as linear combinations the photon number operator (7), ata,

(2-l)

of eigenvectors of

1 wi8 > = 1 x > = exp (-;14r)&~In~~ (2.2)

< reie 1 = < 2 1 = exp (+)g&~nl,~ 2 By a single mode single momentum-spin

radiation state k,

field s.

we shall

understand

from

now

on a radiation

field

of

STATIONARITY

AND

THE

RANDOM

PHASE

CONDITION

131

where 1n > is an eigenstate of the number operator N, (N = ata, N 1n > = n 1n >). These states are all normalized but not orthogonal and form an overcomplete set. Making use of the overcompleteness of these states one can express every density matrix, p = 2 2 p(n,n’) n=O n’=O

in the “diagonal”

In>
(2.3)

form (I, 2) P=

s

d2zr#B(z)

12 > < 2 1 ,

where d’z is defined by d2z = r dr de. b(z) may be interpreted as a phase space distribution function (8) which, however, is not necessarily nonnegative. It may be expressed as a symbolic function (1, .2),

We wish to determine what restriction is imposed on the distribution function (2.5) if we assume that the ensemble of the pure radiation fields is stationary. Now the Hamiltonian of a pure radiation field consisting of a single mode is given by H = fiwa+a.

(2.6)

If the radiation field is stationary, then the density matrix describing the ensemble commutes with the Hamiltonian. Using Eqs. (2.3) and (2.6) we can calculate the commutator [H, p], W, PI = fiu %gon$dn,

n’>l n>.

(2.7)

The stationarity condition requires that the commutator (2.7) be a zero operator, i.e., all the matrix elements of the operator must vanish. Let us calculate the matrix element of [H, p] between two eigenstates of the number operator

= &I 2 2 p(n, n’)(n - n’) n=On’=O =

~WPO,

/J>o

-

PL).

In order for the matrix element to be zero p(X, P) must vanish for X z p, so that PO,

CL) =

Poe,

~)4A

*

(2.8)

132

KANO

On substituting from (2.8) into (2.5) we finally obtain the phase space distribution function of a stationary ensemble of radiation fields of a single mode:

(2.9) We see that the distribution function, +(x), does not depend on the phase angle 13(z = reie). Conversely, assuming that the phase space distribution function 4(z) is independent of the phase angle 8, we can easily show that [II, p] = 0 i.e., the ensemble is stationary. Therefore, we conclude that for a single mode radiation field whose Hamiltonian is given by (2.6), the stationarity condition, [H, p] = 0, is the necessary and su#Gent condition for the phase space distribution function C#J(Z) to be independent of the phase angle 0. The above conclusion, however, has no immediate generalization to a multimode radiation field. The Hamiltonian for a multimode radiation field is given by

where the summation is taken over all the momenta fik and spins s of the photons. If we denote the sequence of the eigenvalues of the number operator corresponding to each state (k, s) by {n}, we can express the density matrix in the form P = C C44, Ial In’1

b’H

(2.11)

IbWf4l.

As before the density matrix (2.11) can be expressed in terms of the eigenstates of the annihilation operators in the “diagonal form”

P=sd2L4dW I(21 ><(4I,

(2.12)

where {z) denotes the sequence of the complex eigenvalues of the annihilation operator corresponding to each state (k, s), and the phase space distribution function 4( {x} ) is given by (I)

(2.13)

Using (2.10) and (2.11) we can calculate the commutator,

W, PI= gn {ig di4,b’I We see that [H, p] = 0 if ~({nj, dbd,

{n’)) b’l>

,rz

fiWk(nk.e

-

and obtain

n~,Jll~n~~
takes the form, = m(bl,

~n’lh.~~

(2.15)

STATIONARITY

AND

THE

RANDOM

PHASE

CONDITION

133

where E = xk+ fiwknk,* , E’ = xk,s 6wknk+. . If we substitute from (2.15) into (2.13), we obtain the phase space distribution function for a stationary radiation field. However, this distribution function +( f x) ) still depends on { 0)) since Eq. (2.15) does not, imply that nk,8 = nh,* for all k and s. Hence the stationarity condition (1.2) does not imply that the function +( {z) ) is independent of the phases (01. The statement that +( { x) ) is independent of the phases { e} must not be confused with the statement that the distributions of the phases of the eigenvalues are mutually independent. The former statement implies that the distribution function 4((z)) does not depend on the phaseangles, i.e., 4((x}) = 4(( lzl]). In other words the phases 6ks are (for each k, s) distributed uniformly in the interval (0, 2,).” The latter statement implies that the joint, probability distribution function p (( 0) ) of the phases is expressible as a product of probability distributions p&s(&) relating to the individual phases, i.e., p({ 0) ) = -&,S pk,s(@&.s).4 Let us next determine the condition for the distribution function to be independent of the phases, ok,* (for all k, s). We see from (2.13) this will be the case if nk,s

=

for all

&

k

and

s.

