r,

PHYSICAL REVIEW A, VOLUME 63, 062102

Homodyne Bell’s inequalities for entangled mesoscopic superpositions

Andrzej DraganInstytut Fizyki Teoretycznej, Uniwersytet Warszawski, Hoz˙a 69, PL-00-681 Warszawa, Poland

Konrad BanaszekCenter for Quantum Information and Rochester Theory Center for Optical Science and Engineering, University of Rocheste

Rochester, New York 14627~Received 28 October 2000; published 8 May 2001!

We present a scheme for demonstrating the violation of Bell’s inequalities using a spin-12 system entangled

with a pair of classically distinguishable wave packets in a harmonic potential. For electromagnetic fields, suchwave packets can be represented by coherent states of a single radiation mode. The proposed scheme involvesstandard spin-12 projections and continuous measurements of the position and the momentum of the harmonic-oscillator system, which for a radiation mode can be realized by means of homodyne detection. We discusseffects of imperfections, including nonunit efficiency of the detectors, and analyze a direct link between thevisibility of interference and the violation of Bell’s inequalities in the described scheme. We also point out therole of Bell’s inequalities as a sensitive probe for decoherence effects.

DOI: 10.1103/PhysRevA.63.062102 PACS number~s!: 03.65.Ta, 42.50.Dv

gpl

edomphein

kradpi

c

ics,-fic

inds

t obe-li

tionlysisandis-hesethen-een

al-w-arein-er

oftor

s theourd by-ve

pti-

calionich

ribedt-o-as

ura-r

I. INTRODUCTION

The quantum-mechanical superposition principle brinunexpected and counterintuitive consequences when apto macroscopic systems. Presumably the most famousample is Schro¨dinger’s cat, which remains half-alive anhalf-dead while entangled with a decaying radioactive at@1#. Recent experimental advances have opened up newsibilities to study the superposition principle beyond tpurely microscopic domain. It is now possible to producea laboratory states of the type

uK&51

A2~ u↑& ^ ua&1u↓& ^ u2a&), ~1!

whereu↑&, u↓& are two orthogonal states of a spin-12 system,

and ua&, u2a& are two distinguishable coherent wave pacets of a harmonic oscillator. Such states have been genefor a trapped ion@2# and a microwave cavity field entanglewith an atom@3#. They can be considered as mesoscoequivalents of the example used by Schro¨dinger in his origi-nal argument. The entangled states defined in Eq.~1! areclosely related to the issue of generating and detectingherence between classically distinguishable states@4–6#.

In this paper, we show how states described by Eq.~1!can be used to test incompatibility of quantum mechanwith local realism. Specifically, we derive Bell’s inequalitiewhich are violated by the Schro¨dinger cat states. These inequalities are based on the continuous measurements osition and momentum observables for the harmonoscillator subsystem, and standard projections for the sp1

2

subsystem. An interesting feature of our proposal is thattection of continuous variables having a well-defined clascal analog allows one to investigate the macroscopic limiviolating Bell’s inequalities by the entangled states givenEq. ~1!. We demonstrate that in the limit of large wavpacket amplitudes, a substantial violation of Bell’s inequa

1050-2947/2001/63~6!/062102~6!/$20.00 63 0621

siedx-

os-

-ted

c

o-

s

po---e-i-fy

-

ties is possible, provided there is ideal noise-free detecand lack of decoherence. We also perform a general anaof the proposed scheme, including imperfect detectiondissipation, which gives a quantitative description of the dappearance of nonlocal phenomena in the presence of tdeleterious effects. In particular, our analysis shows thatviolation of Bell’s inequalities in the proposed scheme vaishes at the same rate as the visibility of interference betwthe two distinct wave packets given by the statesua& andu2a&. This result illustrates the close link between nonlocity and quantum coherence. As will be clear from the folloing calculations, all these features are universal, i.e., theyindependent of the particular form of the wave packetsvolved in the superposition. We also point out that in ordto demonstrate the violation of Bell’s inequality, only onethe two measurements applied to the harmonic-oscillasubsystem needs to have microscopic resolution, whereasecond one is relatively insensitive to losses. Thus,scheme provides another example of a situation describeYurke and Stoler@7#, whose proposal for observing the violation of local realism employed a combination of sensitiand insensitive detectors.

