Squeezed and Correlated States of Quantum Fields And

ted ode TES orking tion of o, A ultim en and o dd ts of some dis- TIPLICITY ossible extension y distributions in w, Russia TED ST y distributions in high ell as of those w olynkin t momen ultiplicit ulan .G. P ultiplicit yp es of nonclassical states used ts in studying squeezed and cor- 1 t pap er is to attract the atten ysical Institute, t from the electromagnetic eld (gluons, Abstract elopmen o, and P 〉 PostScript processed by the SLAC/DESY Libraries on 28 Feb 1995. v, I.M. Dremin, O.V. Man'k

HEP-PH-9502394 ed. Their distribution functions are analyzed ac- odinger cat states) for one{mo de and m Leb edev Ph hr TICLE DISTRIBUTIONS teractions. The phenomenon of oscillations of particle V.I. Man'k AR ysicists to new dev hromo dynamics. The new t P tum optics as squeezed states, correlated states, ev t states (Sc Leninsky Prosp ekt, 53, 117924, Mosco tum c V.V. Do dono The primary aim of the presen teraction are review energy particle in distribution functions of the squeezed elds is describ ed andtributions confron for squeezed andof the correlated metho d eld to states. eldspions, di eren etc.) P is sp eculated. to the phenomenon of oscillations of cum related states of theon the electromagnetic latest eld topic asquan to w new ndingsin ab out quan m coheren particle ph in cording to the metho d used rst for m SQUEEZED AND CORRELA OF QUANTUM FIELDS AND MUL

1. INTRODUCTION

The nature of any source of radiation (of photons, gluons or other entities)

can b e studied by analyzing multiplicity distributions, energy sp ectra, var-

ious correlation prop erties, etc. A particular example is provided by the

coherent states of elds which give rise to Poisson distribution. However,

in most cases one has to deal with various distributions revealing di erent

dynamics. We try to describ e some of them app earing in electro dynamics

and in chromo dynamics but start rst with coherent states. As incomplete

as it is, this review, we hop e, will stimulate physicists working on di erent

topics to think over similarity of their problems and to lo ok for common

solutions.

The coherent states for photons have b een intro duced in [1]. They are

widely used in electro dynamics and quantum optics [2, 3]. The coherent

states of some nonstationary quantum systems were constructed in [4]-[7],

and this construction was based on nding some new time{dep endent in-

tegrals of motion. The integral of motion which is quadratic in p osition

and momentum was found for classical oscillator with time{dep endent fre-

quency a long time ago by Ermakov[8]. His result was rediscovered by Lewis

[9]. The two time{dep endentintegrals of motion which are linear forms in

p osition and momentum for the classical and quantum oscillator with time{

dep endent frequency,aswell as corresp onding coherent states were found

in [5]; for a charge moving in varying in time uniform magnetic eld this

was done in [4] (in the absense of the electric eld), [10,11] (nonstation-

ary magnetic eld with the \circular" gauge of the vector p otential plus

uniform nonstationary electric eld), and [12] (for the Landau gauge of the

time-dep endentvector p otential plus nonstationary elecrtic eld). For the

multimo de nonstationary oscillatory systems such new integrals of motion,

b oth of Ermakov's typ e (quadratic in p ositions and momenta) and linear in

p osition and momenta, generalizing the results of [5]were constructed in [7].

The approach of constructing the coherent states of the parametric systems

based on nding the time{dep endentintegrals of motion was reviewed in [13]

and [14]. Recently the discussed time{dep endentinvariants were obtained

using No ether's theorem in [15]-[17].

The coherent states [1] are considered (and called) as classical states of

the eld. This is related to their prop erty of equal noise in b oth quadrature

comp onents (dimensionless p osition and momentum) corresp onding to the

noise in the vacuum state of the electromagnetic eld, and to the minimiza-

tion of the Heisenb erg uncertainty relation [18]. These two prop erties yield

the Poisson particle distribution function in the eld coherent state, which

is the characteristic feature of the classical state.

On the other hand, the so{called squeezed states of the photons were

considered in quantum optics for the one{mo de eld [19]-[23]. The main

prop erty of these states is that the quantum noise in one of the eld quadra-

ture comp onents is less than for the vacuum state. The imp ortant feature

of the nonclassical states is the p ossibility to get statistical dep endence of

the eld quadrature comp onents if their correlation co ecient is not equal

to zero. For the classical coherent state and for the particle numb er state

there is no such correlation. The correlated states p ossessing the quadra-

ture statistical dep endence were intro duced in [24]by usage of the pro cedure

of minimization of the Schrodinger{Rob ertson uncertainty relation [25, 26].

The nonclassical states of another typ e, namely,even and o dd coherent

states, were intro duced in [27] and called as the Schrodinger cat states [28].

The even and o dd coherent states are very simple even and o dd sup erp osi-

tions of the usual coherent states. The particle distribution in these states

di ers essentially from the Poissonian statistics of the classical states. Its

most striking feature is the oscillations of the particle distribution functions

whichischaracteristic prop erty of nonclassical light [29]-[31]. They are es-

p ecially strong for even and o dd coherent states [27,32], moreover, they are

very sensitive to the correlation of the quadrature comp onents [31]. These

phenomena are typical not only for the one{mo de photons pro duction, but

also for the multimo de case [33]-[35]. The multimo de generalizations of the

Schrodinger cat states were studied in [32, 35].

Let us turn now to high energy particle interactions. Several years ago

the exp erimentalists of UA5 Collab oration in CERN noticed [36] a shoul-

der in the multiplicity distribution of particles pro duced in pp collisions at

energies ranging from 200 to 900 GeV in the center of mass system. It

lo oked like a small wiggle over a smo oth curve and was immediately as-

crib ed by theorists to pro cesses with larger number of Pomerons exchanged

in the traditional schemes. More recently, several collab orations studying

e e collisions at 91 GeV in CERN rep orted (see, e.g., [37, 38]) that they

failed to t the multiplicity distributions of pro duced particles by smo oth

curves (the Poisson and Negative Binomial distributions were among them).

Moreover, subtracting such smo oth curves from the exp erimental ones they

found steady oscillatory b ehaviour of the di erence. It was ascrib ed to the

pro cesses with di erentnumb er of jets.

In such circumstances one is tempted to sp eculate ab out the alterna-

tive explanation when considering p ossible similarity of these ndings to

typical features of squeezed and correlated states. Indeed, we know that

the usual coherent states app eared imp ortant for the theory of particle pro-

duction [39]. Moreover, the squeezed states, b eing intro duced initially for

solving the problems of quantum optics, now b egin to p enetrate to di erent

other branches of physics, from solid state physics [40]-[42] to cosmology and

gravitation [43]-[46]. Thus why they could not arise in particle physics? In

slightly di erent context the ideas ab out squeezed states in particle physics

were promoted in [47]-[49]. No attempts to use analogy with other typ es of

nonclassical states likeSchrodinger cat states or correlated states are known.

