PHYSICAL REVIEW A DECEMBER 1999VOLUME 60, NUMBER 6

Quantitative wave-particle duality and nonerasing quantum erasure

Peter D. D. Schwindt,1,* Paul G. Kwiat,1,† and Berthold-Georg Englert2,3

1Physics Division, P-23, Los Alamos National Laboratory, Los Alamos, New Mexico 875452Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany

3Department of Physics, Texas A&M University, College Station, Texas 77843-4242~Received 3 May 1999!

The notion of wave-particle duality may be quantified by the inequalityV21K2<1, relating interferencefringe visibility V, and path knowledgeK. With a single-photon interferometer in which polarization is used tolabel the paths, we have investigated the relation for various situations, including pure, mixed, and partiallymixed input states. A quantum-eraser scheme has been realized that recovers interference fringes even when nowhich way information is available to erase.@S1050-2947~99!02911-X#

PACS number~s!: 03.65.Bz, 42.50.2p, 07.60.Ly

sisa

-

mten

e

noinys

bo

nts.

—p

a

e

t

-t

er

’’

al

rcesee-

es

le

ion

P,

ere—

asere

e0

to

ad

on

INTRODUCTION

Wave-particle duality~WPD! dates back to Einstein’sseminal paper on the photoelectric effect@1#, and is a strikingmanifestation of Bohr’s complementarity principle@2# ~for aformal definition, see Ref.@3#!. The familiar phrase ‘‘eachexperiment must be described either in terms of particlein terms of waves’’ emphasizes the extreme cases and dgards intermediate situations in which particle and wavepects coexist. Theoretical investigations@4,5#, supplementedby a few experimental studies@6,7#, have led to a quantitative formulation of WPD@Eq. ~1! below#. Here we report anexperiment using a single-photon Mach-Zehnder interferoeter in which polarization marks the path. We investigathe entire scope of the duality relation for pure, mixed, apartially mixed input states, and foundabsoluteagreement atthe percent level@8#. We also realized a quantum-erasscheme, whereby interference is recoverable althoughwhich way ~WW! information was available to erase. Iview of the kinematical equivalence of all binary degreesfreedom, our results are directly applicable whenever anterfering particle is entangled with a two-state quantum stem.

To quantify WPD, one needs quantitative, measuracharacteristics for the wavelike and particlelike behaviorquanta. In an interferometer, the former is naturally quafied by thevisibility V of the observed interference fringeThe quantification of the latter is based on thelikelihood Lofcorrectly guessing the path taken by a particular quantumthe better one can guess, the more pronounced are theticle aspects. A random guess givesL5 1

2 , whereasL51indicates that the way is known with certainty. The actuWW knowledge Kis given byK52L21, with 0<K<1. Inanasymmetricinterferometer one way is more likely than thother to begin with (La priori.

12 ); we call WW knowledge

of this kindpredictability (P52La priori21). The statemen

*Present address: Department of Physics, University of ColorBoulder, Colorado 80309.

†Author to whom correspondence should be addressed. Electraddress: Kwiat@lanl.gov

PRA 601050-2947/99/60~6!/4285~6!/$15.00

orre-s-

-dd

rno

f--

lefi-

ar-

l

V21P2<1 has been known for some time, implicitly or explicitly, in various physical contexts@4,6#. Since one cannolosea priori knowledge,P<K; in fact, P'0 in our experi-ments. Nevertheless, owing to anentanglementof the systemwave function with the wave function of some WW mark~WWM!, the knowledge can still be as large as 1.

The actual value ofK depends on the ‘‘betting strategyemployed; the optimal strategy maximizesK and identifiesthe distinguishability D5max$K% — it is the maximumamount of WW knowledge available, although a nonoptimmeasurement may yield less or even zero.~Experimental in-accessibility of some crucial degrees of freedom may fothe experimenter to settle for a nonoptimal measurement;Ref. @9# for further remarks.! Except where noted, our measurements were suitably optimized to maximizeK. The du-ality relation accessible to experimental test then becom@5,9#

V21K2<1. ~1!