For concreteness, we shall consider here a quantum ocal realization of the entangled statesuK&. In the case of aradiation mode, which will serve in this paper as a physirealization of the harmonic-oscillator subsystem, positand momentum correspond to a pair of quadratures, whcan be measured with the help of homodyne detection@8#.The imperfect measurement of quadratures can be descby an efficiency parameterh. This parameter can be straighforwardly generalized to include interaction of the electrmagnetic field with an external environment, which servesa standard model for decoherence@9–11#. The quantum op-tical context will be used here just to fix the notation, and ocalculations retain validity for an arbitrary physical realiztion of a spin-12 particle entangled with a harmonic-oscillatosystem.

©2001 The American Physical Society02-1

e

hIVe-ope-o

a

staonm,rtht

th

nc

erac-stt

esti

-ha

s.dth

eiono-r-

buraio

tive

onheally

op-ringitive

a-tiched in.ure-alss

byes

si-

ANDRZEJ DRAGAN AND KONRAD BANASZEK PHYSICAL REVIEW A 63 062102

This paper is organized as follows. First, in Sec. II, wpresent a simple heuristic idea behind the constructionBell’s inequalities for mesoscopic superposition states. Tidea is elucidated in quantitative terms in Sec. III. Sectiondiscusses the violation of Bell’s inequality including the ralistic case of losses. In Sec. V, we consider the macrosclimit of the proposed scheme. Finally, Sec. VI briefly rviews some of the experimental aspects, and Sec. VII ccludes the paper.

II. HEURISTIC CONSIDERATIONS

We shall start by giving a simple heuristic argument thmotivated us to formulate Bell’s inequalities for Schro¨dingercat states. For concreteness, we shall assume that theua& andu2a& describe two Gaussian wave packets in a hmonic potential centered around dimensionless positiA2a and2A2a, respectively, with zero average momentuand ground-state widths. From the following discussionwill be clear that the violation of Bell’s inequalities in ouscheme is completely insensitive to the specific form ofwave packets. The statesua& andu2a& have a scalar producequal to^2aua&5exp(22a2), and for sufficiently largeathey can be considered as approximately orthogonal. Incase, one can establish a formal analogy@12# between thestateuK& and the singlet state of two spin-1

2 particles used inBell’s original argument, based on the correspondeua&°u↓& and u2a&°2u↑ &. Following this analogy, in or-der to violate Bell’s inequalities one should be able to pform two noncommuting-measurements in the subspspanned byua& and u2a&, which would correspond to projecting the spin onto two different directions. As the firmeasurement on the harmonic-oscillator subsystem, lesimply choose the projection in the basis$ua&,u2a&%. Thismeasurement can be effectively accomplished by the msurement of position: if the distance between the centerthe wave packets is much larger than their spatial extent,sign of the position variable almost unambiguously discrimnates between the statesua& and u2a&. As the second measurement on the harmonic-oscillator subsystem, we stake the projection in the basis of the superpositions:

uC6&51

AN6

~ ua&6u2a&), ~2!

where N652(16e22a2) are the normalization constant

As depicted in Fig. 1, these two superpositions generatetinct interference patterns in the momentum distribution:location of the maxima for the stateuC1& corresponds to theminima for the stateuC2&, and vice versa. Consequently, wcan approximately discriminate between the superposituC1& and uC2& by checking whether the result of the mmentum measurement falls within the vicinity of the inteference fringes either for the stateuC1& or uC2&. Of course,such discrimination is imperfect as the momentum distritions partially overlap; nevertheless, we shall demonstthat the error rate involved is low enough to enable the vlation of Bell’s inequalities.

