Direct measurement of optical quasidistribution functions: Multimode theory and homodyne tests of...

PHYSICAL REVIEW A 66, 043803 ~2002!

Direct measurement of optical quasidistribution functions:Multimode theory and homodyne tests of Bell’s inequalities

Konrad BanaszekClarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom

Andrzej Dragan and Krzysztof Wo´dkiewiczInstytut Fizyki Teoretycznej, Uniwersytet Warszawski, Hoz˙a 69, PL-00-681 Warszawa, Poland

Czesław RadzewiczInstytut Fizyki Dos´wiadczalnej, Uniwersytet Warszawski, Hoz˙a 69, PL-00-681 Warszawa, Poland

~Received 12 April 2002; published 4 October 2002!

We develop a multimode theory of direct homodyne measurements of quantum optical quasidistributionfunctions. We demonstrate that unbalanced homodyning with appropriately shaped auxiliary coherent fieldsallows one to sample point-by-point different phase-space representations of the electromagnetic field. Ouranalysis includes practical factors that are likely to affect the outcome of a realistic experiment, such as nonunitdetection efficiency, imperfect mode matching, and dark counts. We apply the developed theory to discussfeasibility of observing a loophole-free violation of Bell’s inequalities by measuring joint two-mode quasidis-tribution functions by photon counting under locality conditions. We determine the range of parameters of theexperimental setup that enable violation of Bell’s inequalities for two states exhibiting entanglement in theFock basis: a one-photon Fock state divided by a 50:50 beam splitter, and a two-mode squeezed vacuum stateproduced in the process of nondegenerate parametric down-conversion.

DOI: 10.1103/PhysRevA.66.043803 PACS number~s!: 42.50.Dv, 03.65.Ud, 42.50.Ar

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I. INTRODUCTION

Over the years, the field of quantum optics has gainejustified reputation of a practical testing ground for the foudations of quantum physics. The broad class of nonclassstates of optical radiation that can be feasibly generatedlaboratory has allowed one to demonstrate a variety of qutum phenomena. Furthermore, progress in the detection tniques has opened up possibilities to realize a wide rangquantum measurements on the electromagnetic fields. Tadvances have led, in particular, to two interesting devements. On one hand, it has become feasible to charactcompletely the quantum state of optical radiation usingmethods of so-called quantum tomography@1#. On the otherhand, the development of sources of correlated photbased on parametric down-conversion has resulted in emous progress in tests of Bell’s inequalities@2# and moregenerally provided practical means to realize experimenta number of theoretical proposals in the field of quantinformation processing@3–5#.

Among different representations that can be used to cacterize the quantum state of optical radiation, phase-spquasidistribution functions possess several interestingtures@6#. They are a convenient tool in the analysis of quatum interference, as well as provide insight into the classlimit of quantum theory, including the phenomenon of decherence@7#. Quasidistribution functions have played an important role in the field of quantum state measurement sthe first experiments on the reconstruction of the Wigfunction by means of optical homodyne tomography@8# andalso the earlier work on the determination of theQ functionusing double homodyne detection@9#. It was subsequentlyrealized that standard experimental techniques such as h

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dyning and photon counting can be combined into an alnative, more direct method for measuring quasidistributfunctions@10,11#. This method directly provides the value oa quasidistribution function at a specific point of the phaspace. The coordinates of that point are defined by theplitude and the phase of an auxiliary coherent field usedhomodyning. The basic idea of the direct method is to pform photocounting on the signal field superposed at a higtransmissive beam splitter with a relatively strong coherfield. With a good approximation, such a procedure addcoherent amplitude to the signal field, while retaining all tquantum fluctuations it initially contained. The unbalanchomodyning scheme for measuring the Wigner functionbeen realized in a proof-of-principle experiment in Ref.@12#.

The development of unbalanced homodyning has lednovel proposal for testing Bell’s inequalities in optical sytems@13#. The proposal was based on an observation thattwo-mode generalization of the direct scheme for measuquasidistribution functions could be straightforwardly reiterpreted as a standard arrangement for measuring cortion functions between two spatially separated apparatusatisfying the locality conditions. The role of dichotomvariables in this proposal is played either by the parity ofregistered number of photons, or by the presence ofnumber of photons. It was demonstrated theoretically foselection of input states that such correlation functionscapable of violating Bell’s inequalities@13,14#. In addition,the character of measured observables does not reqsupplementary assumptions that were necessary in the pous proposals for testing Bell’s inequalities using balanchomodyning@15#. This has opened up yet another routeoptical testing of Bell’s inequalities, alternative to thobased on polarization@2# or frequency-time entanglemen

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BANASZEK, DRAGAN, WODKIEWICZ, AND RADZEWICZ PHYSICAL REVIEW A66, 043803 ~2002!

@16#. Recently, the possibility of testing Bell’s inequalitiebased on homodyning and photon counting has beenplored experimentally by Kuzmichet al. @17#.

The purpose of the present paper is twofold. In the fipart, we provide a complete multimode theory of the dirmethod for measuring quasidistribution functions. Previotheoretical discussions of unbalanced homodyning wbased on the assumption that the beams interfered at a bsplitter contained single excited modes with perfectly mating spatiotemporal characteristics. Here we will analyzecase when both the beams are described by general elemagnetic field operators. We will demonstrate that if the snal field comprises several excited modes, the full multimoquasidistribution function of the signal field can be measudirectly by an appropriate spatiotemporal shaping of the ailiary coherent field used for homodyning. Interestingly, thmeasurement scheme does not depend on whether thcited modes can be separated before the detection or notonly observable that needs to be determined is the statiof the total number of photons in all the modes of intereThe general multimode treatment will also allow us to adress rigorously the problem of imperfect mode matchbetween the signal and probe fields. We will show thateffect of mode mismatch is essentially different from thoccurring in balanced homodyne detection@18#, where itsimply contributes to the overall detection efficiency. Fcompleteness, we will also discuss here other practicalpects that are likely to affect a realistic experiment, suchdark counts.

In the second part of this paper, we apply the multimotheory to perform a feasibility study of the tests of Belinequalities based on unbalanced homodyning. We focusattention on two quantum states of optical radiation: a sinphoton divided by a 50:50 beam splitter and a two-mosqueezed vacuum state. Both these states can be practgenerated using spontaneous parametric down-conversiomedia withx (2) nonlinearity. We determine here the rangeexperimental parameters such as detector efficiencymode matching, which enable violation of Bell’s inequalitieGeneration of a one-photon Fock state in a controlled stiotemporal mode has been recently demonstratedLvovsky et al. @19#, who were able to achieve a sufficieoverlap with the local oscillator field to demonstrate tnegativity of the corresponding Wigner function. This sinificant experimental result encourages us to examine mclosely the possibility of testing Bell’s inequality by homodyning a single photon divided on a beam splitter. Genetion of the second state studied here, i.e., the two-msqueezed vacuum state, also attracts presently a great dinterest as one of the constructional primitives in quantinformation processing based on continuous variables@5,20#.We note that in contrast to a typical scenario consideredthis context, involving measurements of continuous quadture observables, we will be dealing with a discrete, bindetection scheme. This example suggests that continuvariable entanglement could perhaps be useful alsoimplementations of other quantum information protocolslying on binary logic@21#.