In the ab ove approach to exp erimental data, however, the form of oscil-

lations dep ends on the background subtracted. The new sensitive metho d

was prop osed in [50,51] (for the review see [52]). It app eared as a bypro duct

of the solution of the equations for generating functions of multiplicity dis-

tributions in quantum chromo dynamics (QCD). It app eals to the moments

of the multiplicity distribution. According to QCD, the so{called cumulant

moments (or just cumulants), describ ed in more details b elow, should reveal

the oscillations as functions of their ranks, while they are identically equal

to zero for the Poisson distribution and are steadily decreasing functions for

Negative Binomial distribution so widely used in phenomenological analysis.

Exp erimental data show the oscillatory b ehaviour of cumulants (see [53])

of multiplicity distributions in high energy inelastic pro cesses initiated by

various particles and nuclei, even though some care should b e taken due

to the high multiplicity cut-o of the data. When applied to the squeezed

states, the metho d demonstrates [54] the oscillations of cumulants in slightly

squeezed states and, therefore, can b e useful for their detection.

As wehave already p ointed out ab ove, the aim of this article is to review

the prop erties of the nonclassical states (squeezed, correlated, even and o dd

coherent ones) b oth for one- and multimo de cases in the context of p ossible

applications of them to high energy physics, as well as to acquaint with

new metho ds of analysis of multiplici ty distributions. However, we will not

sp eculate here ab out any application in more details, leaving the topic for

future publications.

The article is organized as follows. In sections 2 and 3 we describ e the

quantities used in analysis of multiplicity distributions for one{dimensional

and multidimensional cases, resp ectively. In sections 4 and 5 weintro duce

the quadrature comp onents of b oson elds, describ e their prop erties and

discuss their connection with exp erimentally measured quantities for elec-

tromagnetic eld. In section 6 the formalism of Wigner function and Weyl

transformation is brie y discussed. In section 7 we give the explicit formulas

describing the p olymo de mixed b oson eld (photons, phonons, pions, glu-

ons, and etc.). Section 8 is dedicated to studying the one{mo de squeezed

and correlated light. We discuss the b eha the photon distribution function

and of its moments and demonstrate their strongly oscillating character for

the slightly squeezed states of a eld. Even and o dd coherent states of one{

mo de b oson eld are considered in section 9. Concluding remarks are given

in section 10.

2. DISTRIBUTION FUNCTION AND ITS MOMENTS.

ONE{DIMENSIONAL CASE

In this part of the review we will brie y summarize de nitions and nota-

tions for the values that are used to characterize various pro cesses of inelastic

scattering according to numb er of particles pro duced in these pro cesses. We

will also p oint out the relations connecting these values to each other and

give the examples related to some distributions typical in the probability

theory.

Any pro cess of inelastic scattering (that is the scattering with new par-

ticles pro duced) can b e characterized by the function P , the multiplicity

distribution function. The value of P denotes the probability to observe n

particles pro duced in the collision. It is clear that P must b e normalized to

unity:

P =1: (2.1)

n=0

Sometimes the multiplicity distribution of particles pro duced can b e conve-

niently describ ed by its moments. It means that the series of numb ers P is

replaced by another series of numb ers according to a certain rule. All these

moments can b e obtained by the di erentiation of the so{called generating

function G(z ) de ned by the formula:

G(z )= P z : (2.2)

n=0

Thus, instead of the discrete set of numb ers P we can study the analytical

function G(z ).

We will use the factorial moments and cumulants de ned by the following

relations:

P (n 1) (n q +1) 1 d G(z )

n=0

F = = ; (2.3)

q q

( P n) hni dz

n=0

z =1

1 d ln G(z )

K = ; (2.4)

q q

hni dz

z =1

where

hni = P n (2.5)

n=0

is the average multiplicity.

Recipro cal formulas expressing the generating function in terms of cu-

mulants and factorial moments can also b e obtained:

hni F G(z )=

q !

q=0

(F = F =1); (2.6)

0 1

ln G(z )= hni K

q !

q=1

(K =1): (2.7)

The probability distribution function itself is related to the generating func-

tion in the following way:

1 d G(z )

P = : (2.8)

n! dz

z =0

Factorial moments and cumulants are connected to each other by the follow-

ing recursion relation:

q 1

F = C K F ; (2.9)

q q m m

q 1

m=0

where

(q 1)!

C =

q 1

m!(q m 1)!

are binomial co ecients. This formula can b e easily obtained if we remind

that F and K are de ned as the q {th order derivatives of the generating

q q

function and its logarithm, resp ectively, and use the well{known formula for

di erentiation of the pro duct of two functions:

q 1

C K F

q m m

q 1

m=0

q 1

q m m

1 d d 1

ln G(z ) G(z ) = C

q 1

q m q m m m

hni dz hni dz

z =1

m=0

q 1

m q 1m

1 d=dz G(z ) d d

G(z ) = C

q 1

q q 1m m

hni dz G(z ) dz

z =1

m=0

q 1

1 d d=dz G(z )

= G(z )

q q 1

hni dz G(z )

z =1

q 1

1 d

= d=dz G(z ) = F : (2.10)

q q 1

hni dz

z =1

Relation (2.9) gives the opp ortunity to nd the factorial moments if cumu-

lants are known, and vice versa.

It must b e p ointed out that cumulants are very sensitive to small varia-

tions of the distribution function and hence can b e used to distinguish the

distributions which otherwise lo ok quite similar.

Usually, cumulants and factorial moments for the distributions o ccuring

in the particle physics are very fast growing with increase of their rank.

Therefore sometimes it is more convenient to use their ratio:

; (2.11) H =

which b ehaves more quietly with increase of q remaining a sensitive measure

of the tiny details of the distributions.

In what follows we imply that the rank of the distribution function mo-

ment is non{negativeinteger even though formulas (2.3) and (2.4) can b e

generalized to the non{integer ranks.

Let us demonstrate twotypical examples of the distribution.

1: P oisson distr ibution:

The Poisson distribution has the form:

hni

P = exp (hni) : (2.12)

Generating function (2.2) can b e easily calculated:

G(z ) = exp (hniz ) : (2.13)

According to (2.3) and (2.4) wehave for the moments of this distribution:

F =1; K =H = : (2.14)

q q q q1

2: N eg ativ e binomial distr ibution:

This distribution is rather successfully used for ts of main features of

exp erimental data in particle physics. It has the form:

n (n+k)

hni (n + k ) hni

1+ P = ; (2.15)

(n + 1)(k ) k k

where is the gamma function, and k is a tting parameter.

At k =1 wehave the usual Bose distribution. The Poisson distribution

can b e obtained from (2.15) in the limit k !1.

Generating function for the negative binomial distribution reads:

zhni

; (2.16) G(z )= 1

and the moments of this distribution are:

(k + q )

F = ;

(k )k

(q )

K = ;

q 1

(q )(k +1)

H = : (2.17)

(k + q )

3. DISTRIBUTION FUNCTION.

N {DIMENSIONAL CASE

Supp ose that in a pro cess of inelastic collision the particles of N di erent

sp ecies app ear. Such a situation can b e met, for example, in considering the

electromagnetic eld, when these \particles of di erent sp ecies" are photons

corresp onding to the di erent mo des of a eld. Tocharacterize the pro cess

of such a kind it is necessary to intro duce the values analogous to those

intro duced in the previous section generalizing them to the multidimensional

case.