06210

ofis

ic

n-

t

atesr-s

it

e

is

e

-e

us

a-ofhe-

ll

is-e

s

-te-

III. QUANTITATIVE ANALYSIS

Let us now discuss the idea sketched above in quantitaterms. As a concrete physical realization, we will takeua&and u2a& to be two coherent states of a single radiatimode, witha assumed to be real. The optical analog of tposition and momentum observables is a pair of canonicconjugated quadratures defined in general asxu5(eiua†

1e2 iua)/A2. Two values of the phase—u50 andu5p/2—correspond to the position and the momentumerators, respectively. The standard technique for measuquadratures is homodyne detection, described by the posoperator-valued measure@13#:

H~x;u!51

Ap~12h!expS 2

~x/Ah2 xu!2

1/h21D , ~3!

whereh is the detection efficiency. More generally, the prameter h can include dissipation of the electromagnefield generated by an interaction with an environment. Tperfect noise-free measurement of quadratures is obtainethe limit h51, whereash,1 describes nonideal detectionSuch nonideal detection corresponds to a blurred measment of position and momentum with finite resolution equto A(1/h21)/2, expressed in the canonical dimensionleunits of the harmonic oscillator.

The statesuC6& defined in Eq.~2! generate the followinginterference patterns foru5p/2:

K C6UHS x;p

2 D UC6L5

2

ApN6

e2x2@16e22(12h)a2

cos~A8hax!#. ~4!

It is easily seen that within the Gaussian envelope giventhe factore2x2

, the spacing between the interference fringis given byT5p/A2ha. For the stateuC1&, the interfer-

FIG. 1. Plot of the momentum distributions for the superpotions uC1& ~solid line! and uC2& ~dashed line!, assuming perfectnoise-free measurement anda56. The location of maxima for onestate corresponds to the location of minima for the other one.

2-2

ifyelngioto

tio

ur

iohe

-

r

th

ed

atm

e--deeal

n-

this

-

ellthe

the

ion,

re

-

in

HOMODYNE BELL’S INEQUALITIES FOR ENTANGLED . . . PHYSICAL REVIEW A63 062102

ence pattern has maxima for integer multiples ofT, whereasthe interference maxima for the stateuC2& are shifted byT/2.

With the above definitions in hand, we can now specthe measurement scheme that leads to the violation of Binequalities for Schro¨dinger cat states. The party measurithe spin-12 subsystem performs standard spin projectalong a direction selected randomly between two unit veca5(ax ,ay ,az) or a85(ax8 ,ay8 ,az8). The party measuring theharmonic-oscillator subsystem applies homodyne detecwith the phaseu adjusted to either 0 orp/2. For u50, thisrealizes the quantum optical analog of the position measment. In this case, the continuous outcomex of homodynedetection needs to be converted into its sign, as the regx.0 and x,0 correspond, respectively, to detecting tstatesua& and u2a&. Consequently, foru50 we effectivelyperform the measurement of the following operator:

C05Ex.0

dxH~x;0!2Ex,0

dxH~x;0!. ~5!

For the settingu5p/2, it is necessary to discriminate between the location of fringes for the superpositionsuC1& anduC2&. For this purpose, let us define two disjoint subsetsL1

andL2 of the possible outcomes of the homodyne measument:

L65 øn52`

`

@~n71/4!T,~n11/271/4!T#. ~6!

These two subsets are obtained by comparing which oftwo values computed in Eq.~4!, ^C1uH(x;p/2)uC1& or

^C2uH(x;p/2)uC2&, is larger for eachx, assuming that thenormalization factorsN6 are approximately equal.

The results11 and 21 are assigned to the measurquadraturesx, which belong, respectively, to the subsetsL1

andL2 . Thus we measure the operator

Cp/25EL1

dxH~x;p/2!2EL2

dxH~x;p/2!. ~7!

The measurements performed on the harmonic-oscillsubsystem are correlated with the standard spin measurealong directions defined by the unit vectorsa and a8. Thuswe are interested in the correlation functions of the form

E~a,u!5^Ku~a•s! ^ CuuK&, ~8!

whereu stands for 0 orp/2. The outcomes of the measurments performed both on the spin-1

2 and on the harmonicoscillator subsystems correspond to local realities bounby 21 and 1. Consequently, any of Bell’s inequalities drived for a pair of spin-12 projections can be used to test locreality in our measurement scheme.

