Corrections to the theory of photocounting

Corrections to the theory of photocounting

Volume 144, number 8,9 PHYSICS LETTERS A 19 March 1990 CORRECTIONS TO THE THEORY OF PHOTOCOUNTING V.1. TATARSKII Institute of AtmosphericPhysics, M...

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Volume 144, number 8,9

PHYSICS LETTERS A

19 March 1990

CORRECTIONS TO THE THEORY OF PHOTOCOUNTING V.1. TATARSKII Institute of AtmosphericPhysics, Moscow 109017, USSR Received 10 December 1989; accepted for publication 10 January 1990 Communicated by V.M. Agranovich

A new mode expansion is introduced which allows us to investigate all possible states with the same electromagnetic field spectrum as the states of a single-mode system. This approach is applied to the well-known Glauber theory of photocounting statistics and to its modification proposed recently. As was shown before, the standard theory leads to causality violation. A new parameter a, which has zero value in the Glauber theory, arises in the modified theory, for which causality is restored. The 2 ~ 1, p1 being the mean value ofthe photon number for the considered state ofthe field. Glauber theory is valid ifa ~ 1 and iia The modified theory is also restricted by the condition n> a 2/ (1 — a2) which appears if the theory is applied to squeezed states. The new theory reduces to Glauber’s theory under appropriate circumstances.

E~—~(t)—

1. Introduction

r,

x

The well-known quantum theory ofphotocounting

J



2i~

d3k k 112e 1a(k) exp [i(k r—ckt) ]â,~(k)

.

(1.3)

statistics was developed about 25 years ago. It is discussed in the literature (see, for example, refs. [l 4]), the most detailed development given 4his theoryofit the being probability in ref. [5]. According to of recording the arrival of n photons by a small ideal [1] detector in the time interval (0, T) is

Here ~a(k) and a~ (k) are the annihilation and the creation operators corresponding photons with wave vector k and polarization alongtothe direction of the unit vectors ea (k) (a = 1, 2). They obey the commutation rules

P(n)= ~Tr{~o:~exp(—4):}.

The vectors ea(k) satisfy the relations

(1.1)

[âa(k),âp~(k’)]=öapô(kk’).

kea(k)=0 Here p is the density matnx of the electromagnetic field state; A: denotes the operation of normal ordering of the operator A;

(1.4)

ea(k)ep(k)=öa

eja(k)eja(k)=5y—kjkj/k2.

(1.5)

The operator ~ is found from the relation ~ = (E~~ )t The operator of the total electric

~o=vJniE(r,t)niEj()(r,t)dt

(1.2)

o

(1,1= 1, 2, 3); v is the detector sensitivity; the unit vector ii specifies the direction of polarization of the field incident on the detector. The positive-frequency part of the field operator in the Heisenberg picture is given by

fieldis

~

.

(1.6)

Let us consider the case when p= I z> is the coherent state of the field: aa(k)Iz>=za(k)Iz>, 1. From (1.1) we then obtain

0375-9601 /90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)

(1.7)

491

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PHYSICS LETTERS A

1
P~ ( T) = (n! ) = (n! ) — ‘N ‘~exp (

Here N= V( r,

x

n, V~(r,

t)

$

=

d3k

v

$



N)

V~(r,

1) =



£~):I z>

is that in the original Glauber theory the term (1.8)

.

t)

V(r,

t)

exp [i(k’ r— ck

E~ appears after discarding the terms ~(E~~~)2and ~(E(~)2 in the expression ~ )2, The ratio of these small terms to —)

the main term E + is characterized by a new small parameter a2, which has zero value in the

dt,

—)

~~jc 2it

,Ji~n, e,a (k)

19 March 1990

Glauber theory but is finite in the theory suggested in ref. [6]. The operator was used in ref. [7] to

A

(1.9)

investigate the states. statistics of photocounting for coherent and Fock It was shown that causality is then restored. But for the Fock states additional

