lundeen steinberg 2008 0810.4229v1 hardy's

8/10/2019 Lundeen Steinberg 2008 0810.4229v1 Hardy's

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Experimental joint weak measurement on a photon pair as a probe of Hardys Paradox

J. S. Lundeen and A. M. Steinberg

Department of Physics, University of Toronto,60 St. George Street, Toronto ON, M5S 1A7, Canada

(Date textdate; Received textdate; Revised textdate; Accepted textdate; Published textdate)

AbstractIt has been proposed that the ability to perform joint weak measurements on post-selected systems would allow us to study

quantum paradoxes. These measurements can investigate the history of those particles that contribute to the paradoxicaloutcome. Here, we experimentally perform weak measurements of joint (i.e. nonlocal) observables. In an implementationof Hardys Paradox, we weakly measure the locations of two photons, the subject of the conicting statements behind theParadox. Remarkably, the resulting weak probabilities verify all these statements but, at the same time, resolve the Paradox.

Retrodiction is a controversial topic in quantum me-chanics [1]. How much is one allowed to say about thehistory (e.g. particle trajectories) of a post-selected en-semble? Historically this has been deemed a questionmore suitable for philosophy (e.g. counterfactual logic)than physics; since the early days of quantum mechan-

ics, the standard approach has been to restrict the basisof our physical interpretations to direct experimental ob-servations. On the practical side of the question, post-selection has recently grown in importance as a tool inelds such as quantum information: e.g. in linear op-tics quantum computation (LOQC) [2], where it drivesthe logic of quantum gates; and in continuous variablesystems, for entanglement distillation [3]. Weak mea-surement is a relatively new experimental technique fortackling just this question. It is of particular interest tocarry out weak measurements of multi-particle observ-ables, such as those used in quantum information. Here,we present an experiment that uses weak measurement toexamine the two-particle retrodiction paradox of Hardy[4, 5], conrming the validity of certain retrodictions andidentifying the source of the apparent contradiction.

Hardys Paradox is a contradiction between classicalreasoning and the outcome of standard measurementson an electron E and positron P in a pair of Mach-Zehnder interferometers (see Fig. 1). Each interferome-ter is rst aligned so that the incoming particle alwaysleaves through the same exit port, termed the brightport B (the other is the dark port D). The interfer-ometers are then arranged so that one arm (the Innerarm I) from each interferometer overlaps at Y. It is as-sumed that if the electron and positron simultaneouslyenter this arm they will collide and annihilate with 100%probability. This makes the interferometers Interaction-Free Measurements (IFM) [ 6]: that is, a click at the darkport indicates the interference was disturbed by an ob- ject located in one of the interferometer arms, withoutthe interfering particle itself having traversed that arm.Therefore, in Hardys Paradox a click at the dark port of the electron (positron) indicates that the positron (elec-tron) was in the Inner arm. Consider if one were to detect

both particles at the dark ports. As IFMs, these resultswould indicate the particles were simultaneously in theInner arms and, therefore should have annihilated. Butthis is in contradiction to the fact that they were actu-ally detected at the dark ports. Paradoxically, one doesindeed observe simultaneous clicks at the dark ports [7],

just as quantum mechanics predicts.Weak measurements have been performed in classical

optical experiments [8], as well as on the polarization of single photons [ 9]. Weak measurements of joint observ-ables are particularly important, as this class of observ-ables includes nonlocal observables, which can be usedto create and identify multiparticle entanglement (e.g.in cluster state computing [ 10]). Joint observables alsoinclude sequential measurements on a single particle, al-lowing them to characterize time-evolution in a system[11]. In this experiment, we demonstrate a new tech-nique that for the rst time enables us to perform jointweak measurements. With this technique, we implementa proposal by Aharonov et al. [5] to weakly measure thesimultaneous location of the two path-entangled photonsin Hardys Paradox [4]. This technique opens up thepossibility of in situ interrogation and characterizationof complex multiparticle quantum systems such as thoseused in quantum information.

