Homodyne Bell’s inequalities for optical Schrodinger cat states

Andrzej DraganInstytut Fizyki Teoretycznej, Uniwersytet Warszawski, Hoza 69, PL–00–681 Warszawa, Poland

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

University of Rochester, Rochester NY 14627

(October 19, 2000)

We present a scheme for demonstrating violation of Bell’sinequalities using a a spin-1/2 system entangled with a pair ofclassically distinguishable wave packets in a harmonic poten-tial. In the optical domain, such wave packets can be repre-sented by coherent states of a single light mode. The proposedscheme involves standard spin-1/2 projections and measure-ments of the position and the momentum of the harmonicoscillator system, which for a light mode can be realized bymeans of homodyne detection. We discuss effects of imperfec-tions, including non-unit efficiency of the homodyne detector,and point out a close link between the visibility of interferenceand violation of Bell’s inequalities in the described scheme.

PACS Numbers: 03.65.Bz, 42.50.Dv

Quantum mechanical superposition principle bringsunexpected and counterintuitive consequences when ap-plied to macroscopic systems. Presumably the most fa-mous example is Schrodiger’s cat, which remains half-alive and half-dead while entangled with a decaying ra-dioactive atom [1]. Recent experimental advances haveopened up new possibilities to study the superpositionprinciple beyond purely microscopic domain. It is nowpossible to produce in a laboratory states of the type:

|K〉 =1√2(|↑ 〉 ⊗ |α〉 + |↓ 〉 ⊗ |−α〉) (1)

where | ↑ 〉, | ↓ 〉 are two orthogonal states of a spin-1/2 system, and |α〉, |−α〉 are two distinguishable wavepackets in a harmonic oscillator. Such states have beengenerated for a trapped ion [2] and a microwave cav-ity field entangled with an atom [3]. They can be con-sidered as mesoscopic equivalents of the example usedby Schrodinger in his original argument. The entagledstates defined in Eq. (1) are closely related to the issueof generating and detecting coherence between classicallydistinguishable states [4–6].

In this paper we show how states described by Eq. (1)can be used to test incompatibility of quantum mechanicswith local realism. Specifically, we derive Bell’s inequal-ities which are violated by Schrodinger cat states. Theseinequalities are based on the measurement of position andmomentum observables for the harmonic oscillator sub-system, and standard projections for the spin-1/2 sub-system. For a light mode, which will serve in this paper

as a physical realization of the harmonic oscillator sub-system, position and momentum correspond to a pair ofquadratures, which can be measured with the help of ho-modyne detection [7]. Furthermore, we demonstrate thatviolation of Bell’s inequalities in the proposed scheme isclosely related to the visibility of interference betweenthe two distinct wave packets given by the states |α〉 and|−α〉.

We shall start from giving a simple heuristic argumentwhich motivated us to formulate Bell’s inequalities forSchrodinger cat states. For concreteness, we shall as-sume that the states |α〉 and |−α〉 describe two Gaussianwave packets in a harmonic potential centered arounddimensionless positions α/

√2 and −α/

√2, respectively,

with zero average momentum and ground state widths.From the following discussion it will be clear that vio-lation of Bell’s inequalities in our scheme is completelyinsensitive to the specific form of the wave packets. Thestates |α〉 and |−α〉 have the scalar product equal to〈−α|α〉 = exp(−2α2), and for sufficiently large α theycan be considered as approximately orthogonal. In thiscase, one can establish a formal analogy [8] between thestate |K〉 and the singlet state of two spin-1/2 particlesused in original Bell’s argument, based on the correspon-dence: |α〉 7→ | ↓ 〉 and |−α〉 7→ −| ↑ 〉. Following thisanalogy, in order to violate Bell’s inequalities one shouldbe able to perform two noncommuting measurements inthe subspace spanned by |α〉 and |−α〉, which would corre-spond to projecting the spin onto two different directions.As the first measurement on the harmonic oscillator sub-system, let us simply choose the projection in the basis{|α〉, |−α〉}. This measurement can be effectively accom-plished by the measurement of position: if the distancebetween the centers of the wave packets is much largerthan their spatial extent, the sign of the position variablealmost unambiguously discriminates between the states|α〉 and |−α〉. The second measurement on the harmonicoscillator subsystem we shall select to be the projectionin the basis of the superpositions:

|Ψ±〉 =1

√

N±

(|α〉 ± |−α〉), (2)

where N± = 2(1±e−2α2

) are the normalization constants.As depicted in Fig. 1, these two superpositions generatedistinct interference patterns in the momentum distribu-tion: location of maxima for the state |Ψ+〉 corresponds

