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PHYSICAL REVIEW A, VOLUME 65, 033818

Double-slit quantum eraser

S. P. Walborn,1 M. O. Terra Cunha,1,2 S. Padua,1 and C. H. Monken11Departamento de Fı´sica, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, MG 30123-970, Brazil

2Departamento de Matema´tica, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, MG 30123-970, Braz~Received 17 November 2000; published 20 February 2002!

We report a quantum eraser experiment which actually uses a Young double slit to create interference. Theexperiment can be considered an optical analogy of an experiment proposed by Scully, Englert, and Walther@Nature ~London! 351, 111 ~1991!#. One photon of an entangled pair is incident on a Young double slit ofappropriate dimensions to create an interference pattern in a distant detection region. Quarter-wave plates,oriented so that their fast axes are orthogonal, are placed in front of each slit to serve as which-path markers.The quarter-wave plates mark the polarization of the interfering photon and thus destroy the interferencepattern. To recover interference, we measure the polarization of the other entangled photon. In addition, weperform the experiment under ‘‘delayed erasure’’ circumstances.

DOI: 10.1103/PhysRevA.65.033818 PACS number~s!: 42.50.Dv, 03.65.Ta, 42.50.Ar, 42.25.Kb

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I. INTRODUCTION

Wave-particle duality, a manifestation of the complemetarity principle, proposes many questions about the behaof particles in interferometers. It has long been known twhich-path information and visibility of interference fringeare complementary quantities: any distinguishability betwthe paths of an interferometer destroys the quality~visibility !of the interference fringes. The incompatibility betwewhich-path information and interference effects has bquantified through inequalities by various authors@1–6#.Originally, it was thought that the uncertainty principle wthe mechanism responsible for the absence of interferefringes due to a which-path measurement. The first andhaps most famous example of this idea is the Einstein-Bdialogue at the Fifth Solvay Conference in Brussels conceing Einstein’s recoiling double-slit gedanken experiment,which the momentum transfer from incident particles todouble slit is measured to determine the particles’ trajecto@7,8#. However, Bohr showed that the uncertainty in tknowledge of the double slit’s initial position was of thsame order of magnitude as the space between the inteence minima and maxima: interference fringes w‘‘washed out’’ due to the uncertainty principle@7#.

More recently, Scully and Dru¨hl have shown that, in certain cases, we can attribute this loss of interference not touncertainty principle but to quantum entanglement betwthe interfering particles and the measuring apparatus@9#. Forexample, disregarding internal degrees of freedom, werepresent the state of particles exiting an interferometer

uC&51

&@ uc1~r !&1uc2~r !&], ~1!

where uc1(r )& and uc2(r )& represent the possibility for thparticles to take path 1 or 2, respectively. The probabidistribution for one-particle detection at a pointr is given byz^r uC& z2; the cross termsc1(r )ur &^r uc2(r )& and ^c2(r )ur &3^r uc1(r )& are responsible for interference. The introdution of an apparatusM capable of marking the path taken b

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a particle without disturbinguc1(r )& or uc2(r )& can be rep-resented by the expansion of the Hilbert space of the sysin the following way:

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&@ uc1~r !&uM1&1uc2~r !&uM2&], ~2!

whereuM j& is the state of the which-path marker corresponing to the possibility of passage through the pathj. Thewhich-path marker has become entangled with the two psible particle states. A 100% effective which-path markerprepared such thatuM1& is orthogonal touM2&. In this case,a measurement ofM reducesuC& to the appropriate state fothe passage of the particle through path 1 or 2. However,disappearance of the interference pattern is not dependesuch a measurement. The which-path marker’s presealone is sufficient to make the two terms on the right-haside of Eq.~2! orthogonal and thus there will be no croterms inz^r zC& z2. Therefore, it is enough that the which-painformation is available to destroy interference. Moreovprovided thatuc1(r )& and uc2(r )& are not significantly per-turbed by the observer, one canerasethe which-path infor-mation and recover interference by correlating the partdetection with an appropriate measurement on the whpath markers. Such a measurement is known asquantumerasure. In addition, if the which-path marker is capablestoring information, the erasure can be performed even athe detection of the particle. The possibility of delayed esure generated a discussion with respect to its legitimawith the argument that it would be possible, in this way,alter the past@11,12#. This argument is founded on an erroneous interpretation of the formalism of quantum mechan@13,14#. In recent years, there have been a number of idand experiments~performed and proposed! in which which-path information is accessible without causing severe perbations to the interfering particles@10,9,15–30#. Among theproposals, we distinguish the ones due to Scully and Dr¨hl@9# and to Scully, Englert, and Walther@10# because of theiroriginality and pedagogical content.