IV. VIOLATION OF BELL’S INEQUALITY

We shall consider the following Bell combination costructed from four correlation functions@14#:

06210

l’s

nrs

n

e-

ns

e-

e

orent

d-

S5E~a,0!1E~a,p/2!1E~a8,0!2E~a8,p/2!. ~9!

For local hidden variable theories, the absolute value ofcombination is bounded byuSu<2. Explicit calculation ofthe combinationS in our scheme is simplified by the following symmetries of the operatorsC0 and Cp/2 :

^auC0ua&52^2auC0u2a&,

^auC0u2a&5^2auC0ua&50, ~10!

^auCp/2ua&5^2auCp/2u2a&.

With the help of these identities, one easily obtains the Bcombination expressed in terms of the matrix elements ofoperatorsC0 and Cp/2 :

S5~ax2ax8!Re auCp/2u2a&1~ay2ay8!Im^auCp/2u2a&

1~az1az8!^auC0ua&. ~11!

It is seen that there are two matrix elements relevant toBell combination: the diagonal element of the operatorC0,which can be easily expressed in terms of the error funct

^auC0ua&5erf~A2ha!, ~12!

and the off-diagonal element of the operatorCp/2 ,

^auCp/2u2a&52e22a21

2

Ape22a2(12h)

3 (n52`

` E(n21/4)T

(n11/4)T

e2x2cos~A8hax!dx.

~13!

In order to discuss the violation of Bell’s inequality fothe combinationS, let us first perform maximization over thunit vectorsa anda8 along which the spin-12 system is mea-sured. For this purpose, it is convenient to representS as asum of two scalar products:

S5a•u1a8•u8, ~14!

where the vectorsu and u8 are defined asu5(Re auCp/2

u2a&,Im^auCp/2u2a&,^auC0ua&) and u85(2Re auCp/2

u2a&,2Im^auCp/2u2a&,^auC0ua&). The above representation clearly shows that the maximum ofS is reached whenthe unit vectorsa anda8 are selected in the directions ofuand u8, respectively. Thus, the maximum value ofS readsexplicitly

Smax52A^auC0ua&21 z^auCp/2u2a& z2 ~15!

and it is obtained for the following directions of the spmeasurements:

2-3

t

hiblurr

tp

he-ithe

enp

tho

t

w

hisn

la

ec-

ndto2

r

posi-onstsr

et is

ettherm.l-ity

thefer-ureetionione

d

ANDRZEJ DRAGAN AND KONRAD BANASZEK PHYSICAL REVIEW A 63 062102

ax52ax852

SmaxRe^auCp/2u2a&,

ay52ay852

SmaxIm ^auCp/2u2a&, ~16!

az5az852

Smax^auC0ua&.

In Fig. 2, we plotSmax as a function ofa for severalvalues of the detection efficiencyh. In the case of perfecdetection of quadratures, the value ofSmax tends with in-creasinga to a constant value, equal to about 2.37. Tresult clearly contradicts predictions of local hidden variatheories. In the case of nonideal measurement of quadratthe violation of Bell’s inequality can still be observed fosufficiently high efficiencyh of homodyne detection, buthis effect vanishes with the increasing coherent state amtudea.

V. MACROSCOPIC LIMIT

In order to understand the behavior ofSmax in simpleterms, we will now perform an approximate analysis of texpression~15! valid for largea. Under an additional condition Aha@1, which means that homodyne detection wthe phaseu50 is capable of discriminating between thwave packetsua& and u2a&, we have

^auC0ua&'1. ~17!

The approximately constant value of this matrix elemmeans that the outcome of the homodyne measurementformed foru50 is insensitive to the detection efficiency.

Approximation of the matrix element^auCp/2u2a& givenin Eq. ~13! is slightly more intricate. Largea means that thespacingT between the interference fringes observed inmomentum distribution is small compared to the extentthe wave packets. Consequently, we can assume thaGaussian envelope multiplying the integrand in Eq.~13! isconstant over each of the integration intervals, which allo

FIG. 2. Maximum violation of Bell’s inequality for Schro¨dingercat states as a function of the coherent state amplitudea. Thegraphs correspond to measurements performed using a homodetector having several different efficiencies.

