Violation of Smooth Observable Macroscopic Realism in a Harmonic Oscillator

Amir Leshem and Omri Gat

Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel(Received 5 February 2009; published 13 August 2009)

We study the emergence of macrorealism in a harmonic oscillator subject to consecutive measurements

of a squeezed action. We demonstrate a breakdown of dynamical realism in a wide parameter range that is

maximized in a scaling limit of extreme squeezing, where it is based on measurements of smooth

observables, implying that macroscopic realism is not valid in the harmonic oscillator. We propose an

indirect experimental test of these predictions with entangled photons by demonstrating that local realism

in a composite system implies dynamical realism in a subsystem.

DOI: 10.1103/PhysRevLett.103.070403 PACS numbers: 03.65.Ta, 03.65.Sq, 03.65.Ud, 42.50.Dv

Introduction.—In spite of prolonged effort, the questionof what constitutes a macroscopic object is still not settled.From the operational point of view, the answer is clear:Macroscopic objects are those that are well-described byclassical physics. However, it is also widely believed thatmacroscopic objects are in principle governed by the lawsof quantum mechanics and that classical physics is anemergent phenomenon at an appropriate limit. A commonview holds that a macroscopic object becomes classicalwhen it interacts with an environment that might as wellconsist of its internal degrees of freedom [1]. As a resultof decoherence, quantum uncertainty is translated intoclassical ignorance that can be expressed in terms of clas-sical hidden variables. A different approach places theemergence of macroscopic realism of closed systems onthe impossibility of distinguishing individual quantum lev-els [2].

Classical mechanics postulates more than the existenceof definite values for all of the observables. Since themotion is deterministic, classical observables follow defi-nite trajectories that constrain the different-time jointprobability distribution of classical observables. In otherwords, a hidden-variable underpinning of an evolvingquantum system becomes a hidden trajectory dynamicalmodel. If one adds to the hidden trajectory model theassumption (valid in classical mechanics) that a measure-ment can be performed with an arbitrarily weak perturba-tion of the system, it is possible to derive Leggett-Garginequalities (LGIs) that constrain linear combinations oftemporal correlation functions Cðt; t0Þ ¼ hSðtÞSðt0Þi of abounded observable S [3]. For example, the analog ofthe Clauser-Horne-Shimony-Holt (CHSH) [4] inequalityis the LGI jhLGij ¼ jCðt1; t2Þ þ Cðt2; t3Þ þ Cðt2; t3Þ �Cðt1; t4Þj � 2. In the formulation of this LGI, it is usuallyassumed that the observable S can assume only the values�1, but as with the CHSH inequalities [5], it is onlynecessary to assume that jSj � 1 to derive the LGI [6].

Inasmuch as they apply to microscopic states, the as-sumptions leading to the LGI that we will summarize bythe term dynamical realism (DR) are clearly false, andindeed the LGI can be violated for any closed quantum

system [7]. Furthermore, most quantum systems do notallow for a hidden trajectory underpinning, and the break-down of DR can be demonstrated with a single measure-ment as shown in Ref. [8] for nonlinear oscillators. On theother hand, there are quantum systems that admit an exacthidden trajectory model, the simplest of which is a har-monic oscillator prepared in a positive Wigner functionstate, and the breakdown of DR can be demonstrated withthe successive measurement scheme of Leggett and Garg.The original motivation for the introduction of the LG

inequality was to test macroscopic realism (MR).Breakdown of MR follows immediately from LGI viola-tion if S is a dichotomous observable, with support on twomacroscopically distinct states undergoing von Neumannmeasurements. This case was studied previously inRefs. [2,3,7,9], but such two-state observables must resolveindividual eigenstates and are therefore singular in theclassical limit. In contrast, the harmonic oscillator is asystem with a well-behaved classical limit, and it is naturalto test MR there with smooth observables, such as positionor momentum, that have a good classical limit as well.When S is a bounded smooth observable, the relativedifference in its values for eigenstates that are not macro-scopically distinct is by definition small. Therefore, thebreakdown of MR is implied by a large violation of theLGI that does not tend to zero in the classical limit.For this reason, we introduce a step parameter �1 �

s � 1 such that, when s ¼ �1, S is the singular (squeezed)parity observable, while S becomes smooth when 1� s issmall (Ref. [10] introduced similarly parametrized observ-ables). Our first main result is to show that the LGI isviolated for a wide range of values s and the squeezingparameter � (defined below), shown in Fig. 1(b). We nextfocus on the smooth limit 1� s � 1 and show that theLGI violation persists in this domain and is in fact maxi-mized when s ! 1�, thus demonstrating breakdown ofsmooth observable MR in the harmonic oscillator. Themaximal violation hLGimax � 2:106 occurs in a singularlimit where s ! 1, � ! 1 ,and the time interval betweenthe consecutive measurements tends to 0, where hLGibecomes a scaling function of a single parameter, shown

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in Fig. 2(b). In contrast, if the smooth limit 1� s ! 0 istaken with fixed �, regular semiclassical physics holds,showing a small violation of LGI for any � larger than athreshold value, implying breakdown of dynamic realismbut not of MR. Last, by demonstrating the equivalence ofour LGI test to a Bell-like measurement of a pair ofentangled photons, we propose a quantum optical setupas an indirect method of observing LGI violations. Ourscheme can be generalized to show that local realism in acomposite system implies DR in a subsystem.

Violations of the Leggett-Garg inequality.—A harmonicoscillator offers a natural hidden trajectory description interms of its classical trajectories. Choosing units such that

the Hamiltonian H ¼ !q2þp2

2 , where q and p are the

position and momentum of the oscillator, respectively,and ! its frequency, the trajectories are qt ¼ q cosð!tÞ þp sinð!tÞ and pt ¼ �q sin!tþ p cos!t. If we also pre-pare the oscillator in a state with a non-negative Wignerfunction Wðq; pÞ, such as a coherent state, the trajectoriescan be used to define a time-dependent phase-space proba-bility density function Ptðq; pÞ ¼ Wðq�t; p�tÞ [11].

We therefore choose to test DR in a harmonic oscillatorprepared in its ground state jgi by performing consecutivemeasurements of the operator S ¼ sn, where n ¼ A

@� 1

2

and A ¼ 12 ð��1q2 þ �p2Þ; S depends on the step parame-

ter jsj � 1 and the squeezing parameter � > 0 and obeysjSj � 1. A is the action, and n is the non-negative integer-valued excitation number, of a squeezed harmonic oscil-lator with eigenstates jni. s can also be viewed as a smooth-ness parameter: The relative change in S between adjacenteigenstates is 1� s, so that, for s ! 1�, S becomessmooth, and its measurement can be achieved with aclassical apparatus that does not resolve individual states.In the opposite limit s ¼ �1, S becomes the singular parityoperator that changes between 1 and �1 for consecutiveeigenstates. This property is also reflected in the Weyl

representation of S: Sðq; pÞ ¼ 21þs e

½ðs�1Þ=�ðsþ1Þ�ðq2þ�2p2Þ.Sðq; pÞ is positive for any s >�1, but its maximumSmax ¼ 2

1þs increases as s decreases and diverges as s !�1. In particular, since Smax > kSk ¼ 1, S is improper[12] for any s < 1, allowing for violations of realism.The correlation function of two S measurements

CðtÞ ¼ X

n0nsn

0þnjhnjeð1=i@ÞHtjn0ij2jhn0jgij2 (1)

depends only on the time difference t because the initialstate is an eigenstate ofH. C can be expressed as a contourintegral over the unit circle

CðtÞ ¼I dz

2�izhgj�ðzÞjgiTr�ðsÞeðHt=i@Þ�ðs=zÞe�ðHt=i@Þ;

(2)

where �ðxÞ ¼ Pnx

njnihnj is a squeezed thermal state. Thetrace operations can be carried out, for example, usingphase-space integrations, and the result of the contourintegration is