We would like to stress here that the operational mean

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of two-mode quasidistribution functions as nonlocal corretion functions depends critically on the specific experimenscheme used for the measurement. In order to performvalid test of quantum nonlocality, the measured quantitmust satisfy the local reality conditions that are assumedthe derivation of the corresponding Bell’s inequality. In paticular, if standard Bell’s inequalities derived for binary mesurements on spin-1

2 particles are to be applied to a quantuoptical experiment, the output signals from the photodettors have to be converted into dichotomic variables onshot-by-shot basis. This condition is fulfilled by the direscheme based on unbalanced homodyning, where in eacone either registers the photon number parity, or checksthe presence of any number of photons. In contrast, thinot the case for the optical homodyne tomography schewhere the Wigner function is reconstructed by an applicatof tomographic back-projection algorithms to quadrature stistics obtained from balanced homodyne detection. Obously, such a procedure cannot be performed on the shotshot basis. This difference can be best seen on the exaof the two-mode squeezed vacuum state: it is well knothat detection of quadratures has a straightforward localden variable model based on the Wigner representatwhereas it has been demonstrated that unbalanced homoing can reveal the nonlocality of this state. The above arment invalidates the claim of Ref.@22# that in order to im-prove the detection efficiency, unbalanced homodyningbe simply replaced in tests of Bell’s inequalities by optichomodyne tomography@8# or cascaded homodyning@23#.We note that the balanced homodyne detection can beto test Bell’s inequalities in other proposals, which are,example, based on a legitimate procedure of binary clascation of the result of a quadrature measurement@24,25#.

This paper is organized as follows. We start in Sec. IIanalyzing multimode interference at an unbalanced besplitter, and demonstrate, using an appropriate modal decposition, how the overall statistics of photocounts canrelated to multimode quasidistribution functions of the inpsignal field. In Sec. III we discuss effects of imperfectionsa practical experimental setup, including nonunit detectefficiency, dark counts, and imperfect mode overlap. We fithat these three factors have distinctively different effectsthe measured observables. Next, we review in Sec. IV hthe two-mode case can be used to test Bell’s inequalities,derive realistic expressions for measured correlation futions that include effects of dominant experimental imperftions. These correlation functions are analyzed in detaithe subsequent sections using two specific examples ofmode input states: a one-photon Fock state divided o50:50 beam splitter in Sec. V, and a two-mode squeevacuum state in Sec. VI. In particular, we find the rangeexperimental parameters where a violation of Bell’s inequties can be observed. Finally, Sec. VII concludes the pap

II. MULTIMODE THEORY

In this section we develop the full multimode theorythe direct scheme for measuring the quantum optical qudistribution functions proposed in Refs.@10,11#. A general

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DIRECT MEASUREMENT OF OPTICAL . . . PHYSICAL REVIEW A 66, 043803 ~2002!

arrangement depicted in Fig. 1 consists of two fields, namas the signal and the probe, interfered at a beam splcharacterized by a power transmissionT. The output port ofthe beam splitter is monitored by a photon counting deteintegrating the incident light over its active surface. In tsingle-mode description of light incident on a beam splitteis sufficient to use a pair of annihilation operators referrto the signal and the probe fields, respectively. The stiotemporal characteristics of these two modes are not imtant in this approach as long as it is implicitly assumed tthe modes are matched perfectly at the beam splitter.shall free our further analysis from these simplifying asumptions.

A. Interference at a beam splitter

Let us denote byEout(1)(r ,t) the positive-frequency part o

the electric-field operator at the surface of the detector. Tfield is a superposition of the signal and the probe fiecombined at the beam splitter BS. Mathematical represetion of this combination is a slightly intricate matter. If wwanted to proceed rigorously and expressEout

(1)(r ,t) in termsof the signal and probe field operators before the beam ster, we would need to introduce appropriate propagatorsthe fields in the presence of the beam splitter@26#. Instead,we shall choose different notations for the signal andprobe fields, which will make the discussion more intuitivLet us denote byES

(1)(r ,t) the electric-field operator of thesignal beam that would fall onto the detector surfacein the

absence of the beam splitter BS. Analogously, letEP(1)(r ,t)

be the probe field at the detector surface, assuming thabeam splitter BSwas replaced by a perfectly reflecting mi

ror. With these definitions, the fieldEout(1)(r ,t) resulting from

the interference of the signal and the probe beams is gsimply by

Eout(1)~r ,t !5ATES

(1)~r ,t !2A12TEP(1)~r ,t !, ~1!

whereT denotes the beam splitter power transmission coficient. We have assumed here that the characteristics obeam splitter are constant over the spectral and polarizaranges of the considered fields. Further, we shall assumethe detected fields are quasimonochromatic with the cenfrequencyv0. This will allow us to relate easily the numbe

FIG. 1. Experimental arrangement consisting of signal aprobe fields, interfered at a beam splitter characterized by a potransmissionT. The output port of the beam splitter is monitoreda photon counting detector integrating the incident light overactive surface.

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of photons to the energy of the field absorbed by the detecAssuming that the direction of propagation of the fieEout

(1)(r ,t) is perpendicular to the detector, the operator ofphoton flux through the detector surface reads

J out52e0c

\v0E

DtdtE

Dd2r Eout

(2)~r ,t !Eout(1)~r ,t !, ~2!

where Eout(2)(r ,t)5@Eout

(1)(r ,t)#†, and the temporal and thspatial integrals are performed, respectively, over the detor opening timeDt and its active surfaceD. The probabilitypn of registeringn counts on a photodetector is given by

pn5K :e2hJout~hJ out!

n

n!:L

S,P

, ~3!

where h is the detector quantum efficiency and :•••: de-notes normal ordering. Following the idea of Ref.@11#, weshall use the measured photocount statistics to evaluateexpression of the form

P~s!5 (n50

` S s11

s21D n

pn , ~4!

where s is a real parameter, assumed to be nonpositiveorder to assure convergence of the series on the right-hside of the above formula. We will see that in the limitnegligible losses, the parameters describes the ordering othe measured quasidistribution function. The quantity in E~4! can be written in terms of the photon flux operator as

P~s!5K :expS 22hJ out

12sD :L

S,P

. ~5!

If a coherent field is used as the probe, we may immediaevaluate the quantum expectation value over the probeand obtain

P~s!5K :expS 24he0c

\v0~12s!E

DtdtE

Dd2r @ATES

(2)~r ,t !

2A12TEP* ~r ,t !#@ATES(1)~r ,t !