1

wykres1.psFIG. 1. Plot of the momentum distributions for the super-

positions |Ψ+〉 (solid line) and |Ψ−〉 (dashed line), assuming

perfect, noise-free measurement, and α = 6. Location of max-ima for one state corresponds to the location of minima forthe other one.

to minima for the state |Ψ−〉, and vice versa. Conse-quently, we can approximately discriminate between thesuperpositions |Ψ+〉 and |Ψ−〉 by checking whether theresult of the momentum measurement falls within thevicinity of the interference fringes either for the state|Ψ+〉 or |Ψ−〉. Of course, such discrimination is imperfectas the momentum distributions partially overlap; never-theless, we shall demonstrate that the error rate involvedis low enough to enable violation of Bell’s inequalities.

Let us now discuss the idea sketched above in quanti-tative terms. As a concrete physical realization, we willtake |α〉 and |−α〉 to be two coherent states of a singlelight mode, with α assumed to be real. The optical ana-log of the position and momentum observables is a pairof canonically conjugated quadratures defined in generalas xθ = (eiθa† + e−iθa)/

√2. Two values of the phase:

θ = 0 and θ = π/2 correspond to the position and themomentum operators, respectively. The standard tech-nique for measuring quadratures is homodyne detection,described by the positive operator-valued measure [9]:

H(x; θ) =1

√

π(1 − η)exp

(

− (x/√

η − xθ)2

1/η − 1

)

, (3)

where η is the detection efficiency. The perfect noise-free measurement of quadratures is obtained in the limitη = 1, whereas η < 1 describes non-ideal detection, cor-responding to a blurred measurement of position and mo-mentum with finite resolution equal to

√

1/η − 1. Thestates |Ψ±〉 defined in Eq. (2) generate the following in-terference patterns for θ = π/2:

〈Ψ±|H(x; π2 )|Ψ±〉

=1

N±

e−x2

[1 ± e−2(1−η)α2

cos(√

8ηαx)] (4)

It is easily seen that within the Gaussian envelope given

by the factor e−x2/2, the spacing between the interferencefringes is given by T = π/

√2ηα. For the state |Ψ+〉 the

interference pattern has maxima for integer multiples ofT , whereas the interference maxima for the state |Ψ−〉are shifted by T/2.

With the above definitions in hand, we can now spec-ify the measurement scheme which demonstrates vio-lation of Bell’s inequalities for Schrodinger cat states.The party measuring the spin-1/2 subsystem performsstandard spin projection along a direction selected ran-domly between two unit vectors a = (ax, ay, az) ora′ = (a′

x, a′y, a′

z). The party measuring the harmonic os-cillator subsystem applies homodyne detection with thephase θ adjusted to either 0 or π/2. For θ = 0, this

realizes the quantum optical analog of the position mea-surement. In this case, the continuous outcome x of ho-modyne detection needs to be converted into its sign, asthe regions x > 0 and x < 0 correspond respectively todetecting the states |α〉 and | − α〉. Consequently forθ = 0 we effectively perform the measurement the fol-lowing operator:

C0 =

∫

x>0

dx H(x; 0) −∫

x<0

dx H(x; 0). (5)

For the setting θ = π/2, it is necessary to discriminate be-tween the location of fringes for the superpositions |Ψ+〉and |Ψ−〉. For this purpose, let us define two disjointsubsets of the possible outcomes of the homodyne mea-surement:

Λ+ =

∞⋃

n=−∞

[(n − 1/4)T, (n + 1/4)T ]

Λ− =

∞⋃

n=−∞

[(n + 1/4)T, (n + 3/4)T ] (6)

The results +1 and −1 are assigned to the measuredquadratures x which belong respectively to the subsetsΛ+ and Λ−. Thus we measure the operator:

Cπ/2 =

∫

Λ+

dx H(x; π2 ) −

∫

Λ−

dx H(x; π2 ) (7)

The measurements performed on the harmonic oscillatorsubsystem are correlated with the standard spin measure-ment along directions defined by the unit vectors a anda′. Thus we are interested in the correlation functions of

the form:

E(a, θ) = 〈K|(a · σ) ⊗ Cθ|K〉 (8)

where θ stands for 0 or π/2. The outcomes of the mea-surements performed both on the spin-1/2 and on theharmonic oscillator subsystems correspond to local real-ities bounded by −1 and 1. Consequently, any of Bell’sinequalities derived for a pair of spin-1/2 projections canbe used to test local reality in our measurement scheme.We shall consider the following Bell combination con-structed from four correlation functions [10]:

S = E(a, 0) + E(a, π/2) + E(a′, 0) − E(a′, π/2). (9)

For local hidden variable theories, the absolute value ofthis combination is bounded by |S| ≤ 2. Explicit calcula-tion of the combination S in our scheme is simplified bythe following symmetries of the operators C0 and Cπ/2:

〈α|C0|α〉 = −〈−α|C0|−α〉〈α|C0|−α〉 = 〈−α|C0|α〉 = 0

〈α|Cπ/2|α〉 = 〈−α|Cπ/2|−α〉 (10)

2

With the help of these identities, one easily obtains theBell combination expressed in terms of the matrix ele-ments of the operators C0 and Cπ/2:

S = (ax − a′x)Re〈α|Cπ/2|−α〉

+(ay − a′y)Im〈α|Cπ/2|−α〉 + (az + a′

z)〈α|C0|α〉(11)

It is seen that there are two matrix elements relevant tothe Bell combination: the diagonal element of the oper-ator C0:

〈α|C0|α〉 = erf (√

2ηα) (12)

and the off-diagonal element of the operator Cπ/2:

〈α|Cπ/2|−α〉 = −e−2α2

+2√π

e−2α2(1−η) ×

×∞∑

n=−∞

∫ (n+1/4)T

(n−1/4)T

e−x2

cos(√

8ηαx)dx.

(13)

In order to discuss violation of Bell’s inequality for thecombination S, let us first perform maximization over theunit vectors a and a

′ along which the spin-1/2 system ismeasured. An easy calculation shows that the maximumvalue of S reads:

Smax = 2

√

〈α|C0|α〉2 + |〈α|Cπ/2|−α〉|2 (14)

and that it is obtained for the following directions of thespin measurements:

ax = −a′x =

2

SmaxRe 〈α|Cπ/2|−α〉

ay = −a′y =

2

SmaxIm 〈α|Cπ/2|−α〉

az = a′z =

2

Smax〈α|C0|α〉. (15)

In Fig. 2 we plot Smax as a function of α for severalvalues of the detection efficiency η. In the case of per-fect detection of quadratures, the value of Smax tendswith increasing α to a constant value, equal about 2.37.This result clearly contradicts predictions of local hiddenvariable theories. In the case of non-ideal measurementof quadratures, violation of Bell’s inequality can be stillobserved for sufficiently high efficiency η of homodynedetection, but this effect vanishes with the increasing co-herent state amplitude α.

In order to understand the behavior of Smax in simpleterms, we will now perform an approximate analysis ofthe expression (14) valid for large α. Under an additionalcondition

√ηα � 1, which means that homodyne detec-

tion with the phase θ = 0 is capable of discriminatingbetween the wave packets |α〉 and | − α〉, we have:

wykres2.psFIG. 2. Maximum violation of Bell’s inequality for

Schrodinger cat states as a function of the coherent stateamplitude α. The graphs correspond to measurements per-formed using a homodyne detector having several differentefficiencies.

〈α|C0|α〉 ≈ 1. (16)

Approximation of the matrix element given in Eq. (13) isslightly more intricate. Large α means that the spacingT between the interference fringes observed in the mo-mentum distribution is small compared to the extent ofthe wave packets. Consequently, we can assume that theGaussian envelope multiplying the integrand in Eq. (13)is constant over each of the integration intervals, whichallows us to evaluate the integrals analytically. Further-more, in this regime the sum over n of the remainingGaussian factors can be approximated by an integral.Thus we obtain:

∞∑

n=−∞

∫ (n+1/4)T

(n−1/4)T

e−x2

cos(√

8ηαx)dx

≈ 1

π

∞∑

n=−∞

Te−(nT )2 ≈ 1√π

(17)

This expression yields the following approximate formulafor the off-diagonal matrix element of the operator Cπ/2:

〈α|Cπ/2|−α〉 ≈ 2

πexp[−2α2(1 − η)], (18)

where we have made use of the assumption√

ηα � 1in order to eliminate the first term from Eq. (13). Thuswe finally arrive to the following expression for the Bellcombination:

Smax ≈ 2

√

1 +

(

2

πexp[−2α2(1 − η)]

)2

. (19)

In the case of perfect homodyne detection, the righthand side of the above formula is constant and equalto 2

√

1 + 4/π2. This is the asymptotic value observedin Fig. 2 in the plot of Smax for η = 100%. For im-perfect homodyne detection, violation of Bell’s inequal-ity is damped for large α by the exponential factorexp[−2α2(1 − η)], which decreases with the increasingseparation between the positions of the wave packets.