A standard measurement collapses the measured sys-tem, irreversibly destroying the original quantum stateof the system. Post-selected subensembles are particu-larly difficult to investigate since measurements on theensemble before the post-selection will collapse the sys-tem and, thus, alter the action of the post-selection it-self. Weak measurement was devised by Aharonov, Al-bert, and Vaidman as a way of circumventing these prob-lems [12]. It is an extension of the standard von Neu-mann measurement model [ 13] in which the coupling gbetween the measured system and the measurement de-vice is made asymptotically small. This has the drawbackof reducing the amount of information one retrieves in asingle measurement. The reward is that the consequentdisturbance of the measured system is correspondinglysmall. To extract useful information, one must repeat the

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BS1E

O E

BS1P

IP

e + e -

IEO P

B PD P

BS2P BS2E

Y

D EB E

FIG. 1: Hardys Paradox setup. E and P indicate electron andpositron, respectively, which can collide in region H. BS1P,BS1E, BS2P, and BS2E are 50:50 beamsplitters. D and B arethe dark and bright ports of the interferometers. I and O arethe Inner and Outer interferometer arms.

measurement on a large ensemble of identical quantumsystems. The average result is called the weak value,denoted C

W , where C is the measured operator.

To set up Hardys Paradox we use two photons in-stead of the electron and positron. The experimentalsetup is shown in Fig. 2. A diode laser produces a 30mW 405 nm beam (blue dashed line) which is lteredby a blue glass lter (BF) and sent through a dichroic

mirror (DM). This beam produces 810 nm photon pairs(red solid line) in a 4 mm long BBO crystal through theprocess of Type II spontaneous parametric downconver-sion. These pairs, consisting of a horizontal (E) photonand a vertical (P) photon, take the place of the electronand positron. The pump passes through a second DM, tolater be retroreected. The photon pair passes through alter (F) to remove any residual pump light, followed bya 2mm thick BBO crystal (CC), to compensate for thebirefringent delay in the rst crystal. The photon pairthen meets a 50/50 beamsplitter (BS1EP), which actsas the rst beamsplitter in both the E and P interfer-ometers, so that each photon can either be retroreected

and enter the Inner arm or be transmitted and enter theOuter arm.In place of electron-positron annihilation, a quan-

tum interference effect acts as an absorptive two-photonswitch (Y) [ 14]. Photons reected into the Inner arm passback through the BBO crystal along with the retrore-ected pump beam. The amplitude for the retroreectedpump to create a pair of photons in the crystal is set tointerfere destructively with amplitude for a photon pairin Inner arms. Thus, if both the E and P photons en-

PBS

Dark Port EDark Port P

DMDM

BS1EP

BS2E

BBOH

V CC

PBS

( Y )

BF

/2

/2

/2

/2

BS2P

Q

Q

QQ

PA PA

Laser

N(IP ) N(IE )

N(O P )

N(O E )

F

FIG. 2: The experimental setup. Labels are explained in thetext.

ter their Inner arms they are removed, whereas if only asingle photon enters, it passes through the crystal unim-

peded.Photons transmitted at the rst beamsplitter enter the

Outer arms, which contain a variable delay. Next, boththe Inner and Outer paths encounter polarizing beam-splitters (PBS) so that the E and P photons are splitinto their own spatially separate interferometers. The Pinterferometer contains an additional variable delay sothat both interferometers can be adjusted to have thesame path-length difference. The Inner and Outer pathsof the two interferometers are recombined at two non-polarizing beamsplitters (BS2E and BS2P), taking theplace of the nal Mach-Zehnder beamsplitters for theelectron and positron. Tilted quartz pieces (Q) beforeand after the NPBSs compensate for undesired polariza-tion phase-shifts in them.