06210

sees,

li-

ter-

efthe

s

us to evaluate the integrals analytically. Furthermore, in tregime the sum overn of the remaining Gaussian factors cabe approximated by an integral. Thus we obtain

(n52`

` E(n21/4)T

(n11/4)T

e2x2cos~A8hax!dx'

1

p (n52`

`

Te2(nT)2

'1

Ap. ~18!

This expression yields the following approximate formufor the off-diagonal matrix element of the operatorCp/2 :

^auCp/2u2a&'2

pexp@22a2~12h!#, ~19!

where we have made use of the assumptionAha@1 in orderto eliminate the first term from Eq.~13!. This matrix elementdepends critically on the efficiency of the homodyne dettor.

By combining Eqs.~17! and~19!, we finally arrive at thefollowing expression for the Bell combination:

Smax'2A11S 2

pexp@22a2~12h!# D 2

. ~20!

In the case of perfect homodyne detection, the right-haside of the above formula is constant and equal2A114/p2. This is the asymptotic value observed in Fig.in the plot of Smax for h5100%. For imperfect homodynedetection, the violation of Bell’s inequality is damped folarge a by the exponential factor exp@22a2(12h)#, whichdecreases with the increasing separation between thetions of the wave packets. Let us note that the calculatithat led to the approximate form of the matrix elemengiven in Eqs.~17! and~19! are independent of the particulaform of the wave packetsua& and u2a& involved in thesuperpositionuK&. In order to derive Eq.~17!, one onlyneeds to assume that the spatial extent of the wave packsmaller that the separation between them. Similarly, Eq.~19!follows directly from the assumption that the wave-packenvelope in the momentum domain varies slowly onscale of the oscillations generated by the interference te

It is interesting to note that the violation of Bell’s inequaity in the proposed scheme is directly related to the visibilof interference between the wave packetsua& and u2a&. Ofcourse, homodyne detection performed alone onharmonic-oscillator subsystem does not reveal any interence, as its reduced density matrix is just a statistical mixtof ua& and u2a&. However, conditioning the homodynmeasurement on a specific outcome of the spin projecyields a clear signature of interference. A simple calculatgives the following probability distribution of obtaining thquadraturex for the phaseu5p/2 conditioned on the spin-upoutcome for a projection along the axisa:

yne

2-4

r-

re

e

sey

tieb

thn

hrorkoindisn

etlu

ge

suond

ton

n-dhac

f atoicvo

eussne

si-

c-

sts,

ll

ic-m-

-t

theinge

, as-

geer-

ri-ll’s

al

-6%.

HOMODYNE BELL’S INEQUALITIES FOR ENTANGLED . . . PHYSICAL REVIEW A63 062102

pS x;p

2 U↑aD}e2x2$11e22a2(12h)@axcos~A8hax!

1ay sin~A8hax!#%. ~21!

The above formula shows that the visibility of the inteference fringes is proportional to the factor exp@22a2(12h)#. It is exactly the same factor that appears in the expsion for the Bell combination given in Eq.~20!. Thus, thebetter the visibility of the interference fringes is, the strongthe violation of Bell’s inequalities takes place. Ifh is inter-preted as the parameter describing interaction with a revoir, it is clearly seen that the violation of Bell’s inequalitand the interference visibility decay at the same rate.

When passing to the mesoscopic domain, demonstraof quantum nonlocality in our scheme requires use of a msuring apparatus that is capable of detecting interferencetween the components of the superposition. However,sensitivity is important only in the homodyne measuremeperformed for the phaseu5p/2. The other half of the ho-modyne measurements can be realized in principle witdetector that does not have microscopic sensitivity. This pvides another example of a situation described first by Yuand Stoler@7#, who presented a scheme for the violationlocal realism employing a combination of sensitive andsensitive detectors. Analogously to their proposal, thecrimination between two amplitudes of coherent states issensitive to the efficiency of the detector. The second typthe measurement used by them is the determination ofphoton number parity, which requires single-photon resotion. In our case, the sensitive measurement has the formhomodyne detection capable of resolving interference frinin the superpositionsuC6&5(ua&6u2a&)AN6. Let us notethat the task of distinguishing the statesuC1& and uC2& inour scheme could in principle also be performed by meaing the photon number parity, as these two superpositihave nonzero occupation probabilities only for even or oFock states, respectively.