The function V is known as the analytical signal associated with the real signal. N is the mean number of detected photons, To derive (1.1) several assumptions have to be used. One ofthem is the use ofthe rotating wave approximation in the interaction Hamiltonian. it was shown in ref. [6] that this approximation leads to violation of causality in photocounting probabilities, i.e. the probability of photon observation becomes nonzero before the signal reaches the detector. This effect is especially important for short pulses and short observation times. If either of these is of the order of the period ofoscillation, the oscillating terms are not averaged out; the contribution of the neglected terms is rather essential and they should be taken into account. It is also important that normal ordering be used in (1.1). Such forms of operators result quite naturally from the rotating wave approximation. But they appear first in the form

bunching ofphotocountings was obtained. If the field is in the n-photon state I n>, the conditional probability P( n n) for a>0 is greater than for a = 0. The same is true for P( 01 n). But in the case of intermediate values of m the probabilities P( m I n) for a>0 are less than for a = 0. This effect can change the sub-Poissonian statistics into the super-Poissonian one for a2> 2/ (n 1). Section 3 of the present paper is devoted to other essentially nonclassical states of the field, the squeezed states. In section 2 we introduce a new mode representation of the field. This representation makes it rossible to describe the quantum states of the field with a given (though arbitrary) spectrum, which are multimode in the usual sense, as states of a single-mode system. On the other hand there is an opportunity to analyze the role of normal ordering in the photocounting theory because for a single-mode system this operation does not lead to infinities (see section 4).

t)

]Za(k).

(E~~> +E~—~) (E~~ +E~—~)...(E~) +E~—),



2. Description of different quantum states of a

tl>t 2>...>tn.

If we reduce this product to the normal form by means of commutation relations and then discard the terms which do not correspond to the rotating wave approximation, we will see that some of them include infinite coefficients. This step deserves, therefore, a more thorough investigation. Toreplaces restore causality it was£2~by suggested in ref. [6] that one the operator the operator ~i=~v$E2(r,t)dt,

(1.10)

0

where E is given by (1.6). The reason for this step 492

multi-mode system with the same spectrum as the states of a single-mode system As was noted in ref. [8], the coherent state of the field satisfies eq. (1.7) and can be represented in the form 3k IZa(k) I2 Iz> =exp(_ ~ d

$

+$dkZa(k)aa(k))I0>.

(2.1)

It is convenient to introduce the operators (see also ref. [2], ch. 8)

Volume 144, number 8,9

PHYSICS LETTERS A

19 March 1990

or, in more detail Ifl>(fl!)_U2Jd3k1 ~I~I’$d3k~(k)Ôa(k).

...

fd3knFai(ki)...Fa~(kn)

(2.2) xa~1(k1)...a~,,(k~)l0>

The normalization factor z is determined from the condition [â,â~]=l.

It is clear that the states (2.9) are a subset of Fock states of the general form

(2.3) (fl!)h/2d3ki$...Jd3knøai.a~(ki,...,kn)

It follows that

IZI2Jd3kIZa(k)12.

(2.4)

Xã~1(k1)...a~(k~)I0>

It is convenient also to introduce the normalized function Fa (k) by the equations

and correspond to factorization of ck. The mean value ofthe electric field in the state I z>

Za(k)ZFa(k), Jd3kIFa(k)121.

is equal to

(2.5)

It is possible to include the phase factor either in z or F. Using this notation we can write ~

$d3kFa(k)Ôa~(k),

=

$

3kk”2eja(k)

2it ~L~ic d

x exp[i(kr—ckt)]zF~(k) =V

3kF~(k)~a(k).

aJ d

(2.6)

1(r,t)=zv1(r,t). (2.10) It is clear from (2.10) that Fa(k) determines the spectrum of the field. The mean value in the states (2.9) is zero, but it is easy to show that

(2.7)

E~)(r,t)In>—n”2v

Expression (2.1) then takes the form

Iz>=exp(—~IzI2+zâ~)I0> ,

1(r, tfln—1> which is the same as the expression for the coherent state of a single-mode system. But unlike the usual case the operator â~creates a nonmonochromatic mode. It is easy to prove the relations [~a(k), ~] Fa(k), 2]2Ô~Fa(k), [~a(k),(~~)

[~a(k), (d~Y]=fl(d~Y’’Fa(k), [da(k), f( Ô ~)]

f’ (~)F~(k).