To abbreviate the formulas weintro duce the following notations:

n =(n ;n ;:::n )

1 2 N

N{dimensional vector with integer non{negative comp onents;

=( ; ;::: )

1 2 N

N{dimensional vector with complex comp onents;

z =(z ;z ;:::z )

1 2 N

N{dimensional vector with complex comp onents;

n!= n !n !n !;

1 2 N

n n n n

1 2 N

= ;

1 2 n

n n n

1 2

z = z z z ;

1 2 N

1 1 1 1

X X X X

= : (3.1)

n=0 n =0 n =0 n =0

1 2 N

Supp ose that the particles of N sp ecies are obtained as a result of some

pro cess. Similar to the one{dimensional case, such a pro cess can b e charac-

terized by the multiplicity distribution function P . The value of P yields

n n

the probability to observe n particles of the rst kind, n of the second one,

1 2

and so on, up to N . P is normalized:

P =1: (3.2)

n=0

The mean numberofthei{th sort of particles can b e obtained from P in

the following way:

hn i = P n : (3.3)

i n i

n=0

The mean numb er of particles is obtained by summation of (3.3) over the

index i:

N 1 1

X X X

hn i = P n = (n + n + + n )P : (3.4)

total n i 1 2 N n

i=1 n=0 n=0

Just like for the one{dimensional case, instead of set of numb ers P ,we can

analyze the analytical function G(z) analogous to generating function G(z )

intro duced in (2.2):

G(z)= P z : (3.5)

n=0

From the de nition (3.5) it is clear that

n n

1 N

@ @ 1

G (z) : (3.6) P =

n n

1 N

n! @z @z

z=0

In principle, for more detailed analysis of distributions, in the multidimen-

sional case the cumulant and factorial moments can b e de ned in a way

analogous to the metho d used in the previous section

n n

1 N

1 @ @

K = ln G(z) ;

n n

1 N

hn ihn i @z @z

1 2

z=1

n n

1 N

1 @ @

F = G(z) : (3.7)

n n

1 N

hn ihn i @z @z

1 2

z=1

However, these values, unlikemultiplicity distribution function P and gen-

erating function G(z), are not so widely used.

4. QUADRATURE COMPONENTS.

ONE{MODE CASE

Many elds, app earing in physics for description of di erentinteractions,

can b e quantized according to the b oson scheme. As examples we mention

here the electromagnetic eld, the eld of oscillations in a crystal lattice, the

eld of strong interactions, and so on. The technique of such quantization

is well{known [55]. In fact, it app eals to reducing a eld to the set of

noninteracting harmonic oscillators. In this section we will give a de nition

of so{called quadrature comp onents { the variables imp ortant for the theory.

We will also discuss their connection with real physical quantities, measured

in an exp eriment, for the electromagnetic eld.

As we already stated ab ove, the quantization of a eld means reducing

it to the set of harmonic oscillators. Each such an oscillator corresp onds,

as one says, to one mo de of a eld. In this part of the review we consider

the so{called one{mo de eld, namely the eld describ ed in complete analogy

with a single harmonic oscillator.

According to the usual practice we start with the pair of b oson creation

and annihilation op erators with the commutation rule:

^a; ^a =1: (4.1)

We consider here the stationary case in the Schrodinger representation.

Thereforea ^ and ^a are constant op erators (not dep ending on time). By

de nition, weintro duce the quadrature comp onents of a eld. In the case

of one{mo de eld there are two of them:

+ +

a^^a a^ +^a

p p

; p^= : (4.2) x^ =

2 i 2

The following commutation rule for op eratorsx ^ andp ^ is the immediate

consequence of relation (4.1) and de nition (4.2):

[^x; p^]= i: (4.3)

Situation here is entirely analogous to that of a usual harmonic oscillator.

The only di erence is the meaning of op eratorsx ^ andp ^. These op erators

in a general case are not necessarily the usual co ordinate and momentum.

The real physical sense of quadrature comp onents is clari ed when b oth

the nature and the form of the eld under consideration is sp eci ed. For

example, for the electromagnetic eld, the role of quadrature comp onents is

played by the vector p otential and the electric eld intensity.

What is the meaning of the quadrature comp onent's name? To answer

this question consider classical eld oscillations, the physical nature of which

is of no imp ortance for us here. It is well{known that every value, harmoni-

cally oscillating with time, can b e expressed in the form:

i! t i! t

ae + a e ; (4.4) A(t)=

where a = (x + ip);a is complex conjugated to a, x and p are constant real

numb ers, ! is the frequency of oscillations. Then from (4.4) it immediately

follows that

A = x cos(!t)+ p sin (!t); (4.5)

where

1 1

p p

x = (a + a ) ; p = (a a ) :

2 i 2

x and p are called the \quadrature comp onents" by virtue of relation (4.5).

The p oint is that the phase di erence b etween sine and cosine is =2, and

the value of a can b e represented as a hyp otenuse of right triangle with

legs equal to the values x and p. Under canonical quantization of suchan

oscillation the values a, a , x, p, A are replaced by op erators, but the name

\quadrature comp onents" for op erators corresp onding to x and p remains.

Now turn our attention to the problem of describing the eld state. Every

quantum system (in particular, the mo de of a eld) can b e describ ed in terms

of state vector j i in the case of pure state, when the system is isolated, and

in terms of density op erator ^, if the state is mixed and our system is a part

(subsystem) of some larger system. The pure state can b e considered as a

particular case of the mixed one with ^ = j ih j.

Thus, to describ e the state of the mo de of a eld, we can explicitly sp ecify

the state vector j i in some representation for the pure state, or all matrix

elements of density op erator ^ in the case of mixed state.

However, the di erent approach is also p ossible. Every state of a eld can

be characterized by the mean values (moments) of quadrature comp onents

in this state. The rst order moments are the quantum mean values of

op eratorsx ^ andp ^:

hx^i =Tr(^x^);

hp^i =Tr(^p^) : (4.6)

The second order moments (disp ersions and covariances) are de ned by re-

lations:

2 2 2 2

= hx^ ihx^i =Tr x^ ^ hx^i ;

2 2 2 2

= hp^ i hp^i =Tr p^ ^ hp^i ;

Tr[(^xp^+^px^)^]hx^ihp^i: (4.7) = =

xp px

From the commutation rule (4.3) it follows that the uncertainty relation

must b e valid for any state:

: (4.8)

xx pp

As usual, to p erform calculations, wehave to x the basis in a space of

states. The natural basis is that of the Fock states jni, obtained from the

vacuum state j0i de ned by formula

a^j0i =0; (4.9)

with n{multiple action of creation op erator ^a :

jni = j0i: (4.10)

The vector jni corresp onds to the state of a eld with n quanta of oscillations.

It is the eigenvector with resp ect to the op erator of numb er of particles

N =^a ^a:

Njni=njni: (4.11)

The Fock states form the orthonormalized system:

hnjmi = : (4.12)

Another convenient basis is the basis of coherent states. By de nition, a

state j i is the coherent one, if it is the eigenstate of annihilation op erator

a^:

^aj i = j i: (4.13)

Coherent states are parametrized by the continuous complex parameter .

They can b e expressed through the Fock states:

n 2

j j

jni: (4.14) j i = exp

n=0

The uncertainty relation (4.8) b ecomes minimized in coherent states:

; (4.15) =

xx pp

which is the most imp ortant prop erty of these states. Coherent states are

studied in detail and widely describ ed in literature (see [2,3,13]).