It is interesting to note that violation of Bell’s inequal-ity in the proposed scheme is closely related to the vis-ibility of interference between the wave packets |α〉 and|−α〉. Of course, homodyne detection performed aloneon the harmonic oscillator subsystem does not reveal anyinterference, as the reduced density matrix is just a sta-tistical mixture of |α〉 and |−α〉. However, conditioningthe homodyne measurement on a specific outcome of thespin projection yields a clear signature of interference.

3

wykres3.psFIG. 3. Maximum value of the Bell combination for sev-

eral values of α as a function of the detection efficiency η.For α = 2 the minimum efficiency necessary to demostratenonlocality of the cat state is slightly above 66%.

A simple calculation gives the following probability dis-tribution of obtaining the quadrature x for the phaseθ = π/2 conditioned on the spin up outcome for a pro-jection along the axis a:

p(x; π2 | ↑a) ∝ e−x2{1 + e−2α2(1−η) ×

×[ax cos(√

8ηαx) + ay sin(√

8ηαx)]}(20)

The above formula shows that the visibility of the inter-ference fringes is proportional to the factor exp[−2α2(1−η)]. It is exactly the same factor which appears in theexpression for the Bell combination given in Eq. (19).Thus, the better the visibility of the interference fringesis, the stronger violation of Bell’s inequalities takes place.When passing to the mesoscopic domain, demonstrationof quantum nonlocality in our scheme requires use of ameasuring apparatus which is capable of detecting inter-ference between the components of the superposition.

An important practical aspect of experimental schemesfor testing quantum nonlocality is their sensitivity to var-ious imperfections [11]. The effect of non-ideal spin pro-jection is relatively straightforward to describe: if we as-sume that in a fraction of events the measuring devicereturns a flipped value of the spin, then the Bell com-bination S simply becomes rescaled by a certain factorsmaller than one. For homodyne detection performedon the harmonic oscillator subsystem, the role played bythe efficiency parameter η is more involved. In Fig. 3 weplot the maximum value of the Bell combination Smax

as a function of the homodyne detector efficiency η forseveral values of the coherent state amplitude α, underan assumption of perfect spin projections. According tothese numerical results, the lower bound for the homo-dyne detector efficiency enabling violation of Bell’s in-equality is about 66%. We have found that this bound,clearly seen in the plot for α = 2, shifts to even slightlylower values with increasing α. However, for larger α thestrict bound becomes rather meaningless, as in its vicin-ity the violation of Bell’s inequality becomes negligiblysmall. Thus, its observation would require first huge sam-ple of expermental data, and secondly completely perfectspin measurements.

We shall close this paper with a brief review of exper-imental prospects for demonstrating violation of Bell’sinequalities using Schrodinger cat states. Among pos-sible realizations, the physical systems which would bemost likely to achieve strict locality conditions include asingle radiation mode entangled with an atom [3,5], orwith polarization states of a single photon [6]. The ra-diation mode can either be confined in a high-Q cavity

or travel in free space. In the latter case, the homodynemeasurement is a well-established technique used widelyin quantum-optical experiments, with the advantage ofachieving very high detection efficiencies. In the case ofthe measurement of quadratures for radiation fields in acavity, several schemes are available [12], which remainhowever considerably more complicated in practical real-ization.

In summary, we have shown how Schrodinger’s catstates exhibiting quantum entanglement can be appliedto test violation of Bell’s inequalities. The discussion wasbased on a concrete realisation of the scheme using thehomodyne detection technique for a light mode. We haveanalysed the effects of imperfections, including non-unitdetection efficiency, and pointed out a close link betweenthe visibility of interference and the violation of Bell’sinequalities.

We would like to acknowledge useful discussions withJ. H. Eberly, S. Wallentowitz, I. A. Walmsley, andK. Wodkiewicz. This research was partially sup-ported by ARO–administered MURI grant No. DAAG-19-99-1-0125, NSF grant PHY-9415583, and KBN grant2 P03B 089 16.

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4

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