Second-order statistical properties of quantum chaotic beams

C. Bendjaballah*Laboratoire des Signaux et Systèmes, CNRS and École Supérieure d’Électricité, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France

M. Pourmir†

Département des Télécommunications, École Supérieure d’Électricité, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France�Received 15 October 2008; published 15 May 2009�

We consider the situation in which the field of quantum particles such as photons �bosons� or electrons�fermions� undergoes a significant attenuation during the detection process. We characterize the second-orderstatistical properties of the field by the bunching for bosons and the antibunching for fermions. Bunching andantibunching effects are derived from the time intervals distributions of the random point process of detectioninstead of the moments of the number of particles registered in a given time interval. Fields of both beams aresupposed to be in chaotic states and have second-order time correlation functions of exponential profile and ofarbitrary correlation time. A test is proposed for nonclassical states: the behavior of the antibunching functionis a nondecreasing function near the origin of the time axis.

DOI: 10.1103/PhysRevA.79.053826 PACS number�s�: 42.50.Ar

I. INTRODUCTION

Recent experimental results on bunching and antibunch-ing �1� effects have renewed interest in the comparison ofsecond-order properties of bosons and fermions. These ef-fects, as is well known, can be used to distinguish betweenclassical and quantum interpretation of correlation measure-ments �2�.

The tendency of particles to group together, known as thebunching effect �3,4�, is a feature of fields for which a clas-sical analog exists and can be associated, for example, withbosons in a chaotic state. This effect is now well understood�5� and can be explained by the conventional theory of prob-ability. The density operator of such state admits a positive Prepresentation and the corresponding Wigner function is alsopositive. Indeed, its main properties can be calculated bydifferentiating the generating function which is easily calcu-lated for a Gaussian field having a field correlation functionof exponential profile �6�.

The antibunching effect, which is the opposite to bunch-ing, makes particles separate from each other. There is noclassical analog of this effect which can be treated necessar-ily within the framework of the quantum theory. This is thecase for example, for bosons in a squeezed state or fermionsin a chaotic state. Then, there is no positive P representationfor the density operator and at least part of the correspondingWigner function is negative. By using Pauli’s exclusion prin-ciple and the wave packet formalism, Glauber showed thata very simple transformation connects the main results ofthe statistical theory of bosons with those of fermions ��7�,pp. 115–125�.

More recently, Cahill and Glauber �8� established, in anal-ogy with the theory of bosonic coherent states, a theory offermionic coherent states based upon a practical calculus ofanticommuting variables. The basic definitions and formulaof this theory are recalled in Appendix A. For the analysis of

the detection process of quantum particles from a randompoint process point of view, which is the purpose of thispaper, it is, however, more convenient to follow the wavepacket approach used in �9,10�.

Thus, in this approach basically using the Glauber coher-ent states, and for a measurement of the number of particlesin a fixed time interval �0, t�, the average number of particles�n� is proportional to t so that it is a diverging parameter. Inorder to limit the �n� to the finite values when t becomesinfinite, Mollow �11� proposed to take into account effects ofthe attenuation processes which take place after the absorp-tion of a particle by the detector. By assuming that the totalnumber of quanta in the system of field and absorbing detec-tor is conserved, he solved the system of coupled equationsof motion. He also demonstrated that the density of the pro-cess is attenuated by an exponential time decaying function.Consequently he proved that �n� remains finite when t→�.

In this paper, we adapt the Mollow model to calculate thetime second-order properties, such as in number measure-ments, and in time-interval measurements of the fluctuationsof these beams of particles. For short times, it is seen that thebunching and antibunching effects increase when the attenu-ation effects increase.

Section II will review the basic equations required for thecalculations. The main results are summarized in Sec. III andare illustrated with several curves. Appendixes A and C de-velop the methods of the calculations needed in the paper.

II. BASIC EQUATIONS

We consider a beam of quantum particles, e.g., photons orelectrons, which can be treated within the formalism devel-oped for bosons. The beam is described by the instantaneouscomplex random field E���. We assume that only one modeof the field is excited and the bandwidth �� of the field issmall compared to the frequency of the excited mode.

The second-order time correlation function of the fieldwill be supposed homogeneous and of exponential profile. Itis denoted

�E�t1,t2� = �E��t1�E�t2�� = �e−��t1−t2�, �2.1�*cherif.bendjaballah@lss.supelec.fr†mithridad.pourmir@supelec.fr

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where � is the inverse of �c the correlation time of the field.It is here arbitrary compared to �t1− t2�. The instantaneousintensity is given by J���= �E����2.