The distribution function P , discussed in section 2, is de ned by formula:

P =Tr(^jnihnj) : (4.16)

Rewrite this formula using the Fock basis:

P = hmj^jnihnjmi = hnj^jni: (4.17)

m=0

Therefore, P is nothing but the diagonal matrix element of density op erator

in the Fock basis.

Let the system b e in a pure state j i. Then ^ = j ih j, and expanding

j i in the series of Fock states jni,

j i = C jmi; (4.18)

m=0

weverify that P is the squared mo dulo of co ecient C in this expansion:

n n

P = jC j : (4.19)

n n

As a conclusion of this section consider the interpretation of quadrature

comp onents in terms of physically measured values in the case of electromag-

netic eld. In quantum electro dynamics usually the role of co ordinate and

momentum of a eld is played byvector p otential and electric eld intensity,

resp ectively.However, in quantum optical exp eriments the vector p otential

is of no imp ortance. That is whytwovalues of electric eld intensity mea-

sured in the moments distanced by one{fourth of a p erio d of oscillations are

taken as quadrature comp onents. It can b e easily shown that one of these

values is prop ortional to the vector p otential. This question is discussed in

detail in [56], and we will not describ e it any more.

5. QUADRATURE COMPONENTS.

MULTIMODE CASE

In this section we generalize to the multi{dimensional case the de nitions

and formulas of the previous one. Here we start with the Hamiltonian of

b osonic eld typical for the eld theory:

^a (5.1) H = ! a^ +

i i

i=1

(we assume the Planck constanthb e equal to unity). Thus, the eld, con-

crete nature of which is of no imp ortance here, is represented by the in nite

numb er of noninteracting harmonic oscillators, numb ered by index i in ex-

pression (5.1). In (5.1) ! is the frequency of i{th oscillator,a ^ anda ^ are

i i

b oson annihilation and creation op erators for i{th mo de of a eld. They

ob ey the commutation rule:

i h

= ; a^ ; ^a

ij i

i h

+ +

=0: (5.2) [â ; â ]= â ;â

i j

Supp ose, that somehowwe cancel oscillations in all mo des except the nite

numb er of them. Hamiltonian of such a eld is

H = ! a^ a^ + : (5.3)

i i

i=1

In this section we will consider so{called N {mo de eld with Hamiltonian

de ned by (5.3). By de nition intro duce the quadrature comp onents. There

will b e 2N of them, i.e. the pair of canonically conjugated variables p er each

mo de:

+ +

! â a^ â +â

i i i

i i

; p^ = : (5.4) x^ =

i i

2 i

From (5.2) and (5.4) the commutation relations for quadrature comp onents

follow:

[^x ; p^ ]= i ;

i j ij

[^x ; x^ ]= [^p ;p^ ]= 0: (5.5)

i j i j

Hamiltonian (5.3) rewritten in terms of op eratorsx ^ ,^p has the usual oscil-

i i

latory form:

2 2 2

! x^ p^

i i i

+ : (5.6) H =

2 2

i=1

In analogy with section 4, de ne the mean values (moments) of quadrature

comp onents. The moments of the rst order are:

hx^ i =Tr(^x^ );

i i

hp^ i =Tr(^p^ ) ; (5.7)

i i

where ^ { density op erator describing N {mo de quantized eld.

Moments of the second order (disp ersions) are de ned as:

= hx^ x^ i hx^ ihx^ i =Tr(^x x^ ^) hx^ ihx^ i;

x x i j i j i j i j

i j

= hp^ p^ ihp^ihp^ i =Tr(^p p^ ^) hp^ihp^ i;

p p i j i j i j i j

i j

= = Tr[(^xp^+^px^ )^]hp^ihx^ i: (5.8)

p x x p j i i j i j

i j j i

To simplify the following formulas intro duce the N {dimensional vectors:

n =(n ;n ;:::n );

1 2 N

=( ; ;::: ): (5.9)

1 2 N

Intro duce the basis of Fock states for N {mo de eld. They are numb ered

by N integer parameters. The vacuum state is de ned by formulas:

a^ j0i =0; i=1;2;:::N: (5.10)

States with xed numb er of quanta in each mo de are generated from vacuum

(5.9) by the action of creation op eratorsa ^ :

+n +n +n

N 2 1

a^ ^a a^

2 1

j0i: (5.11) jni =

n !n !n !

1 2 N

Fock states form the complete orthonormalized system:

hnjmi = : (5.12)

n m n m n m

1 1 2 2 N N

In analogy with (4.13) intro duce the basis of coherent states j i:

a^ j i = j i: (5.13)

i i

In this case coherent states are parametrized by N continuous complex pa-

rameters , i =1;2;:::N.

The analogue of formula (4.14), expanding the coherent state in the series

of Fock states, for N {mo de case is:

n n n

2 2 2

1 2 N

j j + j j + + j j

1 2 N

1 2

jni: (5.14) j i = exp

n !n !n !

1 2 N

n=0

N {dimensional distribution function P is intro duced according to the

de nition:

P =Tr[^jnihnj]= hnj^jni: (5.15)

6. WIGNER QUASIDISTRIBUTION FUNCTION

Quantum mechanics in its usual formulation deals with op erators acting

in Hilb ert spaces of states. However, there exist the alternative formulations

of quantum mechanics useful in considering concrete problems. An example

of such alternative approach is the formalism of Wigner function and Weyl

transformation.

Let quantum system b e in a state describ ed in terms of the density

op erator ^.We consider the Hermitean op erator A corresp onding to some

physical quantity.InWeyl formulation of quantum mechanics the ob jects

corresp onding to physical quantities are Weyl transformations, and those

corresp onding to the states of a system are Wigner functions. By de nition,

intro duce the Weyl transformation of op erator A :

u u

iqu=h

A p i; (6.1) a(p; q)= du e h p +

2 2

where jpi is the eigenvector of momentum op erator with eigenvalue equal to

p. Another, equivalent representation reads

v v

ipv=h

a(p; q)= dv e hq A q+ i; (6.2)

2 2

where jqi is eigenvector of p osition op erator with eigenvalue equal to q.We

consider N {dimensional case, so q and p are N {dimensional vectors. From

(6.1) and (6.2) it follows that the Weyl transformation of Hermitean op erator is real.

Op erator A can b e reconstructed from its Weyl transformation in the

following way:

^ ^

A = dp dq a(p; q)(p; q); (6.3)

(2 h)

where Hermitean op erator (p; q)is

u u

iqu=h

(p; q)= du e p ih p + ;

2 2

(6.4)

v v

ipv=h

(p; q)= q+ : (6.5) dv e ihq

2 2

The Weyl transformation can b e also written as a trace including op erator

h i

a(p; q)= Tr A(p; q) : (6.6)

The Wigner function is intro duced as a Weyl transformation of the density

op erator ^:

v v

ipv=h

W (p; q) = dv e h q ^ q + i

2 2

u u

iqu=h

= du e h p + ^ p i: (6.7)

2 2

It is real and normalized:

dp dq

W (p; q)= 1: (6.8)

(2 h)

However, it is not always p ositively de nite. That is why it cannot b e

interpreted as a distribution function in the phase space, therefore it is called

the quasidistribution function.