©2002 The American Physical Society18-1

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WALBORN, CUNHA, PADUA, AND MONKEN PHYSICAL REVIEW A 65 033818

Because of their momentum, time, and polarization colation properties, photon pairs generated by spontaneparametric down-conversion play an important role inexperimental demonstrations of quantum eras@17,18,20,21,28,29#. Although the quantum erasure phenomenon is present in all reported experiments, only one@28# canbe considered an optical analog of the original proposaScully and Dru¨hl @9#. In this paper we report a quantumeraser experiment which actually uses a Young double slcreate interference. The experiment is analyzed in connecwith the proposal of Scully, Englert, and Walther~SEW!@10,31#. To the authors’ knowledge, this is the first demostration of a quantum eraser in which interference is obtaifrom the passage of the particles through a real double s

In Sec. II we give a brief summary of the SEW quantueraser. The theory behind our quantum eraser is presentSec. III. The experimental setup and results are presenteSecs. IV and V, respectively.

II. THE SCULLY-ENGLERT-WALTHERQUANTUM ERASER

The experiment reported here is inspired by the propoof Scully, Englert, and Walther@10#, which can be summarized as follows. A beam of Rydberg atoms in an excitstate is incident on a double slit small enough to formYoung interference pattern on a distant screen. In froneach slit is placed a which-path marker, which consists omicromaser cavity of appropriate length such that the emsion probability for an atom traversing the cavity is 1. Ththe presence of a photon in either cavity marks the passof an atom through the corresponding slit and thus destrthe interference pattern, because which-path informationow available. The perturbation to the spatial part ofwave function of the atoms due to the cavities is ignora@10,13,31#. A measurement that projects the state of the caties onto a symmetric~antisymmetric! combination ofu0& ~nophoton present! and u1& ~one photon present! performs theerasure, and an interference pattern is recovered in corredetection.

III. AN OPTICAL BELL-STATE QUANTUM ERASER

Consider the following experimental setup. A linearly plarized beam of photons is incident on a double slit. If tdouble slit is of appropriate dimensions, the probability dtribution for one-photon detection at a distant screen is giby a Young interference pattern. Suppose that in fronteach slit we place a quarter-wave plate, with the fast axian angle of 45°~or 245°! with respect to the photon polarization direction. Upon traversing either one of the waplates, the photon becomes circularly polarized, and acqua well-defined angular momentum@32#. Supposing that thewave plate is free to rotate, it should acquire an angumomentum opposite to that of the photon, and rotate righleft, depending on the chirality of the photon. If we treat eawave plate as a quantum rotor, we can say that the phinduced a transition withD l 561. Since the wave plates dnot significantly modify the propagation of the beam, w

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have, in principle, a which-path marker with necessary chacteristics for a quantum eraser. However, this scheme isfrom being practical. As well as the difficulty of setting thwave plates free to rotate, the separation between the enlevels of a rotor with the mass and dimensions of a waplate is of the order 10240 eV. In addition, decoherence efects may make it impossible to use macroscopic quanrotors to mark the path of a photon. This idea is similar to‘‘haunted measurement’’ of Greenberger and Ya’sin@33#.

By enlarging the system, however, it is possible to crean adequate which-path detector. Let the beam of phoincident on the double slit be entangled with a second befreely propagating in another direction, so as to define a Bstate

uC&51

&~ ux&suy&p1uy&sux&p), ~3!

where the indicess andp indicate the two beams, andx andy represent orthogonal linear polarizations. If beams is inci-dent on the double slit~without wave plates!, state ~3! istransformed to

uC&51

&~ uc1&1uc2&), ~4!

where

uc1&51

&~ ux&s1uy&p1uy&s1ux&p), ~5!

uc2&51

&~ ux&s2uy&p1uy&s2ux&p). ~6!

The indicess1 ands2 refer to beams generated by slit 1 aslit 2, respectively. The probability distribution for onephoton detection on a screen placed in the far-field regionthe overlapping beamss1,s2 will show the usual interfer-ence:

Ps~d!}11cosd, ~7!

where d is the phase difference between the paths sli→detector and slit 2→detector. Introducing thel/4 platesone in front of each slit with the fast axes at anglesu1545° andu25245° to thex direction, statesuc1& anduc2&are transformed to

uc1&51

&~ uL&s1uy&p1 i uR&s1ux&p), ~8!

uc2&51

&~ uR&s2uy&p2 i uL&s2ux&p), ~9!

whereR andL represent right and left circular polarizationSince uc1& and uc2& have orthogonal polarizations, thereno possibility of interference. In order to recover interfe

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DOUBLE-SLIT QUANTUM ERASER PHYSICAL REVIEW A65 033818

ence, let us project the state of the system over the symmand antisymmetric states of the which-path detector. Thiequivalent to transforminguc1& and uc2& in a way that ex-presses them as symmetric and antisymmetric combinatof polarizations, for example,

ux&51

&~ u1&1u2&), ~10!

uy&51

&~ u1&2u2&), ~11!

uR&512 i

2~ u1&1 i u2&), ~12!

uL&512 i

2~ i u1&1u2&), ~13!