2A12TEP~r ,t !# D :L , ~6!

whereEP(r ,t)5^EP(1)(r ,t)&P is the amplitude of the electric

field of the probe beam. Thus, we can simply replacecoherent probe field operators with their expectation valin all normally ordered expressions involving the photocoustatistics.

B. Modal decomposition

We will now consider the signal fieldES(1)(r ,t) in which

a finite number ofM modes is possibly excited. We shadenote the corresponding annihilation operators byai , andthe mode functions byui(r ,t), where i 51,2, . . .M . Our

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goal will be to relate the photon statisticspn to the multi-mode quasidistribution characterizing these modes. Thusdecompose the signal fieldES

(1)(r ,t) in the form

ES(1)~r ,t !5(

i 51

M

aiui~r ,t !1V~r ,t !, ~7!

where the operatorV(r ,t) is a sum of all the other moderemaining in the vacuum state. This part of the field doescontribute to the detector counts in the normally orderedpression given in Eq.~6!.

Further, we shall assume that virtually all the excited pof the signal field is absorbed by the detector within the gopening time. This allows us to write orthonormality reltions for the mode functionsui(r ,t) in the form

2e0c

\v0E

DtdtE

Dd2r u i* ~r ,t !uj~r ,t !5d i j , ~8!

where the integrals are restricted to the domain definedthe detection process. With these assumptions, we mayplify the exponent of Eq.~6!. It is convenient to introducedimensionless amplitudesa i , which are projections of theprobe field onto the signal mode functions,

a i52e0c

\v0E

DtdtE

Dd2r u i* ~r ,t !EP~r ,t !. ~9!

These amplitudes describe the parts of the coherent pfield EP(r ,t) that match the corresponding modes in the snal field. Furthermore, let us also denote byJP the averagetotal number of photons in the probe field,

JP52e0c

\v0E

DtdtE

Dd2r uEP~r ,t !u2. ~10!

Using this notation, we may write the quantityP(s) deter-mined from the photocount statistics as

P~s!5K :expF22hT

12s (i 51

M S ai†2A12T

Ta i* D

3S ai2A12T

Ta i D G :L expF2

2h~12T!

12s

3S JP2(i 51

M

ua i u2D G . ~11!

The expectation value appearing in the above expressionbe identified, up to a normalization factor, as anM-modequasidistribution function. This equivalence becomes cleawe recall the normally ordered definition of the generalizs-ordered quasidistribution functions@27#,

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M

~ ai†

2g i* !~ ai2g i !D :L . ~12!

After the identification of the parameters we obtain that

P~s!5S p~12s!

2hT D M

WSSA12T

Ta1 , . . . ,A12T

TaM ;

212s2hT

hT D expF22h~12T!

12s S JP2(i 51

M

ua i u2D G .

~13!

The exponential factor multiplying the quasidistributiofunction involves the difference between the average tonumber of photonsJP in the probe field and the sum( i 51

M ua i u2 describing the number of the probe field photothat match the corresponding modes of the signal field. Tfactor results from the part of the probe field that is orthognal @in the sense of spatiotemporal overlap defined by~8!# to the mode functions describing the excited componof the signal beam, and it becomes identically equal to 1the probe field exactly matches theM signal modes of inter-est. The matching condition can be written as

EP~r ,t !5(i 51

M

a iui~r ,t !. ~14!

In such a case of perfect mode matching on the beam spthe quantityP(s) calculated from the photocount statisticsequal, up to a constant multiplicative factor, to the valof a multimode quasidistribution function with thordering 2(12s2hT)/hT, taken at the point@A(12T)/Ta1 , . . . ,A(12T)/TaM#. This result implies asimple recipe for measuring multimode quasidistributifunctions point-by-point. What one needs to do in orderdetermine the value of a quasidistribution function at a scific point, is to prepare the coherent probe field in an apppriate superposition defined by Eq.~14!, and to measure thephotocount statistics of the probe and signal fields combiat an unbalanced beam splitter. The sum over the photocstatistics evaluated according to Eq.~4! yields directly thevalue of the quasidistribution function. The ordering of tmeasured quasidistribution depends on the overall lossethe signal field characterized by the producthT, as well asthe parameters used to evaluate the sum over the coustatistics. The allowed range of the parameters has beendiscussed previously@28# in the context of the statistical uncertainty. It has been shown that in order to keep the statical error bounded, the parameters needs to be taken nonpositive. As in the single-mode case, the highest attainaordering of the measured quasidistribution function is eqto 2(12hT)/hT, which in the limit of ideal lossless detection yields zero, thus corresponding to the detection ofWigner function. An interesting feature of this scheme is th

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there is no need to resolve contributions to the photocostatistics from each of the modes separately; the only obsable that needs to be reconstructed is the sum over thetistics of the total number of photocounts.

One should note that in a realistic situation it is notways possible to measure the complete photocount statisMost of the single-photon detectors used presently, sucavalanche photodiodes operated in the Geiger mode, yonly a binary response telling whether any number of ptons has been registered or not. We can easily specializeprevious considerations to include this case by takinglimit s→21, in which the sumP(s) reduces to

lims→21

P~s!5p0 , ~15!

i.e., the probability of registering zero counts on the dettors. Thus, if only the information about the presence orsence of photons is available, we can reconstruct a quastribution function with the ordering2(22hT)/hT, whichin the limit of lossless detection reduces to21 correspond-ing to the measurement of aQ function.

More generally, one can consider application of a dettion scheme that can resolve the number of photons upcertain value, such as the visible light photon counter@29#,detector cascading@30#, or the loop detector@31#. In thiscase, one is limited to sampling these regions of the phspace where the photon statistics effectively vanishes abthe cutoff value of the used detector. Several examplephoton statistics have been analyzed in detail in Ref.@28#. Ithas been found in certain typical cases that the occurrencstructures in the phase space can be related to destruinterference at the unbalanced beam splitter, resulting inphoton numbers seen by the detectors. These structuresthus be measured using detectors with restricted multiphoresolution.

C. Multiple detectors

The direct scheme for measuring multimode quasidisbution functions can be easily extended to the case wsome of the modes are separated in space. In this case,mode ~or a group of spatially overlapping modes! needs tobe displaced in the phase space by combining it at a bsplitter with a coherent probe field, and then measured ua photon counting detector. For simplicity, let us assumewe are dealing with two separate beams labeledA and B.Such a setting, with the detection procedures satisfyingcality conditions, will serve as a scheme for testing Beinequalities discussed in the subsequent sections of thisper. Generalization of our results to more than two beawill follow in a straightforward manner. The two detectorsour setup yield photocount statisticspA,n andpB,n . We willshow that Eq.~4! applied to the statistics of the total numbof countspn , defined as

pn5 (m50

n

pA,mpB,n2m ~16!

yields the joint quasidistribution function of the signal fiel

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In order to fix the notation, let us assume thatMA andMBmodes are excited in the beamsA and B, respectively, anddenote the corresponding annihilation operators byai andbi . The displacement of the signal field can be performedpreparing the probe field in the form of two beams interfing with the beamsA andB. A calculation completely analogous to the one presented in the preceding section showsP(s) evaluated from the overall count statistics has the flowing form:

P~s!5K :expS 22hT

12s (i 51

MA

@ ai†2A~12T!/Ta i* #

3@ ai2A~12T!/Ta i # D ^ expS 22hT

12s (i 51

MB

~ bi†

2A~12T!/Tb i* !~ bi2A~12T!/Tb i !D :L3expF2

2h~12T!