(2.16)

This condition (2.16) implies that the density matrix (2.11) is diagonalized in the of the number operators, Nk,a = atk ,&,a , so that p andNk+ commute with each other; eigen&ateS

k’,

Nk,81

=

0

for all

k

and

S.

(2.17)

Conversely, if the condition (2.17) holds, then p and Nk,s can be diagonalized simultaneously. Using the representation which diagonalizes p and Nk,l simultaneously, we can form the density matrix (2.11) and then the distribution function (2.13). However, in this representation p((n}, {n’}) in (2.13) is diagonal din), In’)> =

p0({4,

~4P(4,w)

.

(2.18)

Consequently the distribution function 4( (2) ) does not depend on (0). Therefore, we conclude that the condition (2.17) is the necessaryand sujicient condition for 4((z) ) to be independent of { t9}, i.e., for the phasesok.a of each mode to be distributed uniformly in the interval (0, 27r). The condition (2.17) necessarily implies that the field is stationary, i.e., that [H, p] = 0, since the Hamiltonian (2.10) is a linear combination of the number operators, Nk,(l . An important result which follows from this discussionis that if an ensemble of a One then often speaks of the “Random Phase Assumption” or “Random Phase Distribution.” 4 This statement must, of course, be interpreted with caution since in the present representation 0( { z}) is a symbolic function.

134

KANO

radiation fields for which the Hamiltonian is of the form (2.10) satisfies the condition (2.17), then the expectation value of any operator, 0, b = TrjpO}

(2.19)

always vanishes if the operator 0 contains unpaired annihilation operators in the product. The statimarity condition is not sufkGnt,6 ditim (2.17) is necessary and su$icient to obtain the above result. III.

THE

PHYSICAL

MEANING

OF

THE

CONDITION,

or creation but the cm-

[p, Nk,J

In this section we shall consider the physical meaning of the condition To this end let us introduce the photon number operator, N, N = i / 8x( @j-‘(x) d,+j+‘(x)

- do&‘(x)

= 0

(2.17).

.4$+‘(x) ] (3.1)

=z

aLsak,s ,

where the summation is taken over the componentsj = x, y, z, and the integration is carried out over the whole normalization volume L3. c#$+‘(z) and 4j-j (x) are defined by the expression

and (3.3) where $‘(k) denotes the jth component of the real unit polarization vector, and k.x=kfio-k.x(Ico= jk[ = wk/ c, x0 = ct). If in Eq. (3.1) the integration is taken only over a local volume Z”whose size is much smaller than that of the whole volume L3, but whose linear dimensions are much greater than the average wavelength X(L >> I >> X), then the integral (3.1), extended over the volume Z3,may 6 Recently this statement has been criticized by L. Mandel (Phys. Letters 10, 166 (1964)). He claims that the stationarity condition is suficient for the statistical expectation value of any product of unpaired annihilation and creation operators to vanish. However, this is not so, as can readily be seen from the following simple example: Consider the statistical expectation

value

atk,dxk~.~~ ukl.8

. Straightforward

t ak.aak’,.‘ak”.s” when

the

field

is stationary

and

calculation

shows

that

f 0,

when

wk - ok’ - ‘ok” = 0

(Ok = c Ik 1).

STATIONARITY

AND

THE

RANDOM

PHASE

135

CONDITION

be interpreted as the photon number operator in the local volume. Denoting integral by n(x, t) we have

this

n(x, t) = i I z3d3z{&‘(2) .do$jf’(z) - a,&‘(x) Y$;+‘(x>]

x

e-i(k-k’)yi(k-k’)

.I/:!

sin{ (Ici - ki’)/2’1] II i=2.1/,2 (ki - i&9/2* 1

.