FIG. 1. Experimental setup for the Bell-state quantum eraQWP1 and QWP2 are quarter-wave plates aligned in front ofdouble slit with fast axes perpendicular. POL1 is a linear poizer.

FIG. 2. Coincidence counts vs detector Ds position with QWP1and QWP2 removed. An interference pattern due to the doubleis observed.

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nswhere ‘‘1’’ and ‘‘ 2’’ represent polarizations145° and245°with respect tox. Rewriting the complete stateuC&, we have

uC&51

2@~ u1&s12 i u1&s2)u1&p1 i ~ u2&s11 i u2&s2)u2&p].

~14!

According to the above expression, we can recover interence projecting the state of photonp over u1&p or u2&p .Experimentally, this can be done by placing a polarizer inpath of beamp and orientating it at145° to selectu1&p or at245° to selectu2&p . The interference pattern is recoverethrough the coincidence detection of photonss andp. Noticethat the fringes obtained in the two cases are out of phThey are commonly calledfringesandantifringes.

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FIG. 3. Coincidence counts when QWP1 and QWP2 are plain front of the double slit. Interference has been destroyed.

FIG. 4. Coincidence counts when QPW1, QWP2, and POL1in place. POL1 was set tou, the angle of the fast axis of QWP1Interference has been restored in thefringe pattern.

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WALBORN, CUNHA, PADUA, AND MONKEN PHYSICAL REVIEW A 65 033818

A. Obtaining which-path information

Which-path information can be obtained by considerthe polarization of both photonss and p. The process ofobtaining information can be separated into two schemdetectingp befores, or detectings beforep, which we referto as delayed erasure. This can be done by changing threlative lengths of beamss and p. We will assume that onephoton is detected much earlier than the arrival of the otphoton at the measuring devices. Let us consider thepossibility. If photonp is detected with polarizationx ~say!,then we know that photons has polarizationy before hittingthel/4 plates and the double slit. By looking at Eqs.~4!, ~8!,and~9!, it is clear that detection of photons ~after the doubleslit! with polarizationR is compatible only with the passagof s through slit 1 and polarizationL is compatible only with

FIG. 5. Coincidence counts when QPW1, QWP2, and POL1in place. POL1 was set tou1p/2, the angle of the fast axis oQWP2. Interference has been restored in theantifringe pattern.

FIG. 6. Coincidence counts in the delayed erasureup. QWP1, QWP2, and POL1 are absent. A standard Young inference pattern is observed.

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the passage ofs through slit 2. This can be verified expermentally. In the usual quantum mechanics language, detion of photonp before photons has prepared photons in acertain state.

B. Delayed erasure

The possibility of obtaining which-path information aftethe detection of photons leads to delayed choice@34#. De-layed choice creates situations in which it is importanthave a clear notion of the physical significance of quantmechanics. A good discussion can be found in Refs.@11–14#.In as much as our quantum eraser does not allow the expmenter tochooseto observe which-path information or ainterference pattern after the detection of photons, it does

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FIG. 7. Coincidence counts in the delayed-erasure setupQWP1 and QWP2 in place in front of the double slit. No interfeence is observed.

FIG. 8. Coincidence counts in the delayed-erasure setup wQPW1, QWP2, and POL1 are in place. POL1 was set tou, theangle of the fast axis of QWP1. Interference has been restorethe fringe pattern.

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DOUBLE-SLIT QUANTUM ERASER PHYSICAL REVIEW A65 033818

allow for the detection of photons before photonp, a situa-tion which we refer to as delayed erasure. The question‘‘Does the order of detection of the two photons affect texperimental results?’’

IV. EXPERIMENTAL SETUP AND PROCEDURE

For certain propagation directions, type-II spontaneoparametric down-conversion in a nonlinear crystal creathe state

uc&51

&~ uo&sue&p1eifue&suo&p), ~15!

whereo and e refer to ordinary and extraordinary polariztions. f is a relative phase shift due to the crystal birefrgence. Iff50 or p we have the Bell statesuC1& anduC2&,respectively.