12s S JP,A2(i 51

MA

ua i u21JP,B

2(i 51

MB

ub i u2D G . ~17!

Herea i andb i are the probe field amplitudes matching tmodes of the signal field. They are defined analogouslyEq. ~9!, with the integration over the detection time and tactive area of the appropriate detectorA or B. Similarly, JP,AandJP,B are the intensities of the two probe beams expresin the photon number units. For simplicity we have assumhere that the two detectors have the same quantum efficieh, and also that both the beam splitters are characterizethe same transmission coefficientT. As before, the expectation value in Eq.~17! can be identified as proportional tomultimode quasidistribution function, and the second exnential factor results from imperfect matching of the profield to the signal modes.

III. PRACTICALITIES

In this section, we will discuss the effect of experimenimperfections that are likely to occur in a practical realiztion of the proposed scheme. First, we shall analyze theof dark counts in the described measurement scheme.ond, we will study on a simple example the effect of impefect matching between the signal and probe fields atbeam splitter.

A. Dark counts

The first effect that needs to be taken into account isdark noise of the detectors. Apart from the photocounts ornating from the measured field, the detector can geneadditional counts. These can result from thermal excitatiin the active material of the photodetector, or from absotion of stray light. We shall make a simplifying assumptioin our analysis that the dark counts are not correlated sta

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tically with the photocounts generated by the field of intereThus we neglect, for example, the phenomenon of afterping in avalanche photodiodes, which consists in generasignal by carriers trapped after detection of preceding ptons. Under this assumption the statistics of detector copn can be represented as a discrete convolution of the sttics of counts originating from the measured fieldpF,n andthe statistics of dark countspD,n ,

pn5 (m50

n

pF,mpD,n2m . ~18!

It is then straightforward to check that the sum overcount statistics factorizes into the product

(n50

` S s11

s21D n

pn5 (m50

` S s11

s21D m

pF,m(n50

` S s11

s21D n

pD,n .

~19!

This expression means that as long as the dark countsuncorrelated with the counts generated by the photons fthe measured fields,P(s) simply becomes rescaled by a costant factor defined only by the dark-count statistics. Thefore in order to include the effect of dark counts in our anasis we only need to multiply the previous formulas by thconstant factor. Furthermore, if we chooses521, this mul-tiplicative factor simply becomespD,0, i.e., the probability ofa zero-dark-count event.

B. Mode mismatch

In a realistic situation, it is virtually impossible to achievideal matching between the signal and probe fields at a bsplitter. Consequently, experimental results will usuallyaffected by imperfect mode matching. In order to analythis effect in quantitative terms and estimate its importanwe will now consider the practical measurement of a sinquasidistribution function. For this purpose, we will assuthat M51, i.e., only one mode is excited in the decompotion of the signal field defined in Eq.~7!, and for simplicitywe shall drop the index labeling its annihilation operatoraand the corresponding mode functionu(r ,t). In the furtheranalysis, it will be convenient to introduce the mode mating parameterj defined by

j5uau2

JP5

2e0c

\v0

U EDt

dtED

d2r u* ~r ,t !EP~r ,t !U2

EDt

dtED

d2r uEP~r ,t !u2. ~20!

It follows directly from the Schwarz inequality that the prameterj defined in the above way lies between 0 and 1. Tvalue j51 corresponds to the perfect overlap betweensignal and the probe modes, whereas the valuej50 meansthat the signal and probe fields are completely orthogona

Using the mode matching parameter, we can writeP(s)as

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P~s!5p~12s!

2hTWSA12T

Ta;2

12s2hT

hT D3expS 2

2h~12T!

12s

12j

juau2D . ~21!

It is seen that in the case of imperfect mode matchingmeasured quasidistribution function is multiplied byGaussian envelope centered at the origin of the phase spThe width of the Gaussian depends on the factor (12j)/j,which grows with the increasing mismatch between the snal and probe fields. Thus, the effect of the mode mismais more severe in the outer regions of the phase space, wthe amplitude of the signal quasidistribution function is supressed by the Gaussian envelope resulting from the mmismatch.

It is interesting to note that the effect of the mode mmatch in our scheme is quite different from that occurringthe balanced homodyne detection@18#. In the latter case, themode matching parameter simply multiplies the overall eciency of the homodyne detector, and is equivalent to adtional losses of the signal field. In contrast, Eq.~21! includesthe quantum efficiency and the mode matching parametetwo nonequivalent ways, and the strength of the effect ofmode mismatch depends on the selected point of the pspace. This difference between balanced and unbalancedmodyning can be easily understood by comparing the rtive intensities of the signal and local oscillator fields at tdetectors. In the balanced case, only the part of the sigfield matching the shape of the local oscillator affects notiably the photocount statistics. In the case of unbalancedmodyning, both the signal and local oscillator fields hacomparable intensities at the exit of the superposing besplitter and contribute with similar strengths to the coustatistics.

In a practical situation, it is convenient to relate the momatching parameterj to the interference visibility on thebeam splitter used to combine the signal and probe fieSuppose that we can replace the signal field of interest bcoherent stateuaS& prepared in exactly the same spatioteporal mode. A simple calculation shows that the averalight intensityJout received by the detector, expressed in tphoton number units, is given by

Jout5~12j!TuaSu21uAjTaS2A~12T!jau2. ~22!

The minimum intensity of the output light that can be otained by varying the phase and the amplitude of the prfield while keepingaS fixed is given by

Jmin5~12j!TuaSu2. ~23!

If we now change the phase of the probe field byp, this willmaximize the output intensity for the fixed absolute valuethe probe field amplitude, thus yielding

Jmax5~113j!TuaSu2. ~24!

3-6

et

thalapeen

in

radetho

aucaOo

thnn

. 2nto

wipuminoti

m

ly topo-

o-of

t arengherh ass.

sta-ex

atdes

tionali-mlynseer-ame, ofnge ofun-

e-yze

is-ndnrocan

kingm

cess,icalali-bili-te,

y ab-0

tionser-

e

r-

fante


Consequently, the interference visibilityy defined in thestandard way can be expressed in terms of the paramused thorough our discussion as

y5Jmax2Jmin

Jmax1Jmin5

2j

11j. ~25!