For chaotic fields that we only deal with here �the fieldE��� has the Gaussian properties�, the second-order timecorrelation function of the intensity is then given by

�J�t1,t2� = �J��t1�J�t2�� = �J�2 + ��E��t1 − t2���2, �2.2�

assuming the stationarity in time of the field. This is neces-sarily a nonincreasing function of �t1− t2�. Its definition interms of operators is given in �12�, pp. 88–92, pp. 148–162,

G�2��t1, . . . ,t4�

= Tr�E�−��t1�E�−��t2�E�+��t3�E�+���4�� �2.3a�

=G�1��t1,t3�G�1��t2,t4� + G�1��t1,t4�G�1��t2,t3� �2.3b�

=G�1��t1,t1�G�1��t2,t2� + G�1��t1,t2�G�1��t2,t1� , �2.3c�

where t3= t1 , t4= t2. The expression given by Eq. �2.3c� isthe quantum analog of Eq. �2.2�.

In Eq. �2.3a�, the operator E�+� is proportional to a†, the

creation operator of a photon while the operator E�−� is pro-portional to a, the annihilation operator of one photon. Theyobey �a , a†�=1, a��=�� where �� is the coherent state.

The number operator is given by N= a† a. The density op-erator , such that �0, Tr =1, describes the state of thebeam of particles.

The process of measurement of such a field, at low levelof power and using an ideal detector, is a set of time instants�tj , j=1, . . . ,n occurring within the time interval �0, t�. Sucha time sequence constitutes a random point process �RPP�.

For bosons, the random process of detection �RPP� is of-ten called a compound Poisson process. For fermions, it is arenewal process called sometimes a determinantal process�13� or more simply a fermion process �14�.

One method of characterizing such a RPP is to evaluatethe probability distribution function �PDF� that the numberof particles N�t�=n are recorded within the time interval t.This is given by the Poisson transform, known as the Man-del’s equation ��5�, p. 625�

p�n;t� =1

n!�Wn exp�− W�� , �2.4a�

W�t� = �0

t

����d� , �2.4b�

where ���� is the non-negative density of the RPP, i.e., theparticle flux �number of particles per unit time�. In whatfollows, the constant � which connects the incoming par-ticles with the registered ones will be taken such that �=1.All the parameters will then be dimensionless. Notice thatthis equation establishes a relation between a discrete vari-able, n with a continuous variable W.

It is convenient to calculate the probability p�n ; t� in termsof the generating function G�s ; t� by means of the relations

p�n;t� =�− 1�n

n!

�n

�snG�s;t��s=1 �2.5a�

G�s;t� � �e−sW�t�� = �n

�1 − s�np�n;t� , �2.5b�

where s is such that 0 s 1. The �Wk�, moments of the timeintegrated density W, can then be calculated by differentiat-ing the generating function given by Eq. �2.5b� with respectto s such as

�Wk� = �− 1�k �k

�skG�s;t��s=0. �2.6�

Likewise, the ordinary moments of the number derive fromEq. �2.5b�

�nk� = �n

nkp�n;t� = �− 1�k �k

�skG�1 − e−s;t��s=0. �2.7�

Now, another method of characterizing the RPP seems moreuseful in the present study. It is the study of the time intervaldistributions �TID�, the statistics of intervals between the in-stants of detection of particles.

Among the types of intervals, two are of interest here. Thefirst one, called residual time, is the interval of length�t− t0� whose origin t0 is arbitrary. The time t being the nextinstant of the RPP just after t0. Its PDF is denoted v�t� and iscalled the waiting time interval distribution. In case where t0is an instant of the RPP, the interval, called lifetime, is of asecond type. Its PDF is a conditional one and is given byw�t � t0�, written simply w�t�. The PDFs of interest here areagain obtained by differentiating the generating functiongiven by Eq. �2.5b� with respect to time such as �15�

v�t� = −1

s

�

�tG�s,t��s=1, �2.8a�

w�t�0� =1

s2

1

����2

�t2G�s,t��s=1. �2.8b�

In fact, we focus on the function

h�t� =w�t�v�t�

, �2.9�

which mainly characterizes the second-order properties ofthe field of particles.

The two types of measurement, namely, the time interval�t or the number of particles �n are of rather complemen-tary properties and are generally not equivalent.

A. Bosons

The time-dependent moment generating function Gb�t�has been calculated by Bedard �6�. It is given by

Gb�s,t� =e�t

cosh�y� + Y sinh�y�, �2.10a�

y = t��2 + 2����s t��s� , �2.10b�

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Y =�

2y+

y

2�, �2.10c�

where �= 1�c

��0� is the inverse of the correlation time �c.The theory leading to Eq. �2.10� is taken from �3,15,16�

and is very briefly recalled in Appendix A. From Eqs. �2.10�,the moments of the density �Wk� are easily deduced. Thus,setting ���=�, �=�t and �=�t, we have

�W� = �, ∀ � , �2.11�

�W2� = 2�2 −2

3�2� +

1

3����2, O��3� , �2.12�

hb�t� =�W2��W�2 = 2 −

2

3� +

1

3�2, O��3� , �2.13�

where hb�t� is the bunching function for bosons. From Eqs.�2.11� and �A4�, we see that

1

��W� = t = �

m=0

�

�m, �2.14�

which diverges.Therefore, in order that the average value of the density of

photons remains finite when t→�, a transformation which isadapted from the theory of Mollow �11�, is applied.