The equality holds for pure initial states of the WWM, whithe inequality applies to~partially! mixed states.

I. EXPERIMENTAL SETUP AND PROCEDURE

In our experiments single photons~at 670 nm) were di-rected into a compressed Mach-Zehnder interferometer@10#~see Fig. 1!. An adjustable half wave plate~HWP! in path 1was used to entangle the photon’s path with its polarizat~i.e., with the WWM! thus yielding WW knowledge@11#.Our adjustable analysis system — quarter wave plate, HWand calcite prism~PBS! — allowed the polarization WWMto be measured in any arbitrary basis. The photons wdetected using geiger-mode avalanche photodiodessingle-photon counting modules~EG&G #SPCM-AQ, effi-ciency ;60%). The input source described below wgreatly attenuated so that the maximum detection rates walways less than 50 000 s21; for the interferometer passagtime of 1 ns, this means that on average fewer than 124

photons were in the interferometer at any time.The probability for having no photon at all is close

o,

ic

4285 ©1999 The American Physical Society

thdcke

ena

ostaic

icto, aicdi

omle

ainn

acm

hpon

s.teel

nd

eens

g

b-he

al-ad,

om-ede-

or-M

fec-des.ed

er-iza-ndeo-

amt

e

rs

elync-theoolidasis

4286 PRA 60SCHWINDT, KWIAT, AND ENGLERT

unity at any arbitrary instant, but state reduction removesparta posteriorias soon as a detector ‘‘clicks.’’ The reducestate is virtually indistinguishable from a one-photon Fostate because the probability for two or more photons is nligibly small. This one-photon-at-a-time operation is esstial to allow sensible discussion of the likely path taken byindividual light quantum@12#.

Perhaps unnecessarily, we emphasize that our experimis not intended to be a direct proof of the quantum naturelight. Rather, we accept the existence of photons as an elished experimental fact@13#. The quantized electromagnetfield has a classical limit as a field~unlike other quantumfields that have, at best, a limit in terms of particles!, andsome properties of the quantum field have close classanalogs. In particular, the counting rates of single-phointerferometers, such as the one used in our experimentproportional to the intensities of the corresponding classelectromagnetic field. But there is no allowance for invidual detector clicks in Maxwell’s equations@14#, nor forthe quantum entanglement of photonic degrees of freedthat we exploit. And clearly the trajectory of a light quantuthrough the interferometer is a concept alien to classical etrodynamics, as is the experimenter’s knowledgeK aboutthis trajectory.

For visibility measurements the polarization analyzer wlowered out of the beam path, and the maximum and mmum count rates on detector 1 were measured as the leof path 2 was varied slightly~via a piezoelectric transducer!.After subtracting out the separately measured detector bground~i.e., the count rate when the input to the interferoeter was blocked, typically 100–400 s21), the visibility wascalculated in the standard manner:V5(Max2Min)/(Max1Min).

For the determination of the likelihood, and hence tknowledge, the following procedure was used. With thelarization analyzer in place, and path 2 blocked, the couon the two detectors were measured. Detector 1~2! looked atpolarization l (l'), determined by the analysis settingAfter subtracting the backgrounds measured for each detor, the count rates from detector 1 were scaled by the rtive efficiency of the two detectors:h2 /h151.1160.01. ~Inthis way our calculated value of the knowledge correspo

FIG. 1. ~a! Experiment to quantify the wave-particle duality.~b!An asymmetric Mach-Zehnder interferometer with polarizing besplitters separates the horizontal and vertical components ofinput light by much more than the;1-cm coherence length; oncan thus make arbitrary quantum states~pure, mixed, and partiallymixed! by varying the input polarization. A single-mode fiber filtethe spatial mode.

is

g--

n

entfb-

alnre

al-

m

c-

si-gth

k--

e-ts

c-a-

s

to what would have been measured if our detectors had bidentical and noiseless.! Call the resulting scaled rateR1l[R (path 1, polarizationl) and R1l'[R (path 1,polarization l'). Next, we measure the correspondinquantities for path 2:R2l andR2l'. The betting strategy isthe one introduced by Wootters and Zurek@4# and optimizedin Ref. @5#: Pick the path which contributes most to the proability of triggering the detector that has actually fired. Tlikelihood is then

L5max$R1l ,R2l%1max$R1l',R2l'%

R1l1R2l1R1l'1R2l'

. ~2!