Inverting this relation we obtain thatj5y/(22y). Thus, themeasurement of the interference visibility based onmethod sketched above can yield an estimate for the quof the mode matching in the experimental setup. Thisproach has been used in Ref.@12# to analyze the shape of thexperimentally measured Wigner function of a coherstate.

IV. TESTING BELL’S INEQUALITIES

We will now specialize the general theory developedsections herebefore to the case when two light modes inentangled state are sent towards a pair of spatially sepaunbalanced homodyne detectors. It has been suggesteprevious theoretical works@13,14# that such a setup can bused to test Bell’s inequalities. In the proposed approach,standard spin measurements are replaced by dichotomicservables derived from the photocount statistics, and theiliary coherent fields provide adjustable parameters thatbe selected by the observers under locality conditions.goal here will be to perform a complete realistic analysisthe proposed scheme, including all the practical factorsare likely to affect the outcome of the proposed experime

Let us first summarize briefly the unbalanced homodysetup proposed to test Bell’s inequalities, depicted in FigTwo entangled and spatially separated light beams fall ounbalanced homodyne detectors composed of phocounters preceded by high-transmission beam splittersauxiliary coherent fields entering through the sideway inports. The operation of such a setup can be easily reforlated in terms of a standard scheme for testing Bell’sequalities with measuring apparatuses that provide dichmic outcomes. The binary variable we will be interested inwhetherany number of photons has been registered byphoton counting detector, without resolving the actual nu

FIG. 2. Two entangled and spatially separated light beamsonto unbalanced homodyne detectors composed of photon coupreceded by high-transmission beam splitters with auxiliary cohent fields entering through the sideway input ports.

04380

ers

eity-

t

anted

in

eb-x-n

urfatt.e.

ton

thtu--o-sa-

ber of photons. Such a measurement corresponds formala standard test whether a particle has passed through alarization analyzer. It is important to note that this dichotmic measurement can be feasibly realized with the helpavalanche photodiodes operated in the Geiger mode thanot capable of providing the number of photons triggerithe avalanche signal. At the same time, we can utilize otfavorable characteristics of avalanche photodiodes, sucrelatively high quantum efficiency and low dark-count rateThe pair of noncommuting observables at each receivingtion will be established by selecting two different complamplitudes of the coherent field added to the signal beamthe beam splitter. The choice of the coherent state amplituthus corresponds to two possible orientations of polarizaanalyzers used in standard experiments on Bell’s inequties. In each experimental run, the observers select randothe amplitudes of the coherent fields and record the respoof their photon counting detectors. In practice, all the cohent light beams in the setup need to be derived from the ssource in order to ensure fixed phase relationships. Thiscourse, fully complies with the assumptions underlyiBell’s inequalities as long as the amplitude and the phasauxiliary coherent beams are adjusted by the two partiesder locality conditions. We will now discuss in detail corrlation functions constructed from these data and analtheir nonlocal character.

In order to formulate our discussion in terms of quasidtribution functions and utilize results derived in Secs. II aIII, we shall attribute the value11 to events when a photocounting detector did not click and 0 when any nonzenumber of photons was registered. In such a case, wespecialize the results from the sections herebefore by tathe value of thes parameter used in the definition of the suover the photon-count statistics in Eq.~4! equal tos521.Then the whole sum over the photon-count statistics redusimply to p0, i.e., the probability of observing zero photonwhich corresponds exactly to our assignment of numervalues to experimental outcomes. Our test of Bell’s inequties is based on measuring the joint and marginal probaties of such ‘‘photon silence’’ events, which we shall denorespectively, byQ(a,b) andQ(a) or Q(b). Herea and bare complex amplitudes of the coherent fields rescaled bcertain factor whose value we will discuss later. As the proability of a no-count event is obviously bounded betweenand 1, the nonlocal character of the measured correlafunctions can be tested with the help of the standard ClauHorne inequality@32#,

21<CH<0, ~26!

where the combinationCH is constructed with the help of thformula

CH5Q~ a1 ,b1!1Q~ a1 ,b2!1Q~ a2 ,b1!2Q~ a2 ,b2!

2Q~ a1!2Q~ b1!. ~27!

Herea1 ,a2 andb1 ,b2 denote the two settings of the coheent fields used at the respective detectors.

llersr-

3-7

ale

we

ot

.h

.rsn-veases

-Ia

s

il-dth

con

tl tth

It isfec-

ansex-li-s ofles

con-di-

zedex-

ngn-

edli-

co-is

entingi-too-

notock

ent.ob-put

ncann-

gen-twose-m-g aon.plit-tele-ring

es


We will now derive expressions for the joint and marginprobabilities of ‘‘photon silence’’ events that include possibimperfections of the experimental setup. For simplicity,shall assume that the excited parts of the fields travelingthe homodyne detectors can be described by single-mannihilation operatorsa and b. The expression for the joinzero-count event is then simply given by Eq.~17! with s521 and MA5MB51. Following the discussion in SecIII B, it will be convenient to introduce a mode mismatcparameter defined by

j5uau2

JP,A5

ubu2

JP,B, ~28!

assumed to be the same in the both arms of the setupaddition, we will also include dark counts on the detectoAs we are interested only in the probability of zero-couevents, it follows from Eq.~18! that the only relevant parameter ispD,0, i.e., the probability that zero dark counts haoccurred. For brevity, we will denote this quantity simplypD . Combining all the above elements yields the full exprsion for the probability of a joint ‘‘photon silence’’ event,

Q~ a,b !5pD2 ^:exp@2hT~ a†2a* !~ a2a !#

^ exp@2hT~ b†2b* !~ b2b !#:&

3expS 2hT12j

j~ uau21ubu2! D

5S ppD

hD 2

WABS a,b;222h

hD

3expS 2h12j

j~ uau21ubu2! D , ~29!

whereWAB(a,b;s) is the two-modes-ordered quasidistribution function describing the entangled signal state.the above formula, we have introduced several tilded qutities that will be convenient as independent parameterthe further discussion. First,a5A(12T)/Ta and b5A(12T)/Tb denote the rescaled amplitudes of the auxiary coherent fields. The producth5hT describes the combined losses of the signal field at a beam splitter and atector, and can be easily generalized to include also osources of losses experienced by the signal beams. Wefind in a similar manner that the zero-count probabilitythe detectorA is equal to

Q~ a !5ppD

hWAS a;2

22h

hD expS 2h

12j

juau2D ,

~30!

and analogously for the detectorB. Thus, in general the joinand marginal photon silence probabilities are proportionacorresponding quasidistribution functions that describe

04380

l

tode

In.t

-

nn-in

-

e-eran

oe

properties of the entangled state used in the scheme.seen that the three parameters related to different impertions: overall efficiencyh, mode mismatch parameterj, andzero dark-count probabilitypD enter these expressions innonequivalent way. As a lower value of any of them meamore imperfections of the experimental setup, one canpect a universal behavior that the violation of Bell’s inequaties becomes more pronounced with the increasing valuethese parameters. This will be confirmed in all the exampthat we will study later.