(2.8)

By means of (2.8) we obtain from (2.7) ~a(1é)I~>

=exp(—3 IzI2)

~a(k)

exp(zd~)I0>

=exp(—flzI2)zexp(zd~)Fa(k)I0>=za(k)Iz>. This relation shows that (2.1) and (2.7) really represent the coherent state. It is also possible to introduce the states

In> =

~

0>,

a In> = .,/~In

a~In>=~/~Tln+l>

Using this relation we find the mean value

=nlv(r, 1)12. Thus, in this case we obtain the same dependence of the field on coordinates and on time, as for the coherentstate. It is clear that for all states which are created by theoperatorâ~anditspowers,thefieldhasthesame spectrum, but is in different quantum states. For a fixed function Fa(k) we can obtain by use of the op erator a + only a subset ofall possible quantum states. It possible, however, to construct complete systemis of operators d~a’,n=0, 1, 2, awhich includes â~,andusingalltheoperatorsâ~ we can construct ...

any arbitrary quantum state of the field, not only the single-mode states. To construct this system ofoperators we introduce a complete orthonormal system of functions

{ W~ (k) }‘ i.e. it satisfies the conditions

— I>

Jd3kWn(k)W~n(k)r~ônm,

(2.11)

(2.9) 493

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~

PHYSICS LETTERS A

W,, (k) W~(k’ ) = ö(k— k’)

(2.12)

.

Let Sap(k) be any unitary 2 x 2 polarization matrix satisfying the relations (2.13) Say(k)S*py(k) ~5afi, Sya(k)S~p(k)=c5afl Let us introduce the operators

in terms of the operators dna. If we multiply eq. (2.14a) by W~(k), sum over n, and take into account the completeness condition (2.12), we find that n~0

W~(k’)dna

Safl(k’ )a~(k’).

The convolution ofthis equation with S~(k’) (tak-

a~=Jd3k W~(k)S~~(k)d~(k), ~na

19 March 1990

$d~kWn(k)Sa~(k)dp(k).

(2.14a)

ing into account (2.13)) leads to the inverse transformation

(2.l4b)

da(k)S*ap(k) ~ 12=0

Using (2.11), (2.13) one can readily derive the commutation relations

d~(k)Sap(k)

~

(2.17)

Wn(k)~in~p.

n=0 [dna,

d~p]

(2.15)

dnmöap.

The requirement for the operators

a

and

01

to be

identical leads to the equation

aJ

We can use these equations to represent any of the

a

operators in terms of dna, ~

~.

E~~(r,t)

3kF~(k)da(k) d

We must choose such functions for which the equation



W0(k)

and Sia(k) (2.16)

is satisfied. We multiply (2. 16) by the complex conjugate equation to obtain 2S I W0(k) I 1aSI’a = I W0(k) I2~(I~)1~a(1~) We can choose W0(k)= [IF1(k)1

J

~

d~kfke 1p(k)exp[i(k’r—ckt)]

2it xS~~(k)W~(k) —

(2.18)

In (2.18) we are mainly interested in E.01d01=

~(r, t )a. For the function tain from (2.18)

1/2,

2(k)1

~(r, 1) = nd’, (r, 1) we ob-

~f(r, t) d~k~fkn1eap(k) exp[i(k’r—ckt)] 2x xSt~(k)W~(k).



2]

we obtain

.

Wo(k)Sia(k)F~(k)

2+ 1F

—)

~ Ena(F, t)dna,

Eina(r, t)

d3k Wo(k)Sia(k)da(k).

For E

~L~1CJ



Sia(k) =F~(k)(IF 1

2+

I~’212)

—1/2,

S~2=Si~1,S21 =—S’~’2. It is thus possible to find W0(k) and Sap(k). The next step is to construct the set of functions W,, (k), n=1, 2, 3 For this we can use one of the wellknown orthogonalization processes. This procedure is not unique but this is not important. It is thus possible to construct a complete orthogonal system of operators dna, d ~ sufficient for our purposes: that the operators a, a + are included in this system. Next we will try to represent the operators da(k) 494

Using the complex conjugate to (2.16), we find that i

~(r,t)=~ Id~k~n1e~p(k) 2i~ ~ xexp[i(k’r—ckt)]Fa(k)~v(r, t)

(2.19)

.

Here v(r, t) = n•v1(r, 1) and the quantity v,(r, 1) is the analytical signal corresponding to the coherent state Iz> =exp(zd~_z*a) 0> with z= 1 (see (2.10)). Thus, the expansion of the operator E = n.E~ in terms of dna has the form E—(r, t)=v(r, t)d+ fljEjmadma. (2.20) —)

—)

~‘

m,a

:1

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PHYSICS LETTERS A

Here denotes the summation over all the modes, except the mode labeled by (0, 1). For the operator ~ which is determined by (1.10) we obtain ~‘

to the formula for the factorial moments N(N—l)...(N—m+1)

~ n(n—l)...(n—m+l)P(n)

T

n=0

J

~ =~v (v(r, t)d+~(r,t)d~

=

2

dt’.