II. EXPERIMENTAL RESULTS

A. Wave-particle duality for pure states

Figure 2 shows the results when a pure verticpolarization state~V! was input into the interferometer, asfunction of the internal HWP’s orientation. As expectewhen the HWP is aligned to the vertical (uHWP50), there-fore leaving the polarization unchanged, we see nearly cplete visibility and obtain no WW knowledge. The measurvalues ofV are slightly lower than the theoretical curve bcause the intrinsic visibility of the interferometer~even with-out the HWP! is only ;98%, due to nonideal optics@15#.Conversely, with the HWP set~at uHWP545°) to rotate thepolarization in path 1 to horizontal (H), the visibility is es-sentially zero, and the knowledge, nearly equal to 1. Fmally, the spatial wave function and the polarization WWwave function are completely entangled by the HWP:uc&}u1&uH&WWM1eifu2&uV&WWM , where f is the relativephase between paths 1 and 2. Tracing over the WWM eftively removes the coherence between the spatial moThat a small visibility persists in our results can be explainby slight residual polarization transformations by the intferometer mirrors and beam splitters, so that the polartions from the two paths are not completely orthogonal; aby the remarkable robustness of interference — both thretically and experimentally,V.4.4% even thoughL.99.9%.

he

FIG. 2. Experimental data and theoretical curves for a purvertically polarized input to the system shown in Fig. 1, as a fution of the orientation of a HWP in path 1. The crosses aremeasured visibilities, the dotted line and triangles correspond tKmeasurements fixed in the horizontal-vertical basis, and the sline and diamonds correspond to measurements in the optimal b@16#.

n

ow

ouhe-in-

mci-r

isth

o

.

ofla-tesmtheut

s

toe a

ofci-

idlyed

re.

e

re-

Wour’’

cen-

uce2ly-s-e

di-terhey

tumble

an-Wles

v-nalwe

bys-

an-

hes

id

iht

PRA 60 4287QUANTITATIVE WAVE-PARTICLE DUALITY AND . . .

In Fig. 2 we also display two sets of knowledge data, otaken in the optimal basis@16#, the other fixed in thehorizontal-vertical basis. These data demonstrate that knedge can depend on the measurement technique. Withoptimal basis, the value ofV21K2 is always very close tothe predicted unit value; to the best of our knowledge,experiment is the first to verify this. The average of all tdata points in Fig. 2 gives 0.97660.017. The slight discrepancy with the predicted value of 1 is mostly due to thetrinsic visibility of the interferometer — for the minimumvisibility arrangement,V21K250.998.

B. Wave-particle duality for „partially …mixed states

Using photons from an attenuated quartz halogen lathat was spectrally filtered with a narrow-band interferenfilter ~centered at 670 nm, 1.5-nm full width at half maxmum! and spatially filtered via a single-mode optical fibewe explored Eq.~1! for mixedstates~slight polarizing effectsfrom the fiber actually led to;4% residual net polarization!.The measurements of visibility and knowledge for thnearly completely mixed input state have values close totheoretical prediction of 0@Fig. 3~a!#. K→0 for a completelymixed WWM state because any unitary transformationsan unpolarized input also yield an unpolarized state~the den-sity matrix is unaffected!, so there is no WW information

FIG. 3. Visibility and knowledge measurements~and theorycurves! for various mixed and partially mixed input states.~a! Amixed state input from the filtered white-light source yieldedaverage value forV21K2 of 0.00360.001; the theoretical prediction based on the measured input state is 0.01760.003~the uncer-tainty in the theory comes from imperfect determination of tstate!. The slight disparity arises from residual polarization tranformations by the empty interferometer.~b! A partially mixed state~purity50.6560.01) was generated using the tunable source.~c! Asummary of allV21K2 data for various input states. The solcurve is the uncorrected theory:V21D252 Tr(r2)21, wherer isthe density matrix of the polarization WWM. The dashed curvethe theory accounting for the maximum visibility and the sligpolarization dependence of the empty interferometer.