This theory deals with situations in which the number ofparticles is subject to a substantial attenuation during themeasurement process. It assumes that the number of ab-sorbed quanta can never be greater than the mean number ofparticles in the initial state of the field. By treating the ab-sorbing atoms as well as the particles of the beam as har-monic oscillators, Mollow established a formula for the gen-erator operator

Q�s,t� = N�e−sf�t�a†a� , �2.15�

where f�t�=1−e−�t and �� �0,�� is the decay rate. In Eq.�2.15�, the symbol N denotes the normal ordering and actssuch as all a† are to the left of all a �disregarding of the

noncommutativity�, e.g., ��Q�s , t����=e−sf�t���� ���.This decay rate is given by �=�M�v��g�v��2 ��11�, �Eq.

�A6��� where M�v� is the number of detector oscillators perunit frequency and g�v� is the coupling parameter per unitfrequency for the field-detector interaction.

This theory has successfully been utilized for calculatingthe TIDs of classical and nonclassical states of light �17�.Finally, the transformation proposed here is such that

y � yt ��t���s� =1

��1 − e−�t���s� , �2.16a�

Gb�s,t;�� =e���t�

cosh�yt� + Y sinh�yt�. �2.16b�

The function Y given by Eq. �2.10c� remains unchanged.Now, the needed expressions for our purpose are easily

derived from Eq. �2.10�. For instance, denoting �=�t, weobtain

�W��t� = ��t�� , �2.17a�

�W2��t� = �W��t�2 −2

3�2��1 −

�

2� + �2�� . �2.17b�

It can now be seen that for �→0, �W�→� as required andfor t→� , �W�→ �

� , ��0.To calculate the moments of the number of bosons, we

need to define the moment generating function gb�s , t ;��=Gb�1−e−s , t ;�� and then �nk�= �−1�k �k

�sk gb�s , t ;�� �s=0,

�n��t� = ��t�� , �2.18a�

�n2��t� = �n��t� + 2�n��t�2 −2

3�2��1 −

�

2� + �2�� .

�2.18b�

The second moment being calculated up toO��5� , O��5� , O�t5�.

Let us now discuss the bunching function. To this end, letus consider first the case ��1 and �→0. The exact result isgiven by Eq. �A7�,

hb�t;�→ 0� =2

1 + �� tanh��t�

, �2.19�

where �=��2+2��. An excellent approximation for Eq.�2.19� is given by

hb�t;�→ 0� � 2 − 2�2� − �� + 2�2� + ��2 �2.20�

up to O��7/2� , O�t3�. The bounds on Eq. �2.19� are such that�−�� hb�t ;0� 2. The lower bound �A9� is taken from �15�.

For arbitrary values of the parameters, the rather complicatedexpressions for the TIDs are given in Appendix B.

The corresponding of the bunching function within theframework of the conventional theory, i.e., ignoring the at-tenuation process, which is to be compared with Eq. �2.20�,can be written as

hb�t� � 2 − 2�� + �� + 2��2 + �� + �2� . �2.21�

There are clearly significant differences. The calculationsyielding the exact expression are presented in Appendix C.

B. Fermions

The statistics of fermions have been modeled by Glauberwho wrote the density operator as the sum of the vacuumstate and a collection of Fock states. He demonstrated thatusing wave packets, the treatment of fermions is no differentfrom the treatment of bosons.

Furthermore, in analogy to the boson coherent states, thetheory of fermion coherent states has been developed �seefor example �8��. However it differs from the latter as theeigenvalues are not ordinary c numbers. They obey a specialalgebra which is briefly presented in Appendix A, Eqs.�A11�–�A14�.

The one mode fermionic oscillator includes just twostates: the vacuum �0� and the excited state �1�. The densityoperator can be written in a diagonal representation of num-ber state

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f = �1 − �n���0��0� + �n��1��1� , �2.22�

with Tr fN� �n��1.

The �N2 , variance of the number operator N is given by

�N2 = �n��1− �n��. A general expression of rth moment of the

number operator is given in Appendix A �Eq. �A17��.For this quantum beam of particles, as described by den-

sity operator �2.22�, only number state �n=0� or �n=1� can beoccupied. Therefore conventional coherent states for fermi-ons, i.e., �n=0,1 , . . . ,��, cannot be constructed. However,other types of coherent states can be defined �8�, �see a briefreview in Appendix A, Sec. II�.

The time-dependent intensity moment generating functionGf�s ; t� for ��t�= t has been calculated by Macchi �10� whoadapted the previous results of Bedard and Glauber.