As we have seen, decreasing the efficiency of detecleads to a progressive disappearance of the interferefringes and consequently diminishing violation of Bell’s iequalities. This effect can be viewed as a model for thecoherence phenomenon. Since Leonhardt and Paulshown@15# that effects of imperfections of homodyne detetors can be described by introducing losses in the form oadditional fictitious beam splitter placed before the detecinefficient detection is equivalent to coupling the harmonoscillator system to an external zero-temperature resergenerating decoherence.

VI. EXPERIMENTAL PROSPECTS

An important practical aspect of experimental schemfor testing quantum nonlocality is their sensitivity to varioimperfections@16#. The effect of nonideal spin projection irelatively straightforward to describe. If we assume that ifraction of events the measuring device returns a flippvalue of the spin, then the ideal spin projections along

06210

s-

r

r-

ona-e-ists

a-ef--

otofhe-ofs

r-s

d

rsce

e-ve

-nr,-ir

s

ada

directiona are replaced by a generalized two-element potive operator-valued measure:

P↑~a!511j

2u↑a&^↑au1

12j

2u↓a&^↓au,

P↓~a!512j

2u↑a&^↑au1

11j

2u↓a&^↓au, ~22!

where u↑a& and u↓a& are the eigenvectors of the operatora•s, and the parameterj, bounded between 0 and 1, charaterizes the efficiency of the spin measurement: forj51, werecover perfect spin projections, whereasj50 correspondsto the completely noisy limit. A simple calculation showthat within such a model of imperfect spin measuremeneach of the correlation functions defined in Eq.~8! is multi-plied by the parameterj. Consequently, the complete BecombinationSbecomes rescaled by the factorj smaller than1. Thus, in the limitAha@1 we obtain

Smax'2jA11S 2

pexp@22a2~12h!# D 2

. ~23!

For homodyne detection performed on the harmonoscillator subsystem, the role played by the efficiency paraeter h is more involved. In Fig. 3, we plot the maximumvalue of the Bell combinationSmax as a function of the ho-modyne detector efficiencyh for several values of the coherent state amplitudea, under the assumption of perfecspin projections. According to these numerical results,lower bound for the homodyne detector efficiency enablthe violation of Bell’s inequality is about 66%. We havfound that this bound, clearly seen in the plot fora52, shiftsto even slightly lower values with increasinga. However,for largera, the strict bound becomes rather meaninglessin its vicinity the violation of Bell’s inequality becomes negligibly small. Thus, its observation would first require a husample of experimental data, and secondly completely pfect spin measurements.

We shall close this section with a brief review of expemental prospects for demonstrating the violation of Be

FIG. 3. Maximum value of the Bell combination for severvalues ofa as a function of the detection efficiencyh, assumingperfect spin projections. Fora52, the minimum efficiency necessary to demostrate nonlocality of the cat state is slightly above 6

2-5

e-t

on

reshit

ieti

ic

thoyn

theniten-cro-isb-the

sibil-in-

J.

by,B

ANDRZEJ DRAGAN AND KONRAD BANASZEK PHYSICAL REVIEW A 63 062102

inequalities using Schro¨dinger cat states. Among possible ralizations, the physical systems that would be most likelyachieve strict locality conditions include a single radiatimode entangled with an atom@3,5#, or with polarizationstates of a single photon@6#. The radiation mode can eithebe confined in a high-Q cavity or travel in free space. In thlatter case, the homodyne measurement is a well-establitechnique used widely in quantum-optical experiments, wthe advantage of achieving very high detection efficiencIn the case of the measurement of quadratures for radiafields in a cavity, several schemes are available@17#, whichremain, however, considerably more complicated in practrealization.

VII. CONCLUSIONS

In summary, we have shown how Schro¨dinger cat statesexhibiting quantum entanglement can be applied to testviolation of Bell’s inequalities. The discussion was baseda concrete realization of the scheme using the homod

.

.

es

s.