+~‘ m,a ni(Eimadma+E~ad~a))

(2.21)

Let us now consider the states created by the operator d~.The general form of such a state is

f(a ~)

I W>

0>

,

(2.22)

wheref(z) is an arbitrary analytical function. Let ~(A) be an arbitrary analytical function in ~,. We will investigate the mean value <~PI:~I(A):I ~P>=

<0~f”(d):~(~j):f(d~)I0>

All the operators dma in :c1: which belong to >‘, are located on the right side and commute with a Thus they can act on 10> and change into their eigenval~.

ues 0. This is equivalent to them disappearing from (2.21 ). The same is true for the operators ~ which act on <01. For the states of the form (2.22) we obtam the relation

~ where

r v

J

19 March 1990

[v(r, t)d+v~’(r,t)a +

2

dt.

(2.23)

< ~I:~:I

(2.26)

~>.

Let us return to (2.23). We introduce the notations 2dt, v2(r, t) dl, K=~V$ Jv(r, t)I

~=~vJ

°

a= fl/K,

0

J=aKexp(iA)

.

(2.27)

In terms of (2.27) the operator ~takes the form

~=K(aa++a+a)+Ja2+J*(a+)2,

(2.28)

:.Q: =2Kd +d+Jd2+J*(d+)2_~K

(2.29)

From the Cauchy—Schwarz inequality we have 0~a~1 If the spectral density Fa (k) is concentrated in a small domain near k= 1k I = (0 0/C then v(r, t) is a quasi-monochromatic signal, i.e. v(r, 1) e~0t. If w0T>> 1, i.e. if the time of observation is much greater than the oscillation period, then IJI ~ K and a << 1. If a0T~ 1 or if v(r, t) is not a quasi-monochromatic signal, the value of a may not be small and the case of a 1 is possible. At a = 0 the operator £~coincides with the operator ~ which corresponds to the Glauber theory. Thus, the value of a is a measure of the accuracy of this approximation.

0

Hence, the unnecessary modes disappear from the expansion. Specifically, P(n)


I

.

The factorial moment generating function 12P(n) n=O (l—A) Q(A)~~ =

(2.24)

3. Examples Let us consider two examples. Let ~>=In>=~IO>. In this case (2.26) gives, with m= 1, N==2Kn.

(2.25)

is also a function of the operator £~for the state I of the form (2.22). Differentiation of Q(A) with respect to )~at the point A = 0 leads, as is well known,

(3.1)

(3.2)

We see from this formula that the quantity p=2K=N/n,

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PHYSICS LETTERS A

which is the ratio of the mean value of detected photons to the total number of photons, has the physical meaning of the probability of detecting one photon, i.e. of the quantum efficiency of the detector. Let us next calculate

N(N— 1) N~— N=

.

For :S~2:we obtain from (2.28)

:~2:_—2(2K2+1J12)(df2)2d2+J2d4+J*2(d+)4 +4K[Ja~a3+r(a~)3a]. Since (d~ )2d2=ñ2—ñ, where ñ—~â~â is the photon

19 March 1990

is obeyed. The inverse transformation is â=ficosh o—fi~sinh Oe~~=e~’de”,

a~=8~ coshO—bsinh6e°=e~”d~e~”. (3.8) The value of 0 determines the degree of squeezing. The state ofsqueezed vacuum I sq, 0> is determined by the equation ~Isq, 0> =0, = 1 . (3.9) One can readily obtain the representations

I sq, 0>

number operator, we obtain

((2n_l)!!\ =(coshO)”2 ~ (—2s

N~—N= _—2(2K2+ 1J12)(n2—n)

n=0

From eqs. (3.2) and (3.3) we find that i~/~—N2=2Kn—4K2n+2a2K2(n2—n)

12n>

)

=(coshe)”2exp(—sa2flo>

(3.3)

.

\ (2n)!!

1/2

e~(~~I0> .