For our model of ��t�, we have

Gf�s,t;�� = Gb−1�− s,t;�� = e−���t��cosh z + Z sinh z� ,

�2.23a�

z = ��t���2 − 2��s ��t���s�, 0� ���

2,

�2.23b�

Z =�

2��s�+��s�2�

. �2.23c�

As before, we can easily calculate the first two moments ofW and obtain

�W��t� = ��t�� , �2.24a�

�W2��t� =2

3�2��1 −

�

2� − �2�� . �2.24b�

We notice that an operator representing the density of theprocess can be well defined. However, for fermions that op-erator cannot be easily interpreted in terms of the instanta-neous intensity although it is simple to calculate its moments.Nevertheless, using the same method for bosons, the first two

moments of N are given by

�n��t� = ��t�� , �2.25a�

�n2��t� = �n��t� +2

3���1 −

��

2+��2

5� − �2���1 −

2

3� +

5

6�� .

�2.25b�

The second moments have been approximated up toO��5� , O��5� , O�t5�.

On the other hand, the expressions of the TIDs have, hereagain, a rather complicated form for arbitrary values of theparameters. However, for ��1, �→0 the TIDs are alreadycalculated �10�

v f�t� = �e−�t�cosh��t� +�

�sinh��t�� , �2.26a�

wf�t� =2��

�e−�t sinh��t� , �2.26b�

where �=��2−2�� is subject to condition �2.23b�. Theseexpressions yield up to O��5�, O�t4�, ��1,

hf�t;0� =2

1 +�

�coth��t�

� 2��1 − �� +4

3�� + ���2.

�2.27�

From these results, it is seen that an antibunching effect cantake place, i.e., 0 hf�t ;0� 1, provided that �� 1

2 , ∀�.This is the illustration of the fact that the fermions statescannot overlap: the instants of the RPP, instants of the detec-tion of the fermions, will tend to avoid one another.

C. Remarks

1. Bosonic quasiprobability functions

In view of Eq. �2.22�, it is seen that p�N=n�=Tr f�n��n�=0, ∀n�0, 1. This is one of the tests for thenonclassical property of any state as established by Mandel��5�, p. 543�. This is with the exception of the vacuum state�0��0� for which p�N=0�=1, and p�N�0�=0. This testcomes from the fact when the state admits a Glauber-Sudarshan P representation =�d�dP������, �P���0�,a state yielding p�N=0� has no classical analog.

One can also calculate the Wigner function, a realquasiprobability ��12�, p. 135–144�, given by W��=�d2�e�

�−��C���, where C���=Tr�e�a†−��a� is the sym-

metrically ordered characteristic function.It is well known that the Wigner function of a boson in

the number state �n� is given by Wbn��=e−2��2�−1�n

�Ln�4��2�, where Ln�x� is the Laguerre polynomial. Part ofthis function is clearly negative. For example for n=1,Wb1��=−e−2��2�1−4��2�, which is negative for 0 �� 1

2 .It has therefore nonclassical properties.

For a bosonic thermal state given by its photon-counting distribution pn= �1−q�qn, where q= �n�

1+�n� , theWigner function is given by Wb��= �1−q��nqnWbn��= 1

1+2�n�exp� −��2

�n�+1 / 2�, which is always positive. This is also

the case for the Q function Qb��= ��b��=e−��2 � pn��2n

n!

= 11+�n�exp� −��2

�n�+1 ��0.

2. Fermionic quasiprobability functions

The symmetrically ordered characteristic functionof Grassmann numbers �� ,��� is given by C�� ,���=Tr� fe

� f†−�� f�, where are f , f† are the annihilation and cre-ation of fermion defined by Eqs. �A11�–�A13�.

Given density operator �2.22�, we obtain the symmetricalWigner function as W fn�� ,���=−��n�− 1

2 �exp� −���

�n�−1 / 2�, where

it must be taken into account that ����R, and ���+���=�2=0. This function cannot be plotted on a graph or evalu-ated numerically. We can just remark that it is negative for12� �n��1. Now, the fermion-counting distribution �A14�,

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given by �n= �1− �n��� �n�1−�n� �

n �n=0,1�, will allow us to cal-culate the Wigner function for a number state �n� which maythus be identified as W fn�� ,���= �−1�n 1

2exp� −���

�n�−1 / 2�. It is

negative for n=1, but positive for the fermionic vacuumstate. The Q function, given by Qf�� ,���= ��� f�−��= �1− �n��exp� ��

�

1−�n� �, is positive.

3. Bunching and antibunching coefficients

For the fermion point processes considered here, it is eas-ily seen that for t� 1

� , we have v�t��w�t�, leading tohf�0;0� 1. On the other hand, it is well known thathC�t ;��=1, ∀ t ,� for bosons in a coherent state. It is alsowell known that hb�t ;0��2− 2

3�t for bosons in a thermalstate with �→0, ��0. Now, for fermions, we havehf�t ;0��2�t when �→0, ��0. Therefore, the bunchingand antibunching coefficients verify hf�0;0��hC�0;0��hb�0;0�.