Cðs; �;!tÞ ¼ cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia� b�

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþ bþ

p KðmÞ; (3)

where KðmÞ ¼ R10½ð1� t2Þð1�mt2Þ��1=2dt is an elliptic

integral of the first kind,

a ¼ �þ 1

ð�� 1Þ ½4�2 þ ð1� s2Þð�2 � 1Þ2sin2ð!tÞ�; (4)

b� ¼ 4s2�2 � sðs2 � 1Þð�2 � 1Þ sinð!tÞ

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4�2 þ ð�2 � 1Þ2sin2ð!tÞ

q; (5)

c ¼ 8�3=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4�2 þ ð1� s2Þð�2 � 1Þ2sin2ð!tÞp

�ð�� 1Þ ; (6)

andm ¼ 2ðb��bþÞaðb��aÞðbþþaÞ . If we choose the three time intervals

in the LG experiment to be of the same length, then the LGexpectation value is hLGi ¼ 3Cðs; �;�Þ � Cðs; �; 3�Þ,where � ¼ !t.hLGi is invariant under each of the transformations s !

�s, � ! 1=�, and � ! ��, so our results are presented

0 0.02 0.04 0.06 0.082

2.00

2.05

2.10

LG

(a)0.2 0.4 0.6 0.8 1.0 1.2

x

2.00

2.05

2.10

LG

(b)

FIG. 2 (color online). (a) A plot similar to Fig. 1(a) with datashown for several values of s between scr � 0:983 (the minimal sallowing LGI violation) and s ¼ 1–10�7 and � that yields themaximal LGI violation for each s. s is increasing in order frombottom to top. The global maximum of hLGi, � 2:106, is shownin the upper dashed purple line. (b) The limit reached by hLGi, ass ! 1 and � ! 1, � ! 0, as a function of the scaling variablex ¼ �2=ð2�3ð1� sÞ2Þ. The maximum is obtained for x � 0:24.

0.1 0.3 0.52

1.4

1.6

1.8

2

LG

(a)

0.985

0.995s5

1022.022.04

(b)

FIG. 1 (color online). (a) A plot of the LG expectation value asa function of the time interval� between measurements (in unitsof oscillator period), for the step parameter s ¼ 0:99 and thesqueezing parameter � � 7:6 that gives maximal violation forthis s. The dashed red line corresponds to the bound hLGi ¼ 2imposed on realistic theories. (b) A plot of max�hLGi as a

function of s and � (only values greater than 2 are shown).The LGI violation is stronger for s close to 1 and � large.

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for positive values of s and � and for values of � greaterthan 1. A typical graph of hLGi as a function of� for fixedvalues of s and � that allow LGI violation is shown inFig. 1(a). The violation occurs in a small window of timesthat are significantly smaller than the period. The region inthe parameter plane that allows LGI violation is shown inFig. 1(b). It is bounded by the rectangle s > scr � 0:98 and� > �cr � 3:2.

The qualitative features of hLGi are as follows. (i) Byfixing one of the parameters s or � at a value that allowsLGI violation, the maximal violation occurs for a finitevalue of the other parameter and �. (ii) max�hLGi in-

creases as s ! 1 and � ! 1, and � ! 0, approachingthe global maximum � 2:106 [see Figs. 1(b) and 2(b)].(iii) If one of the parameters is fixed, and another is taken toits extreme limit s ! 1 or � ! 1, the LGI holds.

Breakdown of macrorealism and the semiclassicallimit.—If the trace in Eq. (2) is evaluated as a phase-spaceintegral, the maximization of hLGi can be understood as atrade-off between the overlap in the supports of the phase-space representations of the observable S and itsHeisenberg-picture time evolution St, that is maximizedfor the trivial values s ¼ 1, � ¼ 0, or � ¼ 1, and thenoncommutativity of H and S, that is maximized for large�. It follows that, in order to maximize the LGI violation, sshould approach 1 to optimize the phase-space overlap,and at the same time � should tend to infinity, and thereforealso � ! 0 to keep the phase-space overlap of the secondmeasurement large.