In the following two sections, we will discuss two specifiquantum states that can be produced with the help of sptaneous parametric down-conversion: a one-photon statevided on a 50:50 beam splitter and a two-mode squeevacuum state. The states used in our discussion formallyhibit entanglement when written in the Fock basis. Beiaware of conceptual difficulties related to this kind of etanglement@33# we shall focus on the violation of Bell’sinequalities as a signature of nonlocality that can be verifiin practice. In order to reveal a violation of Bell’s inequaties, it is necessary to use measurements that probe theherence between different Fock terms. In particular, itwidely recognized that a single particle alone is not sufficito demonstrate quantum nonlocality using particle countdetectors@34#. This is why we need local supplies of addtional photons in the form of auxiliary coherent fieldsdetect nonlocal correlation functions. Similarly, for a twmode squeezed vacuum state, counting photons alone issufficient to probe the coherence between different Fstates in the superposition and to demonstrate entanglemWe need auxiliary coherent fields to make the detectedservables sensitive to the off-diagonal elements of the indensity matrix in the Fock basis.

V. SINGLE-PHOTON STATE

As the first example we will consider a single photodivided at a 50:50 beam splitter. The single-photon statebe produced in a controlled way with the help of the spotaneous down-conversion process in ax (2) nonlinear me-dium assisted by the conditional measurement. Photonserated in this process are always emitted in pairs towardswell-defined directions that are determined by the phamatching conditions inside the nonlinear crystal. One coponent of such a pair can be used as a trigger yieldindefinite information about emission of the second photThe latter can be consequently sent to the 50:50 beam ster. From the formal point of view the output two-mode staof the field leaving the beam splitter resembles a singphoton Fock state entangled with a vacuum mode entethrough the unused port of the beam splitter,

uC&51

A2~ u1&Au0&B1u0&Au1&B). ~31!

The output modes have been denoted here with the indicAandB. We will be interested in correlation functions~29! and

3-8

l

n

-


FIG. 3. Maximized Clauser-Horne combination CH for asingle photon divided at a beamsplitter as a function of the overal

lossesh and the mode mismatchj. Plots for noiseless detectiopD51 ~left! and with a dark-countrate pD50.99 ~right!. The thicklines represent the boundCH.0imposed by local-realistic theories.

rsar

thra

faets

n

nsivener-

de-rall

vio-g.

tical

bi-

en

ntionthein

nothes of

es,srees

c-hillbalhetherum.ap-

our

their marginals~30! of photon silence events on the detectomeasured as a function of the amplitudes of the auxilicoherent beams. Following Eqs.~30! and ~29! we need toevaluate first the generalized quasidistribution function ofstate~31! and its one-mode marginal. After simple algebwe obtain

WABS a,b;222h

hD

5S h

pD 2

^Cu:e2h((a†2a* )(a2a)1(b†2b* )(b2b)):uC&

5S h

pD 2S 12h1

h2

2ua1bu2D e2h(uau21ubu2), ~32!

with the marginal, single-mode quasidistribution

WAS a;222h

hD 5

h

p^Cu:e2h(a†2a* )(a2a):uC&

5h

pS 12

h

21

h2

2uau2D e2huau2. ~33!

Multiplying these expressions by the state-independenttors given in Eqs.~29! and ~30!, we can easily calculate thjoint and marginal probabilities of the photon silence even

Q~ a,b !5pD2 S 12h1

h2

2ua1bu2D expF2

h

j~ uau21ubu2!G ,

~34!

Q~ a !5pDS 12h

21

h2

2uau2D expS 2

h

juau2D . ~35!

04380

y

e

c-

,

As a function ofa and b, these results are the well-knowhat-shaped distributions.

We will now address the question under what conditiothe correlations between detecting zero-count events gby the above formulas lead to the violation of the ClausHorne inequality given in Eq.~26!. As we have discussedearlier, there are three nonequivalent parameters thatscribe imperfections of the experimental setup: the ovedetection efficiencyh, the mode-matching parameterj, andthe zero dark-count ratepD . We would like to identify therange of these parameters that enables one to observe alation of Bell’s inequalities using unbalanced homodyninThe parameters that can be freely adjusted in a pracsetup are the values of the four coherent displacementsa1 ,a2 , b1, andb2 used to construct the Clauser-Horne comnation given in Eq.~27!. We shall therefore optimize themfor a specified set ofh, j, andpD in order to find the maxi-mum possible violation of the Clauser-Horne under givexperimental constraints.

The task of finding the optimal quadruplet of coheredisplacements is a nonlinear multidimensional optimizatproblem and we have approached it numerically withhelp of the standard downhill simplex algorithm describedRef. @35#. It is obvious from the form of Eqs.~34! and ~35!that the value of the Clauser-Horne combination doeschange when all the four amplitudes are multiplied by tsame phase factor. We can therefore assume with no losgenerality that the imaginary part of one of the amplitudwhich we will take to bea1, is equal to zero. This leaves uwith seven independent real numbers that form the fworking parameters for the downhill simplex algorithm. Atypical when optimizing multidimensional nonlinear funtions that may possess multiple local extrema, the downsimplex algorithm does not guarantee to provide the gloextremum of the optimized function. In order to improve tconfidence of our results, we have therefore restartedalgorithm several times with different initial conditions foeach optimization problem, and selected the best extremThe consistency of the obtained results suggests that theplied procedure was adequate to the complexity ofproblem.

3-9

-

e

e


FIG. 4. Coherent amplitudesmaximizing violation of theClauser-Horne inequality for thenoiseless detection (pD51) as a

function of the overall losseshand the mode mismatchj. Formost of the area of violation it issufficient to use just two real am

plitudes ~top a15b1 and a2

5b2). In a small triangular areaseparated by a dashed line thmaximum violation is obtained forcomplex amplitudes, shown in thbottom seven figures.

r-m-

un

bn

ha

ticse

foeeth

rn

nddi---off

effi-in

ty.thearkthetheveis

:g.neser-m-in

In Fig. 3 we depict the optimum violation of the ClauseHorne inequality as a function of the imperfection paraetersh, j, andpD . As the dark-count rates of silicon avalanche photodiodes are relatively low, we have chosenstudy only two rather extreme values of the zero dark-coprobability: the ideal casepD51 andpD50.99 correspond-ing to 1% dark-count rate, which is well above what canexpected from modern detectors. It is seen that the differebetween these two cases is rather minor. Instead, wefocused our attention on the detection efficiencyh and themode mismatchj as more critical parameters in a realissituation. The graphs shown in Fig. 3 were obtained bylecting a 50350 grid in the plotted range ofh andj, and byperforming the optimization procedure described aboveeach point of the grid. The contours in the plots have bdrawn using interpolation, and the thick line separating‘‘nonlocal’’ region has been offset by 1023 in order to avoiddisplay of the effects of numerical errors.