This inequality is, in a sense, the analog of the inequalitynFD nB nBE of the occupancies of a single-particle state�average number of particles in that state� of the Fermi-Dirac, Boltzmann, and Bose-Einstein statistics respectively.They are given by nFD= �e�+1�−1 , nB=e−� , nBE= �e�−1�−1,where �= �−�

kBT , � is the energy, T the temperature, kB the Bolt-zmann’s constant, and � the Fermi energy.

4. Intensity

We can express the TIDs for classical processes such asthe random point process of the detection of bosons, in termsof the intensity of the radiation which is proportional to thedensity of the process. We have �15�

vb�t� = ���t�e−W�t��, wb�t� =1

����0���t�e−W�t�� , �2.28�

yielding the bunching coefficient hb�0;��= ��2����2 , which is

necessarily �1.Notice that only the first moment of the intensity and the

number are proportional �see Eq. �2.18��. Therefore the ex-pression of Hb�t ;��= �n2�

�n�2 is very different from hb�t ;��. Fi-nally, the property 0�hf�0;0��1 cannot be interpreted interms of the moments of the density, contrarily to the case ofbosons. In fact, expressions �2.28� established for bosons arevalid only for compound Poisson processes �see �2��. Theexact expressions of the TIDs are given in Appendix B. Theantibunching function can be written as

hf�t;�� �1 +�

��1 +

2�

�e−�t�tanh����

1 +�

�tanh����

, �2.29�

for ��0, t 1. A discussion based on previous results andon particular expressions of Eq. �2.29� is given in the nextsection.

III. DISCUSSION

Numerical and analytical results of the bunching and an-tibunching functions are summarized in this section. Typicalcurves are plotted in Figs. 1 and 2 on semilogarithmic scales.Note first that both bunching and antibunching functions in-crease at the origin of the time axis either with respect to �,the inverse of the correlation time or to �, the decay rate ofthe function that characterizes the field attenuation of the

10−2

10−1

100

101

102

0

0.5

1

1.5

2

2.5

3

3.5

FIG. 1. Plots in dimensionless units of the variations of thebunching function hb�t ;�� versus �t, the inverse of the correlationtime � times the time interval t separating two detected bosons. Thevalue of the average density is taken such that �� �

2 =4�10−3. Thevalues of the couple of parameters �� ;�� where � is the particleattenuation are such that �10−2 ;10−3� for the curve plotted in solidline �−�, and �10−2 ;10−6� and �1;5�10−3� for the curves plotted indashed lines �−−�, �−.−� respectively.

10−2

10−1

100

101

102

103

0

0.5

1

1.5

2

2.5

FIG. 2. Plots in dimensionless units of the variations of theantibunching function hf�t ;�� versus �t, the inverse of the correla-tion time � times the time interval t separating two detected fermi-ons. The value of the average density is taken such that �� �

2 =4�10−3. The values of the couple of parameters �� ;�� where � isthe particle attenuation are such that �10−2 ;10−6� for the curve plot-ted in solid line �−�, and �10−2 ;5�10−3� and �1;5�10−3� for thecurves plotted in dashed lines �−−�, �−.−� respectively.

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detector. When the time t becomes very long and the corre-lation time very short ��t 1�, the bunching increases andthe antibunching decreases.

The bunching function hb�t ;�� is plotted in Fig. 1 versus�t for the couple �� ;��. They are chosen to take the values�1;10−3�, �0.01;5�10−3�, and �0.01;10−6�. The averagevalue of the density is fixed at �=4�10−3. Although, forbosons, � can be arbitrary compared with � and �, we chosethese values in order that the approximated analytical resultscan be compared with the exact results, and also in order tocompare them with the results obtained for fermions. Thelatter being subject to the conditions mentioned above, i.e.,0��� �

2 , ���, and ���. The results of the conventionaltheory are obtained for �=10−6.

In Fig. 1, it is seen that the function hb�t ;�� hb�t→0;�� is continuously decreasing with a change of regimefor moderate �t. It means that the degree of thermalization ofthe chaotic state of bosons decreases. Furthermore, for higherbut moderate values of �, say 10−3, we have

hb�t → 0;��! 2 +2

3�t , �3.1a�

hb�t → 0;��! 2 +1

3��t2, �3.1b�

the results that are in agreement with the exact numericalresults of Fig. 1. An approximate formula up to O��3�,O��3�, is given by

hb�t;�� � 2 −1

3�2 − ��� +

1

3�1 − ���� −

1

9�1 −

3

2����2, �3.2�

where we recall that �=�t and �=�t. The asymptotic behav-ior of hb�� ;�� is more complicated to describe. However,with the help of the generating function given by Glauber��12�, p. 184� written here as Gb�s , t�=e−���s�−����t�, we canshow that for ��0

hb�t → �;�� = 1 +�

�, �3.3�

which differs from the conventional value �A9�. Notice how-ever that 1 hb�t ;�→0� 2, ∀t in the case that ��1.