This qualitative observation is in accord with the explicitfunctional behavior of C as given in Eq. (3). Indeed, themaximal LGI violation of hLGi is obtained when s is closeto 1, but s must approach 1 in a singular limit where � !1 and � ! 0 according to �� ð1� sÞ�1=2 and �� ð1�sÞ1=4. In this limit hLGi approaches a scaling form �ðxÞ ¼2� ½3Kð�xÞ � Kð�9xÞ� for the scaling variable x ¼�2=½2�3ð1� sÞ2�. �ðxÞ, shown in Fig. 2(b), has a singlemaximum� 2:106 (obtained at x � 0:24) that is the globalmaximum of hLGi.

It is remarkable that the maximal LGI violation, anunequivocal manifestation of quantum mechanics, is ob-tained in the limit when S is smooth, and its measurementis well-modeled classically. In order to understand thissurprising result, it is instructive to compare it to the casewhere s ! 1� and @ ! 0 together in such a way that� @

logs

tends to a finite limit A0 with fixed � and�. In this case, the

Weyl representation of St, e�½Aðqt;ptÞ=A0�þð@=2A0Þ þOð@2Þ, is

smooth and regular semiclassical analysis is applicable sothat LG violation is possible, but it can only be small, ofOð@Þ. Indeed, an explicit calculation gives hLGi¼2þffð�Þþgð�Þ½3cosð2�Þ�cosð6�Þ�g @

A0þOð@Þ2, where

fð�Þ ¼ � ð�2þ1Þð��1Þ4þ4�ð�2�1Þ28�3 and gð�Þ ¼ ð�2�1Þ2ð1þ�2Þ

16�3 .

For � ¼ �=8 and �> �cr, hLGi is larger than 2, but theLGI violation is small and does not imply breakdown ofMR.

This argument fails in the scaling limit: Although Schanges slowly between adjacent states when 1� s � 1,its phase-space representation is a rapidly varying functionwhen � ! 1, and naive semiclassical arguments fail. Putanother way, although S and H are both smooth observ-ables, they are mutually singular for large �, so that acombination of a measurement of S and an evolutionwith H can break MR.Implementation with entangled photons.—Quantum op-

tics offers a natural ground for implementing the LGI testproposed here. However, the standard method of countingphotons is destructive and prevents the second measure-ment in the LG setup from being carried out. We thereforepropose to mimic the LG scheme using an entangledphoton pair and a pair of number-resolving photon coun-ters such that the results effectively measure the LG ex-pectation value.Our proposed scheme is summarized in Fig. 3. We first

reverse the roles of H and S, such that H is the squeezedHamiltonian and S ¼ snphotons . The H ground state is then a

squeezed vacuum state jgi ¼ Pnc2nj2ni, where c2n ¼

½ ffiffiffi2

p�1=4=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ �Þp �½ð2n� 1Þ!!= ffiffiffiffiffiffiffiffiffiffiffið2nÞ!p �ð��1�þ1Þn. Our goal is

to obtain in the wake of a measurement of n photons acollapsed state jni. This is not practical for a single photon,but the same effect can be realized if the initial state is atwo-photon entangled state

Pn¼evencnjnni, jnni being a

state with n photons in both modes. A measurement of n1photons (with probability jcn1 j2) in the first mode collapses

the second to the Fock state jn1i. Next, time evolution bythe squeezed Hamiltonian is applied to the state in thesecond arm, and finally the number of photons in thesecond mode n2 is also measured. Since the photon num-bers are perfectly correlated, the value of sn1þn2 , averagedover a large number of repetitions, constitutes a measure-ment of one of the two correlation functions needed for theLG expectation value, and the second correlation functionis obtained by changing the squeezing interaction.

FIG. 3. A schematic proposal for testing the LGI indirectlywith a pair of entangled photons. The photon pair emitted fromthe source is in an entangled state

Pncnjnni with a fully

correlated photon number, where cn is the n-photon amplitudein the squeezed vacuum state. The number-resolving photoncounter in the top arm effectively performs the first measurementin the LG scheme on the photon state in the bottom arm, whichthen undergoes the second measurement in the bottom photoncounter. Different correlation functions can be measured bytuning the squeezing strength in the second mode.