e

l-the

r

-

pe

,

e

n

That V→0 can be understood by examining the behaviororthogonal pure WWM states, with no definite phase retionship between them. In the basis where the HWP rotathe WWM states by 90°, the orthogonal polarizations fropaths 1 and 2 cannot interfere; in the basis aligned withHWP’s axes, each polarization individually interferes, bthe interference patterns are shifted relatively by 180°~dueto the birefringence of the HWP!, so the sum is a fringelesconstant.

To enable production of an even more mixed input, andallow generation of arbitrary partially mixed states, we us‘‘tunable’’ diode-laser scheme@see Fig. 1~b!#. By rotatingthe ~pure linear! polarization input to the first polarizingbeam splitter, one can control the relative contributionhorizontal and vertical components. For example, for indent photons at 45°, one has equalH and V amplitudes,which are then added together with a random and rapvarying phase to produce an effectively completely mixstate of polarization@17#. With five times more vertical thanhorizontal, the state is then 1/3 completely mixed to 2/3 puThis case is shown in Fig. 3~b!. Note that the maximumvisibility ~and knowledge! is numerically equal to the statpurity, as the mixed component displays no interference~andcontains no WW information!. The data taken for variousinput states show excellent agreement with theoretical pdictions @Fig. 3~c!#.

C. Quantum erasure „erasing and nonerasing…

In contrast to many interference situations where the Winformation may be inaccessible, the quantum state ofWWM is easily manipulated. One can then in fact ‘‘erasethe distinguishing information and recover interferen@3,18# ~though this simple physical picture fails when nopure WWM states are considered!. In our experiments, suchan erasure consists of using a polarization analysis to redor remove the WW labeling. For example, if path 1 andpolarizations are horizontal and vertical, respectively, anasis at645° will recover complete fringes; any photon tranmitted through a 45° polarizer is equally likely to have comfrom either path.

Figure 4~a! shows quantum-eraser data under the contion that a pure vertical photon is input to the interferomeand rotated by the HWP in path 1 by either 90° or 20°. Tvisibility on detector 1after the analyzer can assume anvalue from 0 to 1, the latter case being a complete quanerasure. Even for a completely mixed state, it is still possito recover interference@Fig. 4~b!#. With no WW informationto erase, thisnonerasing quantum erasuremay seem quiteremarkable at first. However, the essential feature of qutum erasure is not that it destroys the possibly available Winformation, but that it sorts the photons into subensemb~depending on the quantum state of the WWM! each exhib-iting high-visibility fringes. Complete interference is recoerable by analyzing along the eigenmodes of the interHWP — along one axis we see fringes, along the othersee ‘‘antifringes,’’ shifted by 180°@19#. More generally, onepostselects one of the WWM eigenstates as determinedthe interaction Hamiltonian of the interfering quantum sytem and the WWM@20#.

-

s

of

l-

anin

nenthme

ocney.i

tio

ity,le-

dis-x-ef.

ibil-

tedre-cee

y ismark-

elyhe

er-or

za-ton

eo-

a-leuch

hat,s of-

ed atwell

elf-detheneeri-

y,ely,e

is

mp-s be-

usthal

1

da

4288 PRA 60SCHWINDT, KWIAT, AND ENGLERT

For a partially mixed WWM state, just as the valueV21K2 lies part way between 1~pure state! and 0 ~com-pletely mixed state!, the analysis angles yielding zero visibiity also fall between those for pure and mixed states@Fig.4~c!#. Quantitatively, for a fractional puritys, the angles areat uHWP6 1

2 arccos@s cos(2uHWP)#; consult Ref.@21# for fur-ther details.