The mean value of op erator A is obtained as integral of its Weyl trans-

formation a(p; q) and Wigner function W (p; q):

dp dq

hAi = W (p; q) a(p; q) : (6.9)

(2 h)

We will not describ e the prop erties of Weyl transformation and Wigner

function, as well as those of some other quasidistribution functions contigu-

ous to W (p; q) refering the reader to the monograph [57] and the review

pap er [58].

The formalism describ ed ab ove turns out to b e very useful in considering

the photon statistics. Its main advantage is that the Wigner function cor-

resp onding to the most general squeezed and correlated state of lightisan

exp onent of quadratic form with resp ect to quadrature comp onents. Cor-

resp onding integrals can b e easily calculated giving the opp ortunityto nd

explicitly characteristics of a system.

7. PHOTON DISTRIBUTION FUNCTIONS

The photon statistics of nonclassical light describ ed by the gaussian

Wigner function has some interesting prop erties related to the squeezing

in one of the quadrature comp onents [19,23]. The problem of nding the

photon distribution function (PDF) for one-mo de gaussian states of elec-

tromagnetic eld was considered in Refs.[19, 21, 23],[59]-[67]. The simplest

expressions in terms of Hermite p olynomials of twovariables with equal

indices were obtained recently in Ref.[33], where the most general mixed

state of one{mo de light describ ed by a generic gaussian Wigner function

was treated. The photon statistics of some sp ecial cases of gaussian pure

states for two{mo de light has b een studied in Refs. [68,69].

The aim of the present section is to discuss the results of Ref.[33] in which

the photon distribution for one{mo de mixed gaussian lightwas obtained ex-

plicitly, and the results of Ref.[34] where the approach develop ed in [33]

was applied to the p olymo de case. All the information ab out the p olymo de

squeezing, mo de correlations and thermal noise is contained for the generic

mixed state of the gaussian light in the quadrature means and matrix ele-

ments of the real symmetric disp ersion matrix which determine the generic

gaussian Wigner function. For N {mo de light the numb er of real parameters

determining the gaussian Wigner function is equal to 2N +3N.We show

that the PDF of the N {mo de eld state describ ed by a generic gaussian

Wigner function can b e expressed in terms of the Hermite p olynomial of 2N

variables with equal pairs of indices.

It should b e noted that the particle distribution function and expression

of density matrix nondiagonal elements for p olymo de systems in terms of

multivariable Hermite p olynomials were given for some gaussian states in

Ref.[70]. However, the physical parameters used in this work were just the

parameters determining the inhomogeneous symplectic canonical transfor-

mation of photon quadratures, which are related to gaussian Wigner function

parameters through a complicated functional dep endence.

The most general mixed squeezed state of the N {mo de light with a gaus-

sian density op erator% ^ is describ ed by the Wigner function W (p; q) of the

generic gaussian form (see, for example, [14])

1 1=2

Q hQi)M (QhQi : (7.1) W (p; q) = (det M) exp

Here 2N {dimensional vector Q =(p;q) consists of N comp onents p ;:::;p

1 N

^ ^

and N comp onents q ;:::;q , op erators p and q b eing the quadrature com-

1 N

^ ^

p onents of photon creation a and annihilation a op erators (we use dimen-

sionless variables and assume h = 1):

+ +

^ ^ ^ ^

a + a a a

p p

^ ^

; q = : p =

i 2 2

2N parameters hp i and hq i, i =1;2;:::;N, combined into vector hQi, are

i i

the average values of quadratures,

^ ^

hpi =Tr(^%p); hqi=Tr(^%q);

A real symmetric disp ersion matrix M consists of 2N + N variances

D E D ED E

^ ^ ^ ^ ^ ^

Q Q + Q Q Q Q ; ; =1;2;:::;2N: M =

They ob ey certain constraints, which are nothing but the generalized uncer-

tainty relations [14].

Of course, one maycho ose di erent representations of the statistical

op erator, for instance, co ordinate representation or various mo di cations of

the coherent state representation (like \normal", \antinormal", \p ositive",

etc.). However, the Wigner function (7.1) seems the most suitable one,

since for gaussian states it is real and p ositive. Moreover, all co ecients of

the quadratic form in the exp onent are real, and they have lucid physical

meaning (see, also ref.[58]).

The photon distribution function is nothing but the probabilitytohave

n photons in the rst mo de, n photons in the second mo de, and so on. To

1 2

p oint out that we discuss namely the photon distribution we shall designate

it in this section by symbol P , where vector n consists of N nonnegative

integers: n =(n ;n ;:::;n ). This probabilityisgiven by formula

1 2 N

P =Tr^%jnihnj;

where% ^ is the density op erator of the system of photons under study, and

the state jni is a common eigenstate of the set of photon numb er op erators

a^ ^a , i =1;2;:::;N:

^a a^ jni=n jni:

i i

The simplest way to nd P is to calculate the generating function for matrix

elements % of the density op erator% ^ in the Fock basis. This generating

function, in turn, is nothing but the matrix element of density op erator in

the coherent state basis,

2 m n 2

j j ( ) j j

% : (7.2) h j%^j i = exp

2 2

(m!n!)

m;n=0

Hereafter and without indices mean N {dimensional vectors with com-

plex comp onents.

To pro ceed from the Wigner function to the matrix element of the den-

sity op erator in coherent state representation one should calculate 2N {

dimensional overlap integral [58]

W (p; q)W (p; q) dp dq; (7.3) h j%^j i =

(2 )

where W (p; q) is the Wigner{Weyl transform of the op erator j ih j which

was found in [14],

2 2

j j j j

N 2 2

W (p; q)= 2 exp p q

2 2

p p

+ 2 (q ip)+ 2 (q+ ip) :

Let us intro duce 2N {dimensional complex vector

=( ; );

which is comp osed of two N {dimensional vectors, and the 2N {dimensional unitary matrix

iI iI

N N

; U =

I I

N N

satisfying relations

0 I

+ T + T

: (7.4) U U = U U = I ; U U = U U =

I 0

+ T

(\ " means the Hermitian conjugation, \ " | complex conjugation, and \ "

| transp osition; I is the N N identity matrix.) Calculating the gaussian

integral on the right{hand side of eq.(7.3) we arrive at the expression

j j 1

h j%^j i = P exp R + Ry ; (7.5)

2 2

where the symmetric 2N {dimensional matrix R and 2N {dimensional vector

y are given by the relations (wehave taken into account the identities from

eq.(7.4)):

R = U (I 2M)(I +2M) U ; (7.6)

2N 2N

T 1

y =2U (I 2M) hQi: (7.7)

Factor P , which do es not dep end on vector , is nothing but the probability

to register no photons. It equals

h i

P = det M + exp hQi (2M + I ) hQi : I

0 2N 2N

If hQi = 0, the probabilitytohave no photons dep ends on 2N 1 parame-

ters coinciding up to numerical factors with co ecients of the characteristic

p olynomial of the variance matrix.