Placing a half-waveplate in an arm allows us to mea-sure whether a photon travelled through this arm. Tounderstand how this functions consider a half-waveplateplaced in the E Outer arm aligned so as rotate the po-larization of a photon passing through it by 90 . Thepolarization of the photon arriving at the E dark portthen perfectly indicates if it was in the E Outer arm ornot. This a measurement of the occupation N (M K )of the M = I or O (Inner or Outer) interferometer armby photon K = E or P (e.g. N (O E ) = |O E OE |). Un-

fortunately this procedure is a standard projective mea-surement and, hence strongly disturbs the system. Inparticular, the interference will be destroyed as the twopaths are now completely distinguishable and, thus, theinterferometer will not function as an IFM. The strengthof the measurement interaction ( U = exp ig N y ) isparameterized by g , the polarization rotation. Inthis experiment, we reduce this disturbance by rotatingthe photons polarization by only 20 , reducing g four-fold, and thereby performing a weak measurement. The

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trade-off is that it is now impossible to know which arma particular detected photon went through. Instead, wemeasure the average polarization rotation at the detectorover many trials to nd what fraction of photons passedthrough that particular arm. If no rotation is observedthen the classical inference would be that the photon wasnever in the arm with the waveplate. Conversely, if wemeasure an average rotation of 20 one might infer thatevery photon passed through the waveplate. Quantummechanically, this rotation constitutes a weak measure-ment of the occupation N of a particular interferometerarm.

The crux of the paradox is that the detected photonscannot have simultaneously been in the Inner arms. Totest this we require a weak-measurement of the joint oc-cupation of two arms. It was previously thought that aphysical interaction between the particles was necessaryto make weak-measurements of joint observables (e.g. theelectrostatic interaction of ions, as in Ref. [15]). In Refs.[16], we theoretically showed that one only needs to per-form single-particle weak measurements on each particle.The joint weak values then appear in polarization corre-lations between the two particles as follows:

N (M K )W

= g 1 Re zK (1)

N (M E ) N (M P )W

= g 2 Re zE zP , (2)

where zK = ( xK i yK ) is the z-basis lowering op-erator for the polarization of photon K = E or P . Inpractice, we independently measure g for each arm to ac-count for polarization-dependent losses. We weakly mea-sure all four combinations of N (M E ) N (M P ) by placinghalf-waveplates ( /2) in all four arms just before the nalbeamsplitters. We measure the occupation of a particularpair of arms by rotating only those two waveplates. Afterthe nal beamsplitters we measure average polarizationrotations as well as the correlations specied in Eq. 2with polarization analyzers (PA) consisting of a quarter-waveplate and polarizer followed by a single-photon de-tector (Perkin Elmer SPCM-AQR). Once the Pauli oper-ators are substituted in Eq. 2 and the real part is found,four Pauli operators remain in the nal expectation value.For each of these Pauli operators, the analyzer must beset to two positions (e.g. for x , 45 and 45 ( and) and for y , right-hand circular and left-hand circular( and )). Thus, each joint weak value requires eightmeasurements of coincidence rates at the two dark ports:

Re zE zP =

R + R R R R + R + R + R

R + R R RR + R + R + R

, (3)

where R sq is the coincidence rate when the P (E ) an-alyzer is set to s(q ). Single weak values for the occu-pation of photon E (P ) are found from these rates by

summing over analyzer settings for photon P (E ). Asan example, we give the measurements contributing to

N (OE ) N (O P )W

: R = 556 , R = 583 , R =834, R = 730 , R = 571 , R = 543 , R =666, R = 750 (all in counts per 420s) and g2 = 0 .365.

In Table 1 we present the weak values for the variousarm occupations. The bottom cells and rightmost cells

give the weak value for the occupation of a single armand the inner cells give the joint occupation of a pair of arms. Error bars are derived from uncertainties in g andstatistical variations in the rates.

N (I P ) N (O P )

N (I E ) 0.245 0.068

[0]0.641 0.083

[1]0.926 0.015

[1]

N (OE ) 0.719 0.074

[1] 0.759 0.083

[ 1] 0.078 0.02

[0]

0.924 0.024[1]

0.087 0.023[0]

Table 1. The weak values for the arm occupations inHardys Paradox.