Using the density matrix given by (2.12) and (2.13), we can calculate the ensemble average of the photon number in the local volume at (z, t) , 4x, t> = Wdx,

t> I

(3.5)

Equation (3.5) is a general formula for the ensemble average of the photon number in the local volume. It is valid for an electromagnetic field, whether it is stationary or nonstationary. If the field is stationary (cf. (1.2)), then Eq. (3.5) gives ~2{+#d{4 x

> k$ g, ~j8’(k)E::8”(k’)Xk*,,Bzkf,sf (Ok’&l)

(3.6)

e-i(k-k’).re-i(k-k’).l/2

i=z,u,r

(ki - lC,‘>/a. 1

’

i.e., for a stationary field the ensemble average (3.6) does not depend on time t, but still depends on the space point x. If we further impose the condition (2.17), then (3.6) simplifies to (3.7)

136

KANO

Equation (3.7) shows that in this case, the ensemble average of the photon number in a local volume depend neither on time nor on position. For this reason we may call (2.17) the “homogeneity” condition. Blackbody radiation is a typical example satisfying the homogeneity condition. For in thiscase the density matrix, p, is given by (9) e--XIRT ’ = Tr(e-HIKT] where K is the Boltzmann Hamiltonian given by

(3.3)

’

constant, T is the absolute temperature,

and H is the

H = g fiw&k,s .

(3.9)

From (3.8) and (3.9) the condition (2.17) readily follows. Let us next consider the second order correlation function, We introduce operator, 0(x, t) whose classical analogue is a field “disturbance”6 0(x, t) = g {f(k, s)ok,8e+w’f+ik’x + f*(k, s)ut,8eiokt--ik’x). Let us define the positive and negative frequency component way : t) = xf(k,

s)ok,.&?+kt+ik’x,

(3(-)(x, t) = gf*(k,

s)ot,8eiokt--ik.x.

0(+)(X,

an

(3.10)

of o in the usual

(3.11)

The second order correlation function associated with the operator at two different space-time points, (x1 , tl) and (x2 , s$), may be defined by the following ensemble average (cf. Glauber (IO) ) : 0(-)(x, , tl) 0(+)(x,,

tz) = Tr {PO(-)(X~ , 4) 8’+‘(xZ,

(3.12)

tz)).

Under assumption of stationarity (condition (1.2) ), we find, on using the “diagonal” representation (2.12) and (2.13) for p, that

(3.13)

fj(k,

0 If we identifyO(x, s), where

t) with

the electric

field

f,(k, d = i

operator

2rhc4 L’l

Ei(x,

t) (j = 1,2,3),

r!‘)(k),

then

f(k,

s) =

(j = 1, 2,3).

STATIONARITY

AND THE RANDOM PHASE CONDITION

This equation shows that the correlation function of a stationary

137

field depends on

tl , and t2 only through t2 - t1 ; however it depends on the two space points x1 and

x2 separately. If the density matrix further satisfies the homogeneity Eq. (3.13) simplifies as follows:

condition

(2.17), then

(3.14) x

1 zk,s

I2 e-

iok(t2-tt)eik.(x2-xI)

Equation (3.14) depends only on the difference between the two time instants and the difference between the two space points. As mentioned above, blackbody radiation satisfies the homogeneity condition. Hence if we calculate the electric correlation tensor, for instance, it will depend on the time separation tl - t2 and the space separation z1 - x2 only, i.e., it will be of the form: Tr (&$-)(x1, Such correlation (4).

t1)Ej+‘(x2,

functions to blackbody

t2)} = Eii(x2 - x1, t2 -

radiation

tl).

(3.15)

have been recently discussed

ACKNOWLEDGMENT

The author wishes to express his sincere thanks to Professor Emil Wolf, who suggested this problem, for continued encouragement. He is also indebted to Professor Susumu Okubo and Dr. C. L. Mehta for helpful discussions. RECEIVED:

March 20, 1964 REFERENCES

E. C. G. SUDARSHAN, Phys. Rev. Letters 10 (1963) 277. E. C. G. SUDARSHAN, Proc. Symp. Optical Masers, New York, New York, p. 45. Polytechnic Inst. Press, Brooklyn, and Wiley, New York, 1963. 3. R. J. GLAUBER, Phys. Rev. Letters 10 (1963) 84. 4. C. L. MEHTA AND E. WOLF, Phys. Rev., in press. 5. M. BORN AND E. WOLF, “Principles of Optics.” Pergamon Press, London, New York, 1959. 6. S. SCHWEBER, J. Math. Phys. 3, 831 (1962). 7. J. R. KLAUDER, Ann. Phys. (N.Y.) 11, 123 (1960). 8. E. WIGNER, Phys. Rev. 40, 749 (1932); see also J. E. MOYAL, Proc. Cambridge Phil. Sot. 16, 99 (1949). 9. A. MESSIAH, “Quantum Mechanics,” Vol. I. North-Holland, Amsterdam, and Wiley, New York, 1961. 10. R. J. GLAUBER, Phys. Rev. 130 (1963) 2529. 1. 8.