Function exp (j j =2) h j%^j i is the generating function for the elements

of density matrix in the photon numb er state basis. Comparing eqs.(7.2)

and (7.5) with the generating function for the Hermite p olynomials of 2N

variables [71],

m n

fRg

exp R + Ry = H (y ); (7.8)

m;n

2 m!n!

m;n=0

we see that the photon distribution function P can b e expressed through

the \diagonal" multidimensional Hermite p olynomials:

fRg

H (y )

: (7.9) P = P

n 0

This formula is a sp ecial case of the matrix element of density op erator

in the Fock basis, whichwas obtained in [70]by the canonical transform

metho d. It may b e shown that the multivariable Hermite p olynomial is an

even function if the sum of its indices is an even numb er and the p olyno-

mial with an o dd sum of indices is an o dd function. This prop erty is the

natural generalization of the parity prop erties of usual Hermite p olynomials

according to which the energy states of harmonic oscillator for even excited

levels are even and for o dd excited levels are o dd ones. Due to this parity

prop erty of p olydimensional Hermite p olynomials the \diagonal" multivari-

able Hermite p olynomial is an even function since the sum of its indices is

always an even numb er. Consequently, the ab ove photon distribution func-

tion is an even function. In the limit case, when the only nonzero elements

of the quadrature disp ersion matrix are the diagonal elements and they are

equal to 1=2, the obtained distribution function (7.9) b ecomes the pro duct

of N one{dimensional Poisson distributions describing the indep endent light

mo des in coherent states.

Let us discuss photon distributions for pure p olymo de states. The formu-

las derived b efore hold for any gaussian state, which is, in general, a mixed

quantum state, i.e.

=Tr(^% )1; Tr^%=1: (7.10)

In terms of Wigner function the \mixing co ecient" can b e expressed as

follows [58, 72]

= W (p; q) dp dq:

(2 )

Evaluating this integral for the gaussian state we arrive at a relation [72, 34]

N 1=2

=2 (det M) :

Consequently, for the gaussian state restriction (7.10) is equivalent to the

inequality

det M (1=4) ; (7.11)

which is nothing but one of the simplest forms of the generalized uncertainty

relations [14]. In particular, for N = 1 inequality (7.11) turns into the

Schrodinger{Rob ertson uncertainty relation [25, 26]:

pp qq

Here we consider the pure gaussian states, when eqs.(7.10) and (7.11)

b ecome strict equalities. In this case the quantum state is describ ed in fact

byawave function, which has twice less variables than the density matrix.

As a consequence, the formulas for the PDF can b e signi cantly simpli ed.

Namely, instead of p olynomials of 2N variables one can manage with the

N {dimensional Hermite p olynomials.

Let us consider an inhomogeneous linear canonical transformation of pho-

ton creation and annihilation op erators,

! ! ! !

b a d

= + ; = ; (7.12)

a d

where is a symplectic 2N x2N {matrix consisting of four N {dimensional

complex square blo cks, and d is a complex N {vector. Designating the eigen-

states of op erators a and b as j i and j i, resp ectively, one can write the

formula (for its derivation see, e.g., Refs.[14, 73]):

2 2

j j j j 1

1 1

exp + h j i = F ( ) exp + ( d) ;

2 2 2

(7.13)

where

1 1

1 1 1 2

+ (d d)+ d djdj : F ( ) = (det ) exp

2 2

(7.14)

Here should b e considered as a lab el of a state, while asavariable.

Expanding the right{hand side of (7.13) in a p ower series of with account

of (7.8), we obtain for the PDF in the state j i (which is, in general, a

squeezed coherent state) an expression through the Hermite p olynomial of

N variables:

P ( ) 1

2 f g 1

; P ( )= jF ( )j : (7.15) P = H [ d]

0 0 n

A similar formula for the transition probabilities b etween the initial and nal

energy states of a multimo de parametric oscillator was obtained in [7,14].

Let us consider the PDF of the squeezed numb er state jmi de ned by

expansion

m 2

( ) j j

) jmi: j i = exp (

(m!)

m=0

Equations (7.13) and (7.14) lead to

fRg

H (L)

1 1 2

P = j det j exp Re(d d) 2jdj :

n!m!

Here m is the lab el of the state, whereas n is a discrete vector variable.

2N x2N {matrix R and 2N {vector L are expressed now in terms of blo cks of

matrix and vector d as follows,

! !

1 1 1

; L = R : R =

1 1 1

d d

So it was demonstrated that in the case of p olymo de light in pure gaussian

state the expression for photon distribution function in terms of Hermite

p olynomial with 2N indices (7.9) may b e replaced by the expression in terms

of the Hermite p olynomial with N indices.

In conclusion let us consider the particular case of one{mo de mixed light.

The mixed squeezed state of light with density op erator ^ is describ ed by

the Wigner function W (p; q ) of the generic gaussian form which contains

ve real parameters. Two parameters are mean values of momentum hpi

and p osition hq i and other three parameters are matrix elements of the real

symmetric disp ersion matrix m

2 2

m = =Tr %^p^ hpi ;

11 pp

2 2

m = =Tr %^q^ hqi ;

22 qq

m = = Tr[^%(^pq^ +^qp^)] hpihq i: (7.16)

12 pq

Belowwe will use invariant parameters

T =Tr m= +

pp qq

and

d = det m = :

pp qq pq

So the generic gaussian Wigner function for one{mo de mixed light has

the form

1 2

expf(2d) (p hpi) W (p; q )= d

+ (q hqi) 2 (p hpi)(q hqi) g: (7.17)

pp pq

The photon distribution function of one{mo de mixed light is describ ed

by formulas (7.9), where matrix elements of the symmetric matrix R are

expressed in terms of the disp ersion matrix m as follows,

( 2i )= R R = T +2d+ ;

pp qq pq 11

1 1

2d ; (7.18) R = T +2d+

2 2

and arguments of Hermite p olynomials are of the form

y = y = T 2d [(T 1)hz i +( +2i )hz i] : (7.19)

1 2 pp qq pq

The complex parameter hz i is given by relation

(hqi+ihpi): (7.20) hz i =2

The probabilitytohave no photons is given by formula

1 1

P = d + T +

2 4

2 2

hpi (2 +1)hqi (2 +1)+4 hpihq i

qq pp pq

exp : (7.21)

1+ 2T +4d

Due to the physical meaning of disp ersions, the parameters and must

pp qq

b e nonnegativenumb ers, so the invariant parameter T is a p ositivenumb er.

Also the determinant d of the disp ersion matrix must b e p ositive.

Thus wehave shown that photon distribution functions for p olymo de

mixed gaussian light, for p olymo de squeezed numb er light and for one{mo de

mixed lightmay b e expressed in terms of Hermite p olynomials of several

variables. The physical meaning of the mixed gaussian state of lightmaybe

understo o d if one takes into account that the pure multimo de gaussian state

corresp onds to the generalized correlated state intro duced by Sudarshan [74], who related these states to the symplectic dynamical group. The one{mo de

correlated state was intro duced in [24] as the state minimizing Schrodinger

uncertainty relation [25,26]. In Ref.[74] the relation of this state to the

symplectic group ISp(2;R)was clari ed and the generalization to multi-

mo de correlated state has b een done on the basis of the symmetry prop er-

ties related to the group ISp(2N; R). As wehave observed, the generalized

correlated state constructed by Sudarshan [74] from the symmetry prop er-

ties satis es also the condition of minimization of multimo de Schrodinger

uncertainty relation, since the determinant of the disp ersion matrix is equal

to the low limit of the uncertainty inequality just for these states. From that

p oint of view the multimo de correlated states of lighthave b oth prop erties

of one{mo de correlated state to give minimum of uncertainties pro duct and

to b e obtained by the full coset of the linear canonical group by the stability

group of the vacuum state. The mixed gaussian states studied ab ovemay

b e considered as the mixture of generalized correlated states plus thermal

noise. But this thermal noise may not b e describ ed by a single temp erature

for all the mo des. For a generic case of mixed gaussian light the distinct

temp eratures may b e prescrib ed to each mo de. So the mixed gaussian light

is the generalized correlated light each normal mo de of whichinteracts with

its own heat bath. The density op erator of such state may b e obtained by

a generic symplectic canonical transform from the pro duct of usual thermal

states of the electromagnetic eld oscillators, each of them b eing describ ed

by the Planck distribution formula with its own temp erature.