(3.10)

(34)

Here s=~e~0tanh0. The general squeezed state Isq, ~> is found by

In the Glauber theory, for which a = 0, we find from (3.4) that in an n-photon state the fluctuations in the number of detected photons is less than Poissonian. But when a 0 and

fiIsq, ~>=~~sq, ~>, = 1. (3.11) It is possible to represent these states in the forms

n>l+2/a2

(3.5)

the fluctuations will according to (3.4) be superPoissonian (see ref. [7]).Hence, even when a<< 1 it is possible to find an n such that the sub-Poissonian statistics changes into the super-Poissonian one. The first term in (3.4), (2K—4K2)n=p(l —p)n, corresponds to the binomial distribution. The restriction p = 2K<< 1 is satisfied due to the smallness of v. As a second example we consider squeezed states. To construct them we use the operators 1”, b=a cosh 0+d÷sinh 0e~’~=e~de fi~d~cosh 0+dsinh 0eiø=ei~d+e~S, (3.6)

Isq, ~>=exp(—~I~I2+~6~)Isq, 0> =exp(~fi~—~’6) Isq, 0> =exp(iO) exp(—~I~I2+~d~)I0> . (3.12) To show the physical meaning of squeezed states we represent E in the form E__v(t)d+v*(t)d±=AiRev(t)+A 2Imv(t). (3.13) Here A12 are the Hermitian operators corresponding to the amplitudes of two “quadrature” components of the electromagnetic field. Note that if

v~exp(—iw

0t)then Re v~cos(w0t)and A=d+d~=fi(cosh0—e~sinh0) +6~(cosh0—e’~sinh0),

where 2e~~—(d~)2e10]

2i [d ñ=D~= -~

Im v= sin ( w0t). For A1,2 we obtain from (3.13), (3.8)

.

2,cosh 0= (n+ 1)1/2, where FT ~Sthe

Here 0=fi~ meansinh photon number in the squeezed vacuum state. The commutation rule

A 1=â+d~=6(cosh0—e’~sinh0) +6 + (cosh 0— e’~sinh 0),

A 2 =i(d—d~)=i[fi(cosh 10sinh0)]. 0+e~ sinh 0)

[6,b~]=l 496

(3.7)

—6~(cosh0+e

(3.14)

Volume 144, number 8,9

PHYSICS LETTERS A

The mean value of the field in the squeezed state has the form =2RevRe[C(cosho—e’°sinhO)] —2 Im vIm [C(cosh 0+ e10 sinh 0)]

.

By means of (3.14) it is possible to find the meansquare fluctuations:

2

a 1,,2 =
CIA,,2 Isq, c>

(3.15)

For the coherent state (corresponding to 0=0) = a~= 1. But for a squeezed state a~ ~&a~. For example, if cos 0 = 1 then

19 March 1990

sinh 20 [tanh0—a cos (A + 0)]

N=K

(It is clear from (3.10) that the squeezed vacuum state always contains some photons.) Ifa=0 (the Glaubertheory) we find from (3.18) N=2Ksinh20~0. But if a>0 and cos(A+Ø)>0 it is possible to obtain from (3.18) N< 0 for tanh 0
or, equivalently fl>.a2/(l—a2)

= e29= 2fl+

1 + 2 [FT(FT+ 1)] 1/2~

Thus, for a squeezed state the mean-square value of one “quadrature” component is less than that of the other one. Therefore, twice during the period of oscillations the fluctuations decrease e times and increase e°times in comparison with the coherent state. Let us return to the mean number of registered photons in the squeezed state. To calculate N it is convenient to represent :~:in terms of 6, 6~.On substituting (3.8) into (2.29) we find that °

(3.18)

.

(3.19)

.

But the restrictions for the Glauber theory are a<
The regions (3.19) and (3.20) are shown in fig. 1. It is clear that the modified theory has a much wider range of validity than Glauber’s theory. It is easy to calculate the mean square ofthe number of detected photons. For the squeezed vacuum state it is given by the formula

50————

:Q:=Ksinh 20 [tanh 0—a cos(A+Ø)] +2R6÷6+762+J*(6+)2,

(3.16) 40

where

ti .t~I

R=K[cosh 20—a sinh 2Gcos(A+Ø)], J=Jcosh2 0+.!” sin2 0e210—Ksinh 20e’0. Now we can obtain the mean value N=
2-,

IC

t-t +

20

C>

IL

+

by means of (3.11):

~

N=Ksinh20[tanh0—acos(A+Ø)] +2~ICI2+2ReJC2.

t-

-~-~

(3.17)

Let us consider more fully the case C= 0, which corresponds to the squeezed vacuum. In this case

°

00

0.6

0.8

Fig. 1. (+ + +) Region of validity of Glauber’s theory. Region of nonvalidity ofthe modified theory.