A convenient geometrical visualization of our results cbe had by considering the polarization analysis in the Pocaresphere representation@22# in which all linear polariza-tions lie on the equator of the sphere, circular polarizatiolie on the poles, and arbitrary elliptical states lie in betweAny two orthogonal states lie diametrically opposed onsphere. For pure polarization input states to our interferoeter, there are, in general, exactly two points on the sphfor which the interference visibility will be exactly equal tzero. These correspond to the polarizations where a detesees light from only one of the interferometer paths. Alothe entire great circle that bisects the line connecting thtwo points, the quantum eraser will yield perfect visibilitCuriously, the situation for a completely mixed input statereversed. Here there are, in general, exactly two polariza

FIG. 4. ‘‘Quantum-eraser’’ data and theory curves for varioinput states. The minima on the curves correspond to analysistransmits light from only one or the other path; the maxima fmidway between these minima.~a! A purely vertically polarizedinput ([90°), with the polarization rotated by the HWP in pathby 90° ~circles, solid line;uHWP545°) or 20° ~triangles, dashedline; uHWP510°); ~b! a completely mixed state, withuHWP545°;and ~c! a partially mixed state~1:2 pure to mixed; circles, solidline!, with the HWP atuHWP522.5° — the dotted and dashecurves show the corresponding theoretical predictions for purecompletely mixed states, respectively.

-

s.

e-re

torgse

sn

states for which the quantum eraser recovers unit visibilcorresponding to the eigenmodes of the polarization ements inside the interferometer; on the great circle equitant from these two points the visibility vanishes. For eample, in some mixed-state experiments described in R@21#, the eigenmodes are the poles on the Poincare´ sphere,and the great circle corresponds to the equator — no visity is observed for any linear polarization analysis.

III. DISCUSSION

Our results demonstrate the validity of Eq.~1! at the per-cent level. Moreover, they highlight some features associawith mixed states, which may not have been widely appciated. Namely, that it is possible for both the interferenvisibility and the path distinguishability to equal zero. Whave also seen that in some cases where the visibilitintrinsically equal to zero, it is possible to perform quantuerasure on the photons and recover the interference. Remably, this is true even when the input state is completmixed and there exists no WW information to erase. Toperation of the polarizer is essentially toselecta suben-semble of photons. Depending on how this selection is pformed, we may recover fringes, antifringes, no fringes,any intermediate case.

The WW labeling in our experiment arose from anen-tanglementbetween the photon’s spatial mode and polarition state. It could just as well have been with another phoaltogether, as in the experiments in@23#, or even with a dif-ferent kind of quantum system@24#. The same results arpredicted, as long as the WW information is stored in a twstate quantum system~e.g., internal energy states, polariztion, spin, etc.!. More generally, our findings are extendibto analogous experiments with quanta of different kinds sas, for example, interferometers with electrons@25#, neutrons@26#, or atoms@7,8#.

To counter a possible misunderstanding let us note tquite generally, entanglement concerns different degreefreedom ~DOF’s!, not different particles. For certain purposes, such as quantum dense coding@27# or quantum tele-portation @28#, it is essential that the entangled DOF’s bcarried by different particles and can thus be manipulatea distance. For other purposes, however, one can just asentangle an internal DOF of the interfering particle itswith its center-of-mass DOF@29#. In our experiment the photon’s polarization DOF is entangled with the spatial moDOF represented by the binary alternative ‘‘reflected atentry beam splitter, or transmitted?’’ Analogously, hyperfilevels of an atom were used to mark its path in the expments of Refs.@7,8#.

In the extreme situation of perfect WW distinguishabilitthe entangled state is of the form stated in Sec. II A, namuc&}u1&uH&1eifu2&uV&. Appropriate measurements on thspatial DOF~defined byu1& and u2&) and the polarizationDOF (uH& and uV&) would show that the entanglementindeed so strong that Bell’s inequality@30# is violated. Ofcourse, inasmuch as one cannot satisfy the implicit assution that the measurements on the entangled subsystemspacelike separated, this violation of Bell’s inequality im

atl

nd

en

serhanofta

forofsityc-is-

PRA 60 4289QUANTITATIVE WAVE-PARTICLE DUALITY AND . . .

plies nothing about the success or failure of local-hiddvariable theories; however, this is not relevant here.