8. OSCILLATIONS OF CUMULANTS IN

SLIGHTLY SQUEEZED STATES

In this section we will repro duce the results of [54] showing the cumu-

lants and the ratio of cumulants to factorial moments exhibit an oscillatory

b ehaviour when one deals with slightly squeezed states of light .

Let us consider the most general mixed squeezed state of one{mo de light

describ ed by the Wigner function W (p; x) of the generic gaussian form with

ve real parameters, hxi, hpi, , , , sp eci ed in the previous section

xx pp px

(in this section we use letter x instead of q for the \co ordinate" quadra-

ture comp onent, while q is reserved for the order of cumulants or factorial

moments). The generating function of the photon numb er distribution was

Note that in [54] the misprint in scale of co ordinate axes for graphs of photon distri-

bution function P app eared. The correct gures are drawn b elow in this section. n

obtained in [33]:

1=2

u u u u

2 1

1 1 exp + , (8.1) G(u)= P

u u

1 2 1 2

where

p p

1 1

= , = , R R R R R + R

1 2 11 22 12 11 22 12

1 R

2 2

, = 1 y R R y y R + y R 2

1 11 22 1 2 11 22

1 2

R R

11 22

1 R

2 2

= 1+ y . R + y R R y y R +2

2 11 11 22 1 2 22

1 2

R R

11 22

It was already mentioned in intro duction that the photon distribution

function exhibits an oscillatory b ehaviour if we deal with highly squeezed

states ( T = + 1 ) for large values of the parameter z . A ques-

pp xx

tion arises: is it p ossible to obtain a similar \abnormal" b ehaviour of other

characteristics of the photon distribution, namely,intro duced in section 2

cumulants, factorial moments and their ratio H ? If yes, then in what

region of parameters can such anomalies reveal themselves?

The direct di erentiation of function ln G(u)at u= 1 yields the cumu-

lants (see section 2)

(q 1)! 1 1 1 1

1 1 2 2

K = + q + q + ; (8.2)

q q q

hni ( 1) 2 1 ( 1) 2 1

1 1 2 2

with the average numb er of photons hni [33]

T 1

+ jz j . hni =

The Schrodinger{Rob ertson uncertainty relation results in restrictions

T 1; d ; (8.3)

which, in turn, lead to inequalities

> 1; (8.4)

< 0 or > 1. (8.5)

2 2

The expression in brackets in eq.(8.2) consists of two terms:

1 1

+ q , (8.6)

( 1) 2 1

1 1

2 2

+ q . (8.7)

( 1) 2 1

2 2

The rst term has constant sign. The second one oscillates in the case < 0.

With the aim to obtain oscillations of the whole function K we will treat

only this case,

< 0.

Then,

(1) 1

= .

q q

( 1) (1 + j j)

2 2

However, then it follows that

j j >1,

2 1

and the alternating term diminishes faster than the constant sign term.

Terms 1=j 1j and 1=( 1) are most close to each other if

2 1

d = (the pure state),

that is used in the following.

First of all we consider the simplest case when value of z as given by

(7.20) equals zero. Then

q=2

K = (q 1)!T ( ),

q q

hni

q=2

q !P ( ), (8.8) F =

q q

hni

where

1 T T 1

= d + , = ,

4 2

4d +12T

T ( ) and P ( ) are the Chebyshev p olynomials of the rst kind and Legen-

q q

dre p olynomials, resp ectively. Let us note that the arguments of the p oly-

nomials are purely imaginary but the whole expressions for the moments are

real, surely.

For H we obtain the expression:

T ( )

. (8.9) H =

qP ( )

Taking d =1=4wehave:

q=2

K = (q 1)!T ( ),

q q

1 T

q=2

q !P ( ), F =

q q

= .

In this case the curve H has step{like shap e at (T 1) ! 0; steps b ecome

smo othed as T grows ( g.1). We should note that direct limit T ! 1 shows

the discontinuous character of the function H (T )atT = 1. The p ointis

that at T =1we are dealing with the usual Poisson distribution (let us

remind that we treat a case d =1=4, jz j = 0), where H = , i.e. H =1,

q q 1 1

H =0at q 6= 1, which di ers from the b ehaviour of H at (T 1)=10

q q

depicted in g.1.

Consider now the case jz j6= 0. Since the photon distribution function is

invariant with resp ect to rotation in a phase space, without loss of generality

we can consider = ( 6= 0 { correlated state). By appropriate

xx pp px

choice of the phase of z (hxi = hpi)we cancel the linearly increasing term

q =(1 ) in (8.6). Moreover, the analogous linear term q =(1 )

1 1 1 2 2 2

in (8.7) b ecomes maximal at xed jz j.Thus wehave left only twovariable

parameters T and jz j, and formula (8.2) has the following nal form:

(q 1)! 1 1 (1) j j

2 2

K = + q , (8.10)

q q

T 1

q q

2( 1) (1 + j j) 2 1+ j j

+ jz j

1 2 2

where

T +1

= = ,

1 2

T 1

=2 +1 jzj .

T 1

In the case of large T (a highly squeezed state) we can obtain the nite

numb er of oscillations of K considering large values of jz j.However, the av-

erage numb er of photons in corresp onding states is large, and the amplitude

of oscillations decreases exp onentially due to the factor 1=hni . Remind that

in this very case the strong oscillations of the photon distribution function

can b e observed.

Now let us consider the case of the slightly squeezed state, y =(T1)

1 when the photon distribution function do es not oscillate. Imp ose also an

additional condition

jz j

1, =

y=2

that makes p ossible to obtain approximate formulas for the functions K ,

F , and H .For K ,wehave the following approximate expression:

q q q

q 1 1q

K = q !(1) . (8.11)

Then recursion relation (2.9) yields:

q q 1

F = q !(1) L ( ), (8.12)

where L (x) are generalized Laguerre p olynomials. For H , with q we

have:

q +1 1q

(1) q ! 1. (8.13) H = K =F =

q q q

L ( )

(If q the term with highest p ower of dominates over the rest of sum

in L ( ), and F ! 1 as for Poisson distribution). The exact shap e of the

function H is shown in g.2. The distribution function P do es not oscillate

q n

( g.2 ).

However, the most abrupt oscillations of the functions K and H have

q q

b een obtained when (T 1) 1, but condition 1 is not valid. The cor-

resp onding curves are shown in gs.3, 3 . Note that the photon distribution

function is smo oth again b eing approximately equal to zero at q 6=1.