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In principle, MR would be tested for strongly squeezedinitial states that have many significant photon state com-ponents, but current number-resolving photon counters candetect only a few photons. A more fundamental limitationis placed by decoherence as discussed below.

An experiment that realizes our proposed scheme is onlyan indirect demonstration of the violation of DR, since itmeasures violations of the LGI in a hypothetical systemwhose dynamics should follow the one studied in theexperiments according to the rules of quantum mechanics.On the other hand, the casting of the LGI into inequalitiesbounding expectation values in an entangled pair has fur-ther consequences of a more fundamental nature. It isrecognized that, in the setting of the Bell inequalities andtheir generalizations, the measurements by the two partiesmay be performed at an arbitrary time delay. Furthermore,the combination of the squeezing action and the secondphoton counting is equivalent to a measurement of a differ-ent observable, unitarily equivalent to S, so that the squeez-ing is analogous to the rotation of the apparatus in the Bellscheme. The upshot is that the violation of the LGI istantamount to a violation of the CHSH inequality in anassociated system such as the one shown in Fig. 3. Hence,DR in a subsystem is implied by local realism in thecomposite system.

The implication of DR by local realism has been dem-onstrated in the specific context of an entangled pair ofharmonic oscillators, but, since the demonstration did notrely on specific properties of harmonic oscillators, it holdsfor any quantum system that can be entangled with anothersystem and prepared in a diagonal state like the one shownin Fig. 3. On the other hand, we conjecture that the con-verse statement is false, since the Hilbert space dimensionof entangled states is greater than that of the subsystem.

Recently, there have been several studies of the break-down of phase-space local realism in two-photon en-tangled states [10,13–16]. In these works the breakdownof local realism was demonstrated using classically singu-lar observables related to the number parity operator. Incontrast, we demonstrate breakdown of MR and local real-ism by measurements of classically smooth observables.

Conclusions.—Our study has shed some light on thepersistent issue of the emergence of classical realism inclosed quantum systems, specifically the breakdown ofMR by the measurement process. We have shown thatthe principles of classical mechanics cannot describeeven the measurement of macroscopic observables with asmooth classical limit.

The results further restrict the validity of MR in closedsystems beyond the analysis of Refs. [2,7]. These worksfocused on the coarse-grained measurement of singularobservables and showed that MR can be broken for singu-lar Hamiltonians. Here the macroscopic nature of themeasurement process is realized with smooth observables,and MR is broken for a Hamiltonian that is itself smooth.This is possible because the Hamiltonian and the observ-able to be measured are mutually singular, but this is not in

principle an obstruction to a realization of the measure-ment and dynamics with a classical apparatus. Still, amacroscopic object is necessarily an open system, and ifthis is taken into account, breakdown of MR can beavoided if decoherence is strong enough. If this is indeedthe case, our results imply a minimum rate of decoherence.Namely, since violation of MR occurs on a time scaleinversely proportional to

ffiffiffiffiffiffiffiffininit

p, where ninit is the number

of S eigenstates in the initial state, the time scale fordecoherence must grow faster than

ffiffiffiffiffiffiffiffininit

p, when ninit 1.

Since the present violation of the LGI stems from thecounterfactual assumption of the possibility of noninvasivemeasurements, the question naturally arises of whetherweak MR, where this assumption is relaxed to allow in-vasive measurements that admit a hidden-variable under-pinning, is valid in the harmonic oscillator. It has beenrecently shown [9] that weak MR also has observableconsequences.A further conclusion from this work is that there is a

logical relation between the postulates of DR and those oflocal realism that facilitates an indirect experimental test ofMR with an entangled photon pair. It is likely connected toa general relation between temporal and spatial nonclassi-cality that has been derived in Ref. [17] by focusing on theunitary evolutions rather than on the quantum states.We thank H. Eisenberg and B. Reznik for helpful dis-

cussions. This work was supported by GIF Grant No. 980-184.14/2007.

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