Using this state in the interferometer described in the pvious section, the probability of detecting photons in coindence is proportional to

1

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f

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~16!

whered is defined right after expression~7!, u is the smallestangle between the fast~slow! axis of the quarter-wave plateand theo axis, anda is the angle of the polarizer in pathp,with respect to theo axis.

The experimental setup is shown in Fig. 1. An argon la~351.1 nm at;200 mW! is used to pump a 1-mm-long BBO(b-BaB2O4) crystal, generating 702.2 nm entangled photoby spontaneous parametric down-conversion. The BBO ctal is cut for type-II phase matching. The pump beamfocused onto the crystal plane using a 1 mfocal length lensto increase the transverse coherence length at the double

FIG. 9. Coincidence counts in the delayed-erasure setup wQPW1, QWP2, and POL1 are in place. POL1 was set tou1p/2, the angle of the fast axis of QWP2. Interference has brestored in theantifringe pattern.

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The width of the pump beam at the focus is approximat0.5 mm @35#. The orthogonally polarized entangled photoleave the BBO crystal each at an angle of;3° with the pumpbeam. In the path of photonp a polarizer cube~POL1! can beinserted in order to perform the quantum erasure. The douslit and quarter-wave plates are placed in paths, 42 cm fromthe BBO crystal. DetectorsDs and Dp are located 125 and98 cm from the BBO crystal, respectively. QWP1 aQWP2 are quarter-wave plates with fast axes at an angl45°. The circular quarter-wave plates were sanded~tangen-tially! so as to fit together in front of the double slit. Thopenings of the double slit are 200mm wide and separatedby a distance of 200mm. The detectors are EG&G SPCM200 photodetectors, equipped with interference filters~band-width 1 nm! and 300mm35 mm rectangular collection slitsA stepping motor is used to scan detectorDs .

The delayed erasure setup is similar, with two changes~i!detectorDp and POL1 were placed at a new distance of 2from the BBO crystal and~ii ! the collection iris on detectoDp has dimensions 600mm35 mm.

V. EXPERIMENTAL RESULTS

Before the quantum eraser experiment was performBell’s inequality tests were performed to verify that etangled states were being detected@36#. Figure 2 shows thestandard Young interference pattern obtained with the douslit placed in the path of photons, without quarter-waveplates QWP1 and QWP2, and with POL1 absent from detor Dp . Next, the path of photons was marked by placingthe quarter-wave plates QWP1 and QWP2 in front ofdouble slit. Figure 3 shows the absence of interference duthe quarter-wave plates. Nearly all interference presenFig. 2 was destroyed. The residual interference present isto a small error in aligning the quarter-wave plates. Twhich-path information was erased and interference recered by placing the linear polarizer POL1 in front of detecDp . To recover interference, the polarization angle of PO~a! was set tou, the angle of the fast axis of quarter-wavplate QWP1. Interference fringes were obtained as showFig. 4. The detection time was doubled in order to compsate for the decrease in coincidence counts due to POL1Fig. 5, POL1 was set tou1p/2, the angle of the fast axis oQWP2, which produced a pattern of interference antifringThe averaged sum of these two interference patterns givpattern roughly equal to that of Fig. 3.

The same experimental procedure was used to prodFigs. 6–9 for the delayed-erasure situation. The experimeresults are comparable to the case in which photonp is de-tected before photons. We use the term ‘‘delayed choiceloosely, in that in our experiment there is no ‘‘choice’’ avaable to the observer in the time period after the detectionphotonss and before the detection of photonp. We simplywish to show that the order of detection is not important,concordance with the literature@13,14#.

VI. CONCLUSION

We have presented a quantum eraser that uses a Ydouble slit to create interference. The quarter-wave plate

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WALBORN, CUNHA, PADUA, AND MONKEN PHYSICAL REVIEW A 65 033818

our experiment served as the which-path markers to desinterference. We recovered interference using the entanment of photonss andp. Our quantum eraser is very similato the that of Scully, Englert, and Walther@10#. We haveshown that interference can be destroyed, by markingpath of the interfering photon, and recovered, by makingappropriate measurement on the other entangled photonhave also investigated this experiment under the conditof delayed erasure, in which the interfering photons is de-tected before photonp. In as much as our experiment did nallow for the observer to choose the polarization angle intime period after photons was detected and before detecti

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of p, our results show that acollapseof the wave functiondue to detection of photons does not prohibit one from observing the expected results. Our experimental data awith the proposal of Scully, Englert, and Walther that quatum erasure can be performed after the interfering parthas been detected@10#.

ACKNOWLEDGMENTS

The authors acknowledge financial support from the Bzilian agencies CNPq, CAPES, FINEP, PRONEX, aFAPEMIG.

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