The most regular oscillating patterns of K and H are seen at (T 1)

q q

0:1, jz j 1 ( gs.4, 4 ).

The alternating sign cumulants are typical also for the xed multiplicity

distribution, i.e. for P = (n = const) [51]. Let us note that there exist

n nn 0

smo oth multiplicity distributions which give rise to cumulants oscillating

with larger p erio d (see, [50]).

Finally we consider the opp osite case when the photon distribution func-

tion P exhibits strong oscillations while K and H b ehave smo othly. Such

n q q

a b ehaviour is typical at T 100, jz j 1 when K exp onentially grows

while H monotonically decreases with q ( g.5).

Thus wehave shown that the cumulants of the photon distribution func-

tion for one{mo de squeezed and correlated light at nite temp erature p os-

sess strongly oscillating b ehaviour in the region of slight squeezing where the

photon distribution function itself has no oscillations. And vice versa in the

region of large squeezing, where the photon distribution function strongly

oscillates, the cumulants b ehave smo othly. Hence, the b ehaviour of cumu-

lants may provide a very sensitive metho d of detecting very small squeezing and correlation phenomena due to the presence of strong oscillations.

9. PHOTON DISTRIBUTION FUNCTION AND ITS

MOMENTS FOR SCHRODINGER CATSTATES

As we said inintro duction, there exists another example of nonclassical

light state (Schrodinger cat state) considered for the rst time in [27]. The

paradoxinvented bySchrodinger is well{known: let's consider a cat in two

states; one corresp onding to alive cat and another | to dead one, b oth

describ ed by their own wave functions. If the wave function of alive cat is

denoted by , and that of dead cat is denoted by (since cat is a macro-

a d

scopic ob ject, b oth of these functions, of course, dep end on a large number

of variables), the sup erp osition principle of quantum mechanics enables the

existence of a cat in the states describ ed by the wave functions

( + ); and =

a d 1

= ( ): (9.1)

2 a d

Certainly, the cat is to o complicated ob ject to b e considered by the quantum

physics. Hence, wehavetocho ose somewhat simpler ob ject.

Consider one{mo de electromagnetic eld in coherent states j i and j i

and form even and o dd normalized combinations of these states:

j i = N (j i + j i) and

+ +

j i = N (j i j i): (9.2)

The normalization constants are given by relations:

2 2

exp (j j =2) exp (j j =2)

p p

N = ; N = : (9.3)

2 2

2 ch j j 2 shj j

The photon distribution function for such states can b e easily calculated

using (4.14) [32]:

1 j j

(+) n

P = [1 + (1) ] ;

chj j n!

1 j j

() n

P = [1 (1) ] : (9.4)

shj j n!

The multidimentional generalization of the ab ove formulas is straightfor-

ward [35]. So, fast oscillations of the photon distribution function for the

Schrodinger cat states are connected with the absence of states with even

numb ers of photons in o dd cat states and states with o dd numb ers in even

cat states.

Using eq.(2.2) one can nd the generating functions for quantum distri-

butions in the even and o dd cat states

2 2

sh (j j z ) ch(j j z)

() (+)

; G (z)= : (9.5) G (z )=

2 2

ch j j sh j j

The average numb ers of photons in these states read

(+) 2 2

hni = j j th j j for even state,

j j

()

hni = for o dd state. (9.6)

th j j

(+) ()

Functions G (z ) and G (z ) can b e easily di erentiated giving the explicit

expressions for factorial moments:

(+) 2

F = th j j ; q =2k;

(q 1)

(+) 2

F = th j j ; q =2k+1; (9.7)

and

() 2

F = th j j ; q =2k;

q 1

() 2

F = th j j ; q =2k+1: (9.8)

(+) ()

The logarithm of G (z ) and G (z ) can not b e so easily di erentiated,

so to nd the cumulants wehave to solvenumerically the system of linear

algebraic equations (2.9). As a result wehave cumulants oscillating with

extremely rapidly growing amplitude in the whole range of values of the

parameter j j. The function H = K =F b ehaves in a similar way. To

q q q

illustrate it we plot here in g.6 the graph of function H for o dd cat state

at j j =2.

10. CONCLUSION

New developments in studies of nonclassical states of the electromagnetic

eld, from one side, and of multiplicity distributions in high energy particle

interactions, from the other side, provide some hop e for p ossible interrelation

of the states under consideration. The problem has matured for further

exploration.

In this brief review we tried to demonstrate the metho ds applied for de-

scription of nonclassical elds. No attempt was made to use them directly in

treatment of high energy multiparticle pro duction even though we consider

it as a very promising problem to b e studied later. We can not resist to stress

that the similarity of Lagrangians of quantum electro dynamics and quantum

chromo dynamics, not to say ab out their obvious di erence, could havemuch

deep er meaning with nonclassical elds playing an imp ortant role. Squeezed

and correlated gluonic elds could b e as crucial as those of photons. Our

hop e for success of the idea is based b oth on the similarity of theoretical

analyses of particle distributions in those cases and on some exp erimental

indications discussed ab ove. In particular, wehave shown that the oscilla-

tory b ehaviour of cumulants of the multiplicity distributions predicted rst

in quantum chromo dynamics for high energy particle pro duction pro cesses

reveals itself also for the slightly squeezed states as well as for even and o dd

states (Schrodinger cats). It could b e a signature of some collective e ects

b eing imp ortant in b oth cases.

Wewould consider our task ful lled if the review helps someb o dy to

promote the analogy further.

ACKNOWLEDGMENTS

This work is supp orted in part by Russian program "Fundamental nuclear

physics", by Russian fund for fundamental research, and by JSPS.

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FIGURE CAPTIONS

Figure 1: The b ehaviour of the function H de ned in (2.8) at d =1=4,

jz j = 0; parameter T is varied: (1) (T 1) = 10 , (2) T =1:1, (3) T =1:2.

Figure 2: The b ehaviour of the function H at d =1=4, T =1:01; the

curves (1) and (2) corresp ond to the values jz j =1:01; 0:8, resp ectively.

Figure 2 : The photon distribution function at d =1=4, T =1:01; the

curves in order of the lowering maxima corresp ond resp ectively to the values

jz j =1:01; 0:8.

Figure 3: The cumulants of the photon distribution function at d =1=4,

5 2

(T 1) = 10 , jz j =0:01.

a 5

Figure 3 : The b ehaviour of the function H at d =1=4, (T 1)=10 ;

2 2 2

parameter jz j is varied: (1) jz j =0:01, (2) jz j =0:005, (3) jz j =0:01.

Figure 4: The cumulants of the photon distribution function at d =1=4,

T =1:1, jz j =1:1.

Figure 4 : The b ehaviour of the function H at d =1=4, T =1:1;

2 2

parameter jz j is varied: (1) jz j = 2, (2) jz j =1:1.

Figure 4 : The photon distribution function at d =1=4, T =1:1; the

curves in order of the lowering maxima corresp ond resp ectively to the values

jz j =2; 1:1.

Figure 5: The smo oth curve for the function H and the oscillating pho-

ton distribution function P at d =1=4, T = 100, jz j =1.

Figure 6: The b ehaviour of the function H for o dd Schrodinger cat state

at j j =2.