Similarly, the antibunching function hf�t ;�� is plotted inFig. 2 versus �t for the couple �� ;�� taking the values�1;10−3�, �0.01;5�10−3�, and �0.01;10−6�. The averagevalue of the density is �=4�10−3 as above. Notice firstwe have 0 hf�t ;�→0� 1, ∀ t, when ��1. We alsohave hf�t→0;���hf�t ;�� , ∀ t only for �, �→0. The func-tion hf�t ;�� with respect to time starts increasing meaningthat the state of fermions starts a thermalization process.For higher values of �, the function hf�t ;�� shows a maxi-mum. The estimation of the abscissa of this maximum

tfm�− 1

�arg tanh��−��2+4�2

2� �→0 when � 1, is shown inAppendix C. It is seen that for t� tf

m, the function hf�t ;��continuously decreases with respect to time and tends to itsasymptotic value �1 in all cases.

It can be useful to interpret this kind of phase transitionfrom the increasing part to the decreasing part appearing inthe behavior of hf�t ;��. Roughly, we may say that this oc-curs when �n� tends to its maximum �

� , ���0�, i.e., t→ 1�

�more precisely tfm which is in fact very close to 1

� when �→0�. On a line x" t, the statistics of fermions change assoon as x�xm where xm" tf

m. This is due to the fact thedistance that keeps fermions apart cannot exceed xm. When�→0, such an effect cannot occur.

Moreover, it can be shown that

hf�t → 0;�� � 2�t , �3.4a�

hf�t → 0;�� ��

�. �3.4b�

It is emphasized that when the attenuation process is takeninto account, the antibunching function hf�t ;�� is alwayspositive. A more general approximate formula is given by

hf�t;�� � 2��1 − �� + ��1

�− 3�� , �3.5�

where the notations are �=�t, �=�t, and �=�t.Here again, adapting the asymptotic form of Glauber to

our case such that the generating function can be written asGf�s , t�=e���−s�−����t�, we obtain for �→�

hf�t → �;��� 1 −�

�, �3.6�

where we recall that ��� by hypothesis.Notice finally that the curves hb�t ;�� and hf�t ;�� inter-

sect at the abscissa t�� 1�+�+2� . In view of the results and

curves, we may conclude that a quantum state for which thetime derivative of the function h�t ;�→0� when 0 t 1

� ispositive is a nonclassical one, i.e., a state without classicalanalog.

ACKNOWLEDGMENTS

One of us �C.B.� would like to thank Don Scarl for help-ful comments. The Laboratoire des Signaux et Systèmesis a joint laboratory �UMR 8506� of C.N.R.S. and ÉcoleSupérieure d’Électricité and associated with the UniversitéParis–Orsay, France.

APPENDIX A: THE GENERATING FUNCTIONFOR CHAOTIC BEAMS

In this appendix, we recall the results of the conventionaltheory �=0. We first present the needed equations for thechaotic bosons because those concerning the chaotic fermi-ons can easily be derived from them.

1. Bosons

The field E��� of the radiation being Gaussian can beexpanded in the time interval �0,��,

E��� = �m

cm#m��� �A1�

with orthonormalized eigenfunctions #m associated with theeigenvalues �m of the integral equation

�m#m�t� = 0

t

�E��1,�2�#m��2�d�2, �A2�

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where �E��1 ,�2� is the normalized second-order time corre-lation function of the field given by Eq. �2.1�. The discretecoefficients of Eq. �A1� are uncorrelated and have a Gaussianprobability distribution

�cm� cn� = �m$mn, �A3a�

p�cm� = �m=0

�1

��me−�cm�2/�m. �A3b�

The generating function of the integrated intensity W�t� canthen be expanded as an infinite product

Gb�s,t� = �m=0

�1

1 + ��ms, �A4�

where �= ���. In the specific case of interest here, the eigen-values are such that �6�

�m = ��t�2���t�2 + �m2 � , �A5�

which are derived from the roots of one of the followingequations, �2�m tan �m =�t ,

2�m cot �m =− �t ,� �A6�

such that only the positive roots of Eq. �A6� are taken intoaccount. The eigenfunctions are not important for the presentcalculations. For ��1 and �→0, the bunching function isgiven by �15�

hb�t� =1

�

�2 − Y2 − 2��% + 2�2%2

�% − �, �A7�

where

� = ��2 + 2��, % =sinh��t� + Y cosh��t�cosh��t� + Y sinh��t�

,

Y =�

2�+�

2�. �A8�

The limiting value of hb�t� for �→� is given by

hb�t → �� →� − �

�� 1 −

�

�. �A9�

The PDF and the moments of the number of bosons havealready been established by Bedard �6�. We briefly establishnow the corresponding results for fermions.