06210

o

edhs.on

al

ene

detection technique for a light mode. We have analyzedeffects of losses and imperfections, including the nonudetection efficiency, and pointed out a direct link betwethe visibility of interference and the violation of Bell’s inequalities. We have found that when passing to the mascopic domain, a substantial violation of Bell’s inequalitiespossible in the limit of perfect noise-free detection and asence of dissipation. In a realistic case, losses destroynonlocal effects at the same rate as they decrease the viity of quantum interference between the classically distguishable wave packets in the superposition.

ACKNOWLEDGMENTS

We would like to acknowledge useful discussions withH. Eberly, S. Wallentowitz, I. A. Walmsley, K. Wo´dkiewicz,and M. Zukowski. This research was partially supportedARO-administered MURI Grant No. DAAG-19-99-1-0125NSF Grant No. PHY-9415583, and KBN Grant No. 2 P03089 16.

-

s.

@1# E. Schrodinger, Naturwissenschaften23, 807 ~1935!; 23, 823~1935!; 23, 844 ~1935!.

@2# C. Monroeet al., Science272, 1131~1996!.@3# M. Bruneet al., Phys. Rev. Lett.77, 4887~1996!.@4# B. Yurke and D. Stoler, Phys. Rev. Lett.57, 13 ~1986!; S.

Song, C.M. Caves, and B. Yurke, Phys. Rev. A41, R5261~1990!; V. Buzek, H. Moya-Cessa, P.L. Knight, and S.J.DPhoenix, ibid. 45, 8190 ~1992!; I.A. Walmsley and M.G.Raymer,ibid. 52, 681 ~1995!; P. Tombesi and D. Vitali, PhysRev. Lett.77, 411 ~1996!.

@5# C.M. Savage, S.L. Braunstein, and D.F. Walls, Opt. Lett.15,628 ~1990!.

@6# M. Genovese and C. Novero, Phys. Rev. A61, 032102~2000!;Phys. Lett. A271, 48 ~2000!.

@7# B. Yurke and D. Stoler, Phys. Rev. Lett.79, 4941~1997!.@8# For use of homodyne detection in testing Bell’s inequaliti

see P. Grangier, M.J. Potasek, and B. Yurke, Phys. Rev. A38,R3132~1998!; S.M. Tan, D.F. Walls, and M.J. Collett, PhyRev. Lett. 66, 252 ~1991!; B.C. Sanders, Phys. Rev. A45,6811 ~1992!; 46, 2966 ~1992!; A. Gilchrist, P. Deuar, andM.D. Reid, Phys. Rev. Lett.80, 3169~1998!; K. Banaszek and

,

K. Wodkiewicz, ibid. 82, 2009 ~1999!; A. Kuzmich, I.A.Walmsley, and L. Mandel,ibid. 85, 1349~2000!.

@9# J.A. Wheeler and W.H. Zurek,Quantum Theory and Measurement~Princeton University Press, Princeton, 1983!.

@10# D.F. Walls and G.J. Milburn, Phys. Rev. A31, 2403~1985!.@11# V. Buzek and P.L. Knight, inProgress in Optics XXXIV, ed-

ited by E. Wolf ~North-Holland, Amsterdam, 1995!.@12# K. Wodkiewicz, Opt. Commun.179, 215~2000!; New J. Phys.

2, 21 ~2000!.@13# S.L. Braunstein, Phys. Rev. A42, 474~1990!; W. Vogel and J.

Grabow, ibid. 47, 4227 ~1993!; K. Banaszek and K. Wo´dk-iewicz, ibid. 55, 3117~1997!.

@14# J.F. Clauser, M.A. Horne, A. Shimony, and R.A Holt, PhyRev. Lett.23, 880 ~1969!; J.S. Bell, inFoundations of Quan-tum Mechanics, edited by B. d’Espagnat~Academic, NewYork, 1971!.

@15# U. Leonhardt and H. Paul, Phys. Rev. A48, 4598~1993!.@16# P. Eberhard, Phys. Rev. A47, R747~1993!.@17# H.M. Wiseman and G.J. Milburn, Phys. Rev. A47, 642

~1993!; M. Fortunato, P. Tombesi, and W.P. Schleich,ibid. 59,718 ~1999!.

2-6