Examining the table reveals that the single-particleweak measurements are consistent with the clicks at eachdark port; as the IFM results imply, the weakly measuredoccupations of each of the Inner arms are close to one andthose of each of the Outer arms are close to zero. Theweak measurements indicate that, at least when consid-ered individually, the photons were in the Inner arms.However, if we instead examine the joint occupation of the two Inner arms, it appears that the two photons areonly simultaneously present roughly one quarter of the

time. This demonstrates that, as we expect, the parti-cles are not in the inner arms together.So far, we seem to have conrmed both of the premises

of Hardys Paradox: to wit, that when D P and D E re,N (I P ) and N (I E ) are close to one (since the IFMs in-dicate the presence of the particles in Y) but thatN (I P &I E ) is close to zero (since when both particlesare in Y, they annihilate and should not be detected).This is odd because in classical logic, N (I P &I E ) mustbe N (I P ) + N (I E ) 1; this inequality is violatedby our results. Although N (I E ) is 93% and N (I P ) is92%, the data in Table 1 suggest that when E is in theInner path, P is not, and vice versa; hence the large

values for N (I E &OP ) = 64% and N (O E &I P ) = 72%.The fact that the sum of these two seemingly disjoint joint-occupation probabilities exceeds 1 is the contradic-tion with classical logic. In the context of weak mea-surements, the resolution of this problem lies in thefact that weak valued probabilities are not required tobe positive denite [5], and so a negative occupationN (O E &O P ) = 76% is possible, preserving the prob-ability sum rules. In an ideal implementation of HardysParadox, the joint probabilities are strictly 0 for both

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particles to be in Inner arms, 1 for both to be in theOuter, and 1 for either to be in the Inner while the otheris in the Outer arm. These are indicated in brackets inTable 1, for comparison with our experimental data. Dis-crepancies are because of the imperfect switch efficiency(85 3%) and IFM probabilities (95 3% for the E IFMand 94 4% for P).

What is the meaning of the negative joint occupation?Recall that the joint values are extracted by studyingthe polarization rotation of both photons in coincidence.Consider a situation in which both photons always simul-taneously passed through two particular arms. When apolarization rotator is placed in each of these arms itwould tend to cause their polarizations to rotate in acorrelated fashion; when P was found to have 45 polar-ization, E would also be more likely to be found at 45

than 45 . Experimentally, we nd the reverse whenP is found to have 45 polarization, E is preferentiallyfound at 45 (and vice versa), as though it had rotatedin the direction opposite to the one induced by the phys-ical waveplate. As in all weak measurement experiments,a negative weak value implies that the shift of a physicalpointer (in this case, photon polarization) has the op-posite sign from the one expected from the measurementinteraction itself.

In summary, Hardys Paradox is a set of conictingclassical logic statements about the location of the par-ticles in each of two Mach-Zehnder interferometers. Itis impossible to simultaneously verify these statementswith standard measurements since testing one statementdisturbs the system and consequently nullies the otherstatements. We attempt to minimize this disturbance byreducing the strength of the interaction used to perform

the measurement. The results of these weak measure-ments indicate that all the logical statements are correctand also provide a self-consistent, if strange, resolutionto the paradox. Since they do not disturb subsequentpost-selection of the systems under study, weak measure-ments are ideal for the interrogation and characteriza-tion of post-selected multiparticle states such as GHZor Cluster states, and processes such as Linear Optics

Quantum Computation. This experiment demonstratesa new technique that, for the rst time, allows for theweak measurement of general multiparticle observablesin these systems.

This work was supported by NSERC, QuantumWorks,CIPI, and the Canadian Institute for Advanced Research.We thank Kevin Resch and Morgan Mitchell for helpfuldiscussions.

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lundeen steinberg 2008 0810.4229v1 hardy's

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