It is seen that the maximum value of the Clauser-Ho

04380

-

tot

eceve

-

rne

e

combination, obtained for perfect detection efficiency amode matching, lies above 0.15. As expected, this valueminishes whenh and/orj decrease, thus implying more imperfections in the setup. One can observe a clear tradebetween these two quantities: the worse the detectionciency is, the better mode matching must be achievedorder to obtain a violation of the Clauser-Horne inequaliThe minimum detection efficiency needed to observeviolation assuming perfect mode matching and zero dcounts is about 84%. In Fig. 4 we show the values ofcoherent displacements that maximize the value ofClauser-Horne combination. Numerical calculations hashown that in most of the area the maximum valueachieved by amplitudes that are real and pairwise equala1

5b1 and a25b2. These two values are depicted in Fi4~a!. Only in a small triangular region in the parameter plathe situation becomes more complicated and the ClauHorne combination is maximized by displacement paraeters with nontrivial imaginary parts. This region is shown

3-10

of


FIG. 5. Optimal violation ofthe Clauser-Horne inequality for asqueezed state in the casenoiseless detectionpD51 ~left!and with a dark-count ratepD

50.99~right!, as a function of the

overall lossesh and the modemismatchj.

ngf eh

thinneget

onthtwav

tt

caleaar

ith

onthe

r-

ee-

rne

re

ted

the

r-

:r-

adeo-t, theer-de, wece-Weiono-

areral

ood

detail in seven graphs in Fig. 4~b!.

VI. TWO-MODE SQUEEZED VACUUM STATE

In this section we will discuss another possibility of usispontaneous parametric down-conversion as a source otanglement for homodyne tests of Bell’s inequalities. Tsingle-photon state discussed previously was obtainedconditional detection on the down-conversion output inweak conversion regime, when the probability of generatsimultaneously two or more photon pairs can be safelyglected. In a general case, the two-mode quantum stateerated in spontaneous parametric down-conversion is ofform

uF&51

coshr (n50

`

tanhnr un&Aun&B , ~36!

where r is a squeezing parameter including informatiabout the pump laser field, interaction time, thickness ofnonlinear crystal, etc. We have assumed here that thebeams are generated in single spatiotemporal modes tring towards directionsA andB.

The state given in Eq.~36! clearly exhibits entanglemenin the Fock basis when the two modes are chosen asseparate subsystems. It is therefore interesting to studyrelations between photon silence events on two spatiseparated unbalanced homodyne detectors fed by the bA and B. We shall analyze whether such correlationscapable of violating the Clauser-Horne inequality~26!. Therelevant probabilities are calculated in the Appendix, wthe final results given by

Q~ a,b !5pD

2

11h~22h !sinh2rexpF2S 11h sinh2r

11h~22h !sinh2r

112j

j D h~ uau21ubu2!1h2sinhr coshr

11h~22h !sinh2r

3~ ab1a* b* !G ~37!

04380

n-ebyeg-n-

he

einel-

heor-lymse

for the probability of a joint photon silence event and

Q~ a !5pD

11h sinh2rexpF2S 1

11h sinh2r1

12j

j D huau2G~38!

describing the marginal probability of a zero-count eventa single detector. Using these expressions we can buildcombination ~27! and check in what regime the ClauseHorne inequality can be violated.

In order to find the optimal violation of the Clauser-Horninequality we have followed the numerical procedure dscribed in Sec. V. It is easy to note that the Clauser-Hocombination is invariant with respect to multiplyinga1 , a2

and b1 , b2 by conjugate phase factors; we can therefoassume with no loss of generality that the amplitudea1 ispurely real. An additional parameter that we have subjecto numerical optimization is the squeezing parameterr, as itshould be possible to tune it rather easily by adjustingintensity of the beam pumping the nonlinear medium.

In Fig. 5 we depict the minimized value of the ClauseHorne combination as a function of the overall efficiencyhand mode matchingj, for two values of the dark-count ratethe ideal casepD51, and the 1% dark-count probability coresponding topD50.99. Once again one can observetrade-off between the detection efficiency and the momatching. However, it is interesting to note that when a twmode squeezed state is used as a source of entanglemenminimum detection efficiency needed to violate the ClausHorne inequality is about 71% in the case of perfect momatching and absence of dark counts. For completenessshow also the optimized values of the coherent displaments in Fig. 6 and the squeezing parameter in Fig. 7.have found numerically that the Clauser-Horne combinatis minimized by pairs of coherent displacements with oppsite signs:a152b1 and a252b2. It is also interesting tonote that optimal values of the squeezing parameterrather moderate, which means that only the first seveterms in the Fock-basis expansion in Eq.~36! are relevant.This observation may be of experimental relevance, as g

3-11

-


FIG. 6. Coherent amplitudesmaximizing violation of theClauser-Horne inequality for thenoiseless detection (pD51). It issufficient to use just two real am

plitudes a152b1 and a25

2b2.

lddsatd

mioiciasim

ll’sanle

fd

t

d.g aisvio-

e

tonthattem--

ve alis-de-li-

orybyco-nalm-ef-the

ark

mode matching between the signal and local oscillator fiemay be more difficult to achieve for strongly squeezed fiel

As it is well known, the two-mode squeezed vacuum stgenerated in spontaneous parametric down-conversion isscribed by a positive-definite Wigner function. Quantustates of this form have been recalled on several occaswhen discussing to what extent the quantum-mechanphase-space formalism can serve as a local hidden-varmodel. As this issue has generated some rather confucomments in previous works, it is instructive to analyze frothis point of view our homodyne scheme for testing Beinequalities. For this purpose, let us use the Wigner formism to rewrite the probability of a joint photon silence eveQ(a,b) to a form that resembles a local hidden-variabmodel,

Q~ a,b !5E d2l1 d2l2 r~a;l1!r~b;l2!W~l1 ,l2!.

~39!

Here r(a;l1) and r(b;l2) are Wigner representations othe observables corresponding to no-count events on the

FIG. 7. Squeezing parameterr maximizing violation of theClauser-Horne inequality for the noiseless detection (pD51).

04380

s.ee-

nsalbleng

l-t

e-

tectors A and B, and W(l1 ,l2) is the positive-definiteWigner function of the stateuF&. An easy calculation com-bining Eq.~29! with Eq. ~A1! from the Appendix shows thar(a;l1) is explicitly given by

r~a;l1!52pD

22hexpS 2

2h

22h~l12a !22h

12j

juau2D

~40!

with an analogous expression forr(b;l2). In order to inter-pret r(a;l1) andr(b;l2) as local realities that are defineby hidden variablesl1 andl2, the expression given in Eq~40! must be bounded between 0 and 1 as representinlocal probability of a zero-count detection event. Itstraightforward to see that the upper bound is in generallated: in the perfect case ofh5j5pD51 the maximumvalue attained byr(a;l1) is equal to 2. Consequently, thWigner representation given in Eq.~39! cannot be usuallyinterpreted as a local hidden-variable model for the phosilence events. It is clearly seen from the above examplepositivity of the Wigner function describing a composiquantum system is in principle no obstacle to use it for deonstrating a violation of Bell’s inequalities. An equally important factor is whether the measured observables haWigner representation that can be interpreted in local reatic terms. Only when both these conditions are met, thetected correlation functions have to satisfy Bell’s inequaties.