2. Fermions

We summarize here the results of the fermion coherentstate theory which are necessary to calculate the quasiprob-ability distributions. We start with the following basic prop-erties for the operators of annihilation and creation:

f �1� = �0�; f†�1� = 0; f �0� = 0; f†�0� = �1�; �A10a�

f f†�1� = 0; f† f �1� = �1�; f f†�0� = �0�; f† f �0� = 0, �A10b�

where f† f is the number operator. It follows that the anticom-mutator and the operators are given by

� f , f†+ = f f† + f† f = 1, f = �0��1�; f† = �1��0� . �A11�

An equivalent expression of the boson coherent state can begiven for the fermion

f ��� = ����, ��� = e−� f†�0� . �A12�

Notice, however, that the eigenvalues � are anticommuting cnumbers and obey the Grassman algebra. For example, fortwo eigenvalues, we have

�i� j = − � j�i, �i2 = 0, ��i, aj+ = ��i, aj

†+ = 0.

�A13�

The following relations are useful for the calculations of in-terest here: �d�=�d��=0, ��d�=1, �g1��1�g2��2�d�1d�2

=�g1��1�d�1�g2��2�d�1d�2, �d��d�e−��F�=F. The fermi-onic pure state density operator �= ����−�� leads to thefermion-counting distribution pn= �n ����−� �n� where theproduct �n ��� can also be written as �−1�n�n �−��. Finally,

we include here the fermionic P representation of the density

operator f =�d��d�P�� ,����, which yields a diagonal rep-resentation for a single mode chaotic fermion �8�

f = − �n� d��d�e−���/�n��

= �1 − �n�� �n=0,1

� �n�1 − �n��

n

�n��n� , �A14�

where P�� ,���=−�n�e−���/�n�.This formalism being recalled, let us focus on N�t�, the

number of fermions measured in a time interval �0, t�. Thecorresponding generating function is simply derived fromEq. �A4� such that

Gf�s,t� = Gb−1�− s,t� = �

m=0

�

�1 − ��ms� , �A15�

which has been shown to obey the condition of a generatingfunction, i.e., �m �

−1 , ∀m �10�. Let gf�s , t� be the fermionnumber generating function, we have

gf�s,t� = Gf�1 − e−s,t� . �A16�

The moments of the number of fermions can easily be de-rived from Eqs. �2.7� and �2.23a� to get

�nr��t� = e−x ��1,. . .,�r

� r!

�1 ! . . . �r!�− x��1+. . .+�r�1+. . .+�r �

m=1

r

qm�m

�A17�

where we denote x=�� and

u��x� =1

2�ex + �− 1��e−x� ,

0 = cosh x, 1 = ex, �A18�

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��2 = ���u��x� + u�+1�x�

+1

2�k=2

�

�− 1�k �!

�� − k�!�x�−ku�+1−k�x�� ,

qm = �s1,. . .,sm

��− 1�s1+. . .+smCs1+. . .+sm

s1 ! . . . sm!�j=1

m � ��j!

�sj

.

�A19�

The notation � stands for ��t� and Cm=1.3.5. . . �2m−1�= �2m−1� ! ! for m�1. The symbol �� means that the sum-mation is constrained such that �1+2�2+ ¯ +r�r=r with 0 �k , . . . ,�r r.

On the other hand, we can write the variance and theprobability distribution function of the number of fermionsregistered in a time interval �0, t�. The utilization of Eq. �2.7�leads to

�N2 �t� = �

m

��m�1 − ��m� = ���e−���2�A1e�� cosh����

+ A2e�� sinh���� − �sinh���� + Z cosh�����2

→�,��1

�2�

4�2 �1 + �t��t2, �A20�

where A1=1+Z� 1�� − 1

�� � and A2=Z+ 1� � 1� − 1

� �. Other nota-

tions are �=��2−2��, ��� �2 �, and Z= �

2� + �2� . A useful ap-

proximation is given by

�N2 �t� →

��1

��� − ��2�

�1 +�2

�2�t + O�t2� . �A21�

Using Eq. �2.5� yields for the special case of interest here,

prob�N = 0� = �j

�1 − �� j� = e−���cosh���� + Z sinh�����

��t�1

1

2e−t��−���1 + Z + �1 − Z�e−2t��

+1

4e−t��−����1 − e−2t����Z − ��

+ �1 + e−2t���� − �Z���t2. �A22�

We also can show

prob�N = 1� = ��j

�1 − �� j��j

� j

1 − �� j

=���

�e−���sinh���� + Z cosh�����

��t�1

��

2�e−t��−���1 + Z − �1 − Z�e−2t��t

+��

4�e−t��−����1 − e−2t���1 + �tZ − �t�

− �1 + e−2t����tZ − Z − �t���t2. �A23�

All these expressions are evaluated up to O��2�. Numeri-cally, it can be shown that the terms in � increase the prob-abilities.

APPENDIX B: THE TIME INTERVAL DISTRIBUTIONS

As above, we set �=�t, �=�t, �=�t and � stands for��t�.