Finally, we would like to mention that further progreswas made since the completion of the work reported hExperimental tests of more sophisticated inequalities tEq. ~1! were performed@31#, and there was progress itheory as well@32#. In particular, the quantitative aspectsquantum erasure were investigated beyond the initial sreached in Ref.@9#.

.

.r,

atte

nb-

,ndec49

c-le

yn

asicll

ri-elhe

-

e.n

ge

ACKNOWLEDGMENTS

B.G.E. is grateful to Helmut Rauch and collaboratorstheir hospitality at the Atominstitut in Vienna, where partthis work was done, and he thanks the Technical Univerof Vienna for financial support. P.G.K. and P.D.D.S. aknowledge Andrew White for helpful discussions and asstance.

p-

icalns.s aised

ent

e-ould

ult

itythe

umear-

@1# A. Einstein, Ann. Phys.~Leipzig! 17, 132 ~1905!; Englishtranslation inThe World of the Atom, edited by H. A. Boorseand L. Motz~Basic Books, New York, 1966!.

@2# N. Bohr, Nature~London! 121, 580 ~1928!.@3# M. O. Scully, B.-G. Englert, and H. Walther, Nature~London!

351, 111 ~1991!.@4# W. Wootters and W. Zurek, Phys. Rev. D19, 473 ~1979!; R.

Glauber, Ann.~N.Y.! Acad. Sci.480, 336 ~1986!; D. Green-berger and A. Yasin, Phys. Lett. A128, 391 ~1988!; L. Man-del, Opt. Lett.16, 1882~1991!; G. Jaeger, A. Shimony, and LVaidman, Phys. Rev. A51, 54 ~1995!.

@5# B.-G. Englert, Phys. Rev. Lett.77, 2154~1996!.@6# H. Rauch and J. Summhammer, Phys. Lett.104A, 44 ~1984!;

J. Summhammer, H. Rauch, and D. Tuppinger, Phys. Rev36, 4447 ~1987!; P. Mittelstaedt, A. Prieur, and R. SchiedeFound. Phys.17, 891 ~1987!; F. De Martiniet al., Phys. Rev.A 45, 5144~1992!.

@7# S. Durr, T. Nonn, and G. Rempe, Nature~London! 395, 33~1998!.

@8# With internal atomic degrees of freedom employed for the pmarking, a recent atom-interferometry experiment investigathe equality of Eq. ~1!, with scaledresults constant to within10%, but achieving unscaled values of only;0.6 @S. Durr, T.Nonn, and G. Rempe, Phys. Rev. Lett.81, 5705 ~1998!#. Forthe record we note that our work was simultaneous with aindependent of Du¨rr, Nonn, and Rempe’s although their pulished account appeared earlier.

@9# G. Bjork and A. Karlsson, Phys. Rev. A58, 3477~1998!.@10# The angle of incidence on the beam splitter was set to 10°

order to minimize polarization variations in the reflection atransmission amplitudes — the resulting beamsplitter refltivities and transmittivities were found to lie in the range 0.to 0.51 for all polarizations.

@11# A HWP reflectslinear polarization about the optic-axis diretion, effectively rotating the polarization by twice the angbetween the incident polarization and this axis.

@12# Note that this same feature enables quantum cryptographbe performed with attenuated coherent states; see, for instaB. Huttner, N. Imoto, N. Gisin, and T. Mor, Phys. Rev. A51,1863 ~1995!.

@13# J. F. Clauser, Phys. Rev. D9, 853 ~1974!; P. Grangier, G.Roger, and A. Aspect, Europhys. Lett.1, 173 ~1986!.