VII. CONCLUSIONS

In this paper, we presented a complete multimode theof measuring quantum optical quasidistribution functionsphoton counting. We demonstrated that by constructing aherent field whose excited modes match pairwise the sigmodes of interest, it is possible to scan point-by-point coplete quasidistribution functions. Our analysis includedfects of experimental imperfections, such as losses ofsignal field, nonunit mode matching, and accidental dcounts.

3-12

mo

n-assieioionfoperal

twa

th-od

filslopa

li-thenlapthth

ceryn

te-esarn

esmOc

glethan

s-ioiffiae

threTh, ia

thpo

vio-al-

factdi-sug-efulsedd

es-l.RI

anwl-ionthe

tetonm

rty-l-l

u-d,


As an interesting application of the developed formaliswe performed a feasibility study of using unbalanced homdyning for testing Bell’s inequalities. As a source of etanglement, we analyzed two states whose generation bon the process of spontaneous parametric down-converseems presently to be most practical. We found the rangexperimental parameters that enable one to observe a vtion of Bell’s inequalities without the standard postselectprocedure. The minimum detection efficiency requiredsuch a loophole-free test, in the absence of any other imfections, is 84% for a one-photon Fock state, and 71% fotwo-mode squeezed vacuum state. Especially the latter vseems to be within the reach of current technologies@36#.Our calculations demonstrated a clear trade-off betweenparameters describing experimental imperfections thatlikely to determine the performance of a realistic setup:detection efficiency and the mode mismatch. Such a tradeis easily understandable: a natural way to improve the mostructure of the signal field is to perform spatiotemporaltering, which, however, inevitably generates excess lossethe signal field. This effect underlines the need to devesources of down-converted radiation with a controlled stiotemporal structure@37#.

Though it does not affect directly tests of Bell’s inequaties, it is worthwhile to note that the optical states used inscheme studied here do not share problems that are currinherent to the standard down-conversion sources of poization entangled states. In the latter case, the entangledton pairs are generated only with a small probability pertime slot defined by the shape of the laser pulse pumpingnonlinear medium. Thus, the actual state of the produlight is given, up to the first order of the perturbation theoby a small two-photon contribution superposed with a strovacuum component@38#. Within current technology it is nopractical to perform a nondestructive test for the presencphotons thus selectinga priori the photon-pair term. Consequently experiments utilizing this type of entanglement,pecially in quantum information processing applications,usually based on the postselection of the detection eveand the strong vacuum component to the produced statremoved by retaining only the cases when a sufficient nuber of photons has been registered by the detectors.might hope that this difficulty could be overcome by produing multiple photon pairs and using the methods of entanment swapping and conditional detection to preparemaximal polarization entanglement in the event-ready mner @39#. However, there exists a rather general proof@38#suggesting that this route is highly difficult, if not imposible, within presently available means. Practical realizatof the states used in our study is not affected by these dculties. The single-photon state divided on a 50:50 besplitter can be produced by conditional detection performon the down-conversion output. Receiving signal fromtrigger detector tells us unambiguously that the requisingle-photon state has been prepared with high fidelity.preparation of the second state that was considered herethe two-mode squeezed vacuum state, does not requirepostselection at all. The state produced from vacuum inprocess of nondegenerate parametric down-conversion

04380

,-

edonofla-

rr-aue

oreeffal-ofp-

etlyr-

ho-eed

,g

of

-ets,is-

ne--e-

n-

mdede.e.,nyes-

sess photon-number entanglement which is sufficient tolate Bell’s inequalities. Of course, this feature is not criticin tests of Bell’s inequalities which can rely validly on random sources of entangled photon pairs. However, thethat we can reveal quantum nonlocality by performingchotomic measurements on continuous-variable systemsgests that continuous-variable entanglement may be a usresource in quantum information processing protocols baon binary logic, and our ability to produce it on demancould be its important advantage.

ACKNOWLEDGMENTS

We thank M. Zukowski for valuable comments on thmanuscript. We would like to acknowledge useful discusions with I. A. Walmsley, S. Wallentowitz, and W. VogeThis research was supported by ARO-administered MUGrant No. DAAG-19-99-1-0125. A.D. thanks the EuropeScience Foundation for financial support. We also acknoedge the financial contribution of the European Commissthrough the Research Training Network QUEST and ofKBN Grant No. PBZ-KBN-043/P03/2001.

APPENDIX

Direct calculation of the quasidistributions of the sta~36! is not as straightforward as in the case of a single phoentangled with vacuum. Therefore we will calculate theusing another method. First we will recall a useful propeof a general,M-dimensional family of quasiprobability distributions: from anys-ordered quasidistribution one can caculate any others8-ordered one using the following integraformula:

W~a1 , . . . ,aN ;s8!5S 2

p~s2s8!D M E d2b1•••d2bM

3expS 22

s2s8(i 51

M

ua i2b i u2D3W~b1 , . . . ,bM ;s!. ~A1!

A two-modeQ function corresponding to the quasidistribtion ordering parameters521 can be very easily calculatefor the NOPA state directly from the Fock representation

WAB~g,d;21!51

p2u^gu^duF&u25

1

p2cosh2rexp@2ugu2

2udu21tanhr ~gd1g* d* !#. ~A2!

We will also need its marginal,

WA~g;21!5E d2d WAB~g,d;21!

51

p cosh2rexpS 2

ugu2

cosh2rD . ~A3!

3-13

e

ua-

rs-


Inserting Eq.~A2! into Eq.~A1! specialized to the two-modcase withM52 one easily obtains that

WABS a,b;222h

hD 5

h2

p2@11h~22h !sinh2r #

3expS 2h1h2sinh2r

11h~22h !sinh2r~ uau2

1ubu2!1h2sinhr coshr

11h~22h !sinh2r~ ab

1a* b* !D . ~A4!

t

-

rd,

r,

r,.

ci--

H.

s

tt.

04380

Inserting the above formula into Eq.~29! yields the probabil-ity of the joint photon silence event given in Eq.~37!. Ananalogous calculation yields the marginal single-mode qsidistribution function,

WAS a;222h

hD 5

h

p

1

11h sinh2rexpS 2

h uau2

11h sinh2rD ,

~A5!

which after multiplying by the state-independent factogiven in Eq. ~30! gives the marginal probability of a nocount event given in Eq.~38!.

v..

tt.

k,

k,

ev.

W.

pl.

.

.

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