1. Bosons

Using Eqs. �2.8a� and �2.16b�, we obtain

vb�t� = e−��+����� + �%��cosh���� + Y sinh�����−1,

�B1�

where Y is given by Eq. �2.10c� and % by Eq. �A8�. Simi-larly, we can show

wb�t� =1

�e−�2�+�����2�2%2 − 2��% + �2� − ��e���cosh����

+ Y sinh�����−1. �B2�

These TIDs can then respectively be approximated

vb�t� ��

�1 + ��2 +�2�3 + ��3�1 + ��3� −

��3 + 3� + ����2 + 4� + 6��3�1 + ��4 �

�B3�

up to O��2� and O��2�. Similarly, we have

wb�t� �2�

�1 + ��3 −2�

�1 + ��4�

+3�1 − � − 2�2� + ����3 + 5�2 + 10� + 18�

�1 + ��5 �

�B4�

up to O��3/2� and O��2�.

2. Fermions

From Eqs. �2.8a�, the exact expressions are easily ob-tained after simple calculations,

v f�t� = �e−��+����cosh���� +�

�sinh����� , �B5�

which is a decreasing function of time around the origin.Similarly, we have

wf�t� = �e−��+����cosh���� +�

��1 +

2�

�e−��sinh����� ,

�B6�

which is an increasing function of time around the origin.A useful form of the antibunching function which has

been given above Eq. �2.29� can be deduced from the fol-lowing approximate expressions of the TIDs:

va�t� = ��1 − ��� +1

3���2�2 +

1

2��

− ���1

�− 2� +

5

3�� +

1

2&�2�� , �B7a�

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wa�t� = 2���1 − � −1

3��� + �1 − ���6 −

20

3� −

5

2�����

�B7b�

up to O��5/2� and O��2�.

APPENDIX C: THE BUNCHING AND ANTIBUNCHINGFUNCTIONS

We set ���=�, and recall the expressions given by Eq.�2.10�,

y = ��t���2 + 2�� ��t�� , �C1�

Y =�

2�+�

2�, �C2�

where ��t�= 1−e−�t

� and � is the inverse of the correlation timeof the field as defined in Eq. �2.1�. We introduce the follow-ing notations:

c = lnY + 1

2, c = ln

Y − 1

2, �C3a�

� = � − �, � = � + � , �C3b�

d = ln �, d = ln � . �C3c�

It is convenient to set H=e��t��+c and H=e−��t��+c. From Eqs.�2.16b�, �2.8a�, �2.8b�, and �2.9�, and after some elementarycalculations, we obtain

hb�t;�� =�

�−

e−�t

���2H − �2H

�H + �H− 2�H + �H

H − H� . �C4�

With '=��t��+ c−c2 , Eq. �C4� becomes

hb�t;�� =�

�+

e−�t

��2� coth ' − � tanh�' −

d − d

2� − �� .

�C5�

For ��1, a good approximation of Eq. �C4� yields

hb�t;�� ��

�+ �� − ��

e−�t

�+ 4��

e−�t−���t�

�, �C6�

where �= Y−1Y+1 �1+ �

2� �.Concerning chaotic fermions, we use the same method.

As above, it is convenient to define K=e��t��f+cf, and K=e−��t��f+cf, where cf =lnZ+1

2 , cf =lnZ−12 , and Z= �

2� + �2� . We

start with Eq. �2.23� and obtain

hf�t;�� =�

�+

e−�t

��� f

2K − � f2K

� fK − � fK� �C7�

where �=��2−2��, � f =�−�, and � f =�+�.A good approximation of Eq. �C7� for ��1, is given by

Eq. �2.29�.Just in order to evaluate the coordinates tm

f , hfm of the

maximum of hf�t ;��, it is useful to set ��t�= t, �=�, and2���� = 2�

� , ��0, such that

hf�t;�� � 1 +2�

�e−�t tanh��t� , �C8�

leading to

tfm � −

1

�arg tanh�� − ��2 + 4�2

2�� �

1

2�ln�4�

�� + O��2� ,

hfm � 1 +

2�

�−�

�+

2�

�ln�1

2��

�� + O��� , �C9b�

where ���.

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�2005�.�3� C. Bendjaballah, Introduction to Photon Communication

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Maitland �Academic Press, New York, 1970�.�8� K. E. Cahill and R. J. Glauber, Phys. Rev. A 59, 1538 �1999�.�9� C. Benard and O. Macchi, J. Math. Phys. 14, 155 �1973�.

�10� O. Macchi, Ph.D. thesis, University of Paris, Orsay, 1972.�11� B. R. Mollow, Phys. Rev. 168, 1896 �1968�.�12� R. J. Glauber, in Quantum Optics and Electronics, edited by C.

DeWitt, A. Blandin, and C. Cohen-Tannoudji �Gordon andBreach, New York, 1965�.

�13� A. Soshnikov, Russ. Math. Surv. 55, 923 �2000�.�14� A. Borodin and G. Olshanski, e-print arXiv:math.RT/9804088.�15� C. Bendjaballah and F. Perrot, J. Appl. Phys. 44, 5130 �1973�.�16� B. E. A. Saleh, Photoelectron Statistics �Springer, New York,

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