@14# It is sometimes claimed that the quantization of matter isone needs to explain detector clicks and that a semiclastheory ~quantized matter interacting with classical Maxwefields! is capable of giving a full account. However, expements ~like those in @13#! have proved that such a modincorrectly predicts physical results in some situations. Rat

A

hd

d

in

-

toce,

llal

r

than adopting a semiclassical approach~which we knoweventually fails! for some experiments and the quantum aproach~which alwaysworks! for others, we feel compelled touse the latter picture throughout. Moreover, a semiclassdescription is unavoidably inconsistent for theoretical reasoEither the charged quantized matter would have to act asource for the classical electromagnetic field or, if thisavoided by construction, action would not be properly pairwith reaction.

@15# This distinguishing WW knowledge in thespatial wave func-tions could possibly be extracted using a suitable measuremthat included spatial-mode information.

@16# For linear polarization states, the optimal knowledgmeasurement axes lie exactly between the axes that wequalize the amplitudes from the two paths~and for which thevisibility is maximum!, i.e., if the light coming from paths 1and 2 is polarized atf1 andf2, then the optimal knowledgebasis is at (f11f2)/2645°.

@17# Individual photons thus manipulated are unpolarized; consJ. Lehner, U. Leonhard, and H. Paul, Phys. Rev. A53, 2727~1996!, for unpolarizedmultiphotonstates.

@18# M. O. Scully and K. Dru¨hl, Phys. Rev. A25, 2208~1982!.@19# In another series of measurements@21#, the internal HWP was

replaced by quartz rotators, relying only on optical activwhose net effect was to rotate the relative polarizations intwo paths by 90°. When a linear polarization atu was input,there was basically never any interference without quanterasure. Complete visibility could be restored using a linanalysis atu, or a circular-polarization analysis. For a completely mixed state input, however,only the circular analysis~i.e., along the eigenmodes of the quartz! recovered completevisibility.

@20# B.-G. Englert, Z. Naturforsch., A: Phys. Sci.54, 11 ~1999!.@21# P. G. Kwiat, P. D. D. Schwindt, and B.-G. Englert, inMyster-

ies, Puzzles, and Paradoxes in Quantum Mechanics, edited byR. Bonifacio, AIP Conf. Proc. No. 461~AIP, Woodbury, NY,1999!, p. 69.

@22# G. N. Ramachandran and S. Ramaseshan, inHandbuch derPhysik, edited by S. Flu¨gge ~Springer, Berlin, 1961!, Vol. 25,Pt. 1.

@23# A. G. Zajoncet al., Nature~London! 353, 507 ~1991!; P. G.Kwiat, A. M. Steinberg, and R. Y. Chiao, Phys. Rev. A45,7729 ~1992!; T. Herzog et al., Phys. Rev. Lett.75, 3034~1995!.

@24# U. Eichmannet al., Phys. Rev. Lett.70, 2359~1993!.@25# E. Bukset al., Nature~London! 391, 871 ~1998!.@26# G. Badurek, H. Rauch, and J. Summhammer, Physica B & C

151, 82 ~1988!; G. Badureket al., Nucl. Instrum. Methods

.

al-op-

t.

4290 PRA 60SCHWINDT, KWIAT, AND ENGLERT

Phys. Res. A~to be published!.@27# C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett.69, 2881

~1992!; K. Mattle, H. Weinfurter, P. G. Kwiat, and AZeilinger, ibid. 76, 4656~1996!.

@28# C. H. Bennettet al., Phys. Rev. Lett.70, 1895 ~1993!; D.Bouwmeesteret al., Nature ~London! 390, 575 ~1997!; D.Boschi et al., Phys. Rev. Lett.80, 1121 ~1998!; A. Furasawaet al., Science282, 706 ~1998!.

@29# For example, such single-particle entanglement recentlylowed the realization of a quantum-search algorithm in antical system with only passive linear elements@P. G. Kwiat, J.R. Mitchell, P. D. D. Schwindt, and A. G. White, J. Mod. Op~to be published!#.

@30# J. Bell, Physics~Long Island City, N.Y.! 1, 195 ~1964!.@31# S. Durr and G. Rempe~unpublished!.@32# B.-G. Englert and J. A. Bergou~unpublished!.