~.0

(— —

—)

497

Volume 144, number 8,9

PHYSICS LETTERS A

7~

P(n)=

=2 =p2FT{2[FT”2—a cos(A+Ø)(FT+ 1)1/212 + 1 —a2}.

19 March 1990

Q(A)=Tr{~5exp(—AS~)}, (3.21)

It is clear from this equation that the inequality — N~ ~ N holds, i.e. the statistics is then superPoissonian. But the expression (3.21) for a>0 can have either a greater or smaller value than for the case when a = 0. This difference depends on the parameters iT, a and A + 0.

(4.4)

which contain no ordering operation. Since [a, a ~] = 1, this expression is finite. All probabilities calculated by means of the preceding formulas are positive and causal. However, it is impossible to explain the existence of sub-Poissonian statistics on the basis of this approach. It is clear that the quantity

(K~-N2)-N=(N~-N)-N2 4. Conclusion is always positive for any Hermitian operator Now the appearance of sub-Poissonian statistics is found in many experiments [9—12]. Thus, the last of formulae (4.4) cannot be correct. Hence we can propose present the theory,ofdiscussed section 3. Inat fact, thisonly modification theory isin based ~.

The new parameter a=

lf~v2dtj I~I v2 (1)1 dt

(4.1)

is very important in the present theory. It has zero value in Glauber’s theory, which can be valid only when the following two conditions are satisfied: a 4( 1, iTa2 << 1

.

(4.2)

These conditions are only necessary but they are not sufficient for the validity of Glauber theory. The modified theory, suggested in refs. [6,7] and discussed in the present paper, is valid under the more general condition ,T>a2/(l —a2)

.

(4.3)

But this condition is also only necessary, not sufficient. Some new conditions can be obtained from more rigorous theory. This restriction is connected with the condition N~0 for squeezed states. From the formal point of view the possibility of negative values for N appearing is connected with the use of the normal ordering form of the operators :~~:.The normal ordering was introduced into the modified theory to eliminate the infinities which appear from the field commutators. But for the single-mode approach we obtain one other possibility: we do not need to employ normal ordering for the relevant mode. In this case we obtain the formulae 498

on the introduction of small corrections into Glauber’s theory. Future approach should be based on the chronologically ordered evolution operator. For the singlemode description, used in this paper, such a theory could be simpler than for the general case. But it will be much more complicated than the present theory.

Acknowledgement I wish to thank Professor Emil Wolf for reading the manuscript and for several useful pieces of advice and critical comments. I am grateful to Professors V.P. Bykov and D.N. Klyshko for fruitful discussions.

References 11] C.

Dc Witt, A. Blandin and C. Cohen-Tannoudji, eds., Quantum optics and electronics, Lectures delivered at the Les Houches Summer School of Theoretical Physics, University ofGrenoble (1964). [2] J.R. Klauder and E.C.G. Sudarshan, Fundamentals of quantum optics (Benjamin, New York, 1968). [3] J. Perina, Coherence oflight (Van Nostrand/Reinhold,New York, 1972).

Volume 144, number 8,9

PHYSICS LETTERS A

[4] R. Loudon, The quantum theory of light (Clarendon, Oxford,l973). [5] P. Kelley and W. Kleiner, Phys. Rev. A 136 (1964) 316. [6] V.P. Bykov and V.1. Tatarskii, Phys. Lett. A 136 (1989) 77. [7] V.P. Bykov and VI. Tatarskii, Zh. Eksp. Teor. Fiz. 97 (1989) 528 [Soy. Phys. JETP (1989), to be published]. [8] VI. Tatarskii, Sov. Phys. JETP 57 (1983) 304.

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[9] H.J. Kimble, M. Dagenais and L. Mandel, Phys. Rev. Lett. 39(1977)961. [101 M. Dagenais and L. Mandel, Phys. Rev. A 18 (1978) 201. [11] R.E. Slusher, L.W. Hollberg, B. Yurke, J.C. Mertz and J.F. Valley, Phys. Rev. Lett. 55 (1985) 2409. [12] D.F. Smirnov and A.S. Troshin, Soy. Phys. Usp. 30 (1987) 851.

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