This paper reports a "delayed choice quantum eraser" experiment proposed by
Scully and Dr\"{u}hl in 1982. The experimental results demonstrated the
possibility of simultaneously observing both particle-like and wave-like
behavior of a quantum via quantum entanglement. The which-path or both-path
information of a quantum can be erased or marked by its entangled twin even
after the registration of the quantum.Comment: twocolumn, 4pages, submitted to PR

A. G. Zajonc

A. Yariv

B. J. Lawson-Daku

B.-G. Englert

C. O. Alley

D. N. Klyshko

J. Baldzuhn

M. H. Rubin

M. O. Scully

Marlan O. Scully

N. Bohr

P. Bogár

P. G. Kwiat

R. Feynman

R. J. Glauber

Rong Yu

S. Dürr

Sergei P. Kulik

T. B. Pittman

T. Hellmuth

T. J. Herzog

U. Eichmann

U. Mohrhoff

Yanhua Shih

Yoon-Ho Kim

Physical Review Letters

English

arXiv

Kim, Yoon-Ho

Yu, Rong

Kulik, Sergei P.

Shih, Yanhua

Scully, Marlan O.

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:quant-ph/9903047v1  13 Mar 1999A Delayed Choice Quantum EraserYoon-Ho Kim, R. Yu, S.P. Kulik∗, and Y.H. ShihDepartment of Physics, University of Maryland, Baltimore County,Baltimore, MD 21250Marlan O. ScullyDepartment of Physics, Texas A & M University, College Station, TX 77842and Max-Planck Institut fu¨r Quantenoptik, Mu¨nchen, Germany(submitted to PRL)This paper reports a “delayed choice quantum eraser” experiment proposed by Scully and Dru¨hl in 1982. The experimentalresults demonstrated the possibility of simultaneously observing both particle-like and wave-like behavior of a quantum viaquantum entanglement. The which-path or both-path information of a quantum can be erased or marked by its entangled twineven after the registration of the quantum.PACS Number: 03.65.Bz, 42.50.DvComplementarity, perhaps the most basic principle ofquantum mechanics, distinguishes the world of quantumphenomena from the realm of classical physics. Quantummechanically, one can never expect to measure both pre-cise position and momentum of a quantum at the sametime. It is prohibited. We say that the quantum ob-servables “position” and “momentum” are “complemen-tary” because the precise knowledge of the position (mo-mentum) implies that all possible outcomes of measuringthe momentum (position) are equally probable. In 1927,Niels Bohr illustrated complementarity with “wave-like”and “particle-like” attributes of a quantum mechanicalobject [1]. Since then, complementarity is often super-ficially identified with “wave-particle duality of matter”.Over the years the two-slit interference experiment hasbeen emphasized as a good example of the enforcementof complementarity. Feynman, discussing the two-slitexperiment, noted that this wave-particle dual behav-ior contains the basic mystery of quantum mechanics[2]. The actual mechanisms that enforce complementar-ity vary from one experimental situation to another. Inthe two-slit experiment, the common “wisdom” is thatthe position-momentum uncertainty relation δxδp ≥ h¯2makes it impossible to determine which slit the photon(or electron) passes through without at the same timedisturbing the photon (or electron) enough to destroythe interference pattern. However, it has been proven [3]that under certain circumstances this common interpre-tation may not be true. In 1982, Scully and Dru¨hl founda way around this position-momentum uncertainty obsta-cle and proposed a quantum eraser to obtain which-pathor particle-like information without scattering or∗Permanent Address: Department of Physics, Moscow StateUniversity, Moscow, Russiaotherwise introducing large uncontrolled phase factorsto disturb the interference. To be sure the interferencepattern disappears when which-path information is ob-tained. But it reappears when we erase (quantum era-sure) the which-path information [3,4]. Since 1982, quan-tum eraser behavior has been reported in several experi-ments [5]; however, the original scheme has not been fullydemonstrated.One proposed quantum eraser experiment very closeto the 1982 proposal is illustrated in Fig.1. Two atomslabeled by A and B are excited by a laser pulse. A pair ofentangled photons, photon 1 and photon 2, is then emit-ted from either atom A or atom B by atomic cascadedecay. Photon 1, propagating to the right, is registeredby a photon counting detector D0, which can be scannedby a step motor along its x-axis for the observation ofinterference fringes. Photon 2, propagating to the left, isinjected into a beamsplitter. If the pair is generated inatom A, photon 2 will follow the A path meeting BSAwith 50% chance of being reflected or transmitted. If thepair is generated in atom B, photon 2 will follow the Bpath meeting BSB with 50% chance of being reflected ortransmitted. Under the 50% chance of being transmittedby either BSA or BSB, photon 2 is detected by eitherdetectorD3 orD4. The registration ofD3 orD4 provideswhich-path information (path A or path B) of photon 2and in turn provides which-path information of photon1 because of the entanglement nature of the two-photonstate of atomic cascade decay. Given a reflection at eitherBSA or BSB photon 2 will continue to follow its A pathor B path to meet another 50-50 beamsplitter BS andthen be detected by either detector D1 or D2, which areplaced at the output ports of the beamsplitter BS. Thetriggering of detectorsD1 orD2 erases the which-path in-formation. So that either the absence of the interferenceor the restoration of the interference can be arranged viaan appropriately contrived photon correlation study. The1experiment is designed in such a way that L0, the opticaldistance between atoms A, B and detector D0, is muchshorter than Li, which is the optical distance betweenatoms A, B and detectors D1, D2, D3, and D4, respec-tively. So that D0 will be triggered much earlier by pho-ton 1. After the registration of photon 1, we look at these“delayed” detection events of D1, D2, D3, and D4 whichhave constant time delays, τi ≃ (Li − L0)/c, relative tothe triggering time ofD0. It is easy to see these “joint de-tection” events must have resulted from the same photonpair. It was predicted that the “joint detection” countingrate R01 (joint detection rate between D0 and D1) andR02 will show interference pattern when detector D0 isscanned along its x-axis. This reflects the wave property(both-path) of photon 1. However, no interference will beobserved in the “joint detection” counting rate R03 andR04 when detector D0 is scanned along its x-axis. Thisis clearly expected because we now have indicated theparticle property (which-path) of photon 1. It is impor-tant to emphasize that all four “joint detection” ratesR01, R02, R03, and R04 are recorded at the same timeduring one scanning of D0 along its y-axis. That is, inthe present experiment we “see” both wave (interference)and which-path (particle-like) with the same apparatus.We wish to report a realization of the above quantumeraser experiment. The schematic diagram of the experi-mental setup is shown in Fig.2. Instead of atomic cascadedecay, spontaneous parametric down conversion (SPDC)is used to prepare the entangled two-photon state. SPDCis a spontaneous nonlinear optical process from which apair of signal-idler photons is generated when a pumplaser beam is incident onto a nonlinear optical crystal[6]. In this experiment, the 351.1nm Argon ion pumplaser beam is divided by a double-slit and incident onto atype-II phase matching [7] nonlinear optical crystal BBO(β−BaB2O4) at two regions A and B. A pair of 702.2nmorthogonally polarized signal-idler photon is generated ei-ther from A or B region. The width of the SPDC regionis about 0.3mm and the distance between the center ofA and B is about 0.7mm. A Glen-Thompson prism isused to split the orthogonally polarized signal and idler.The signal photon (photon 1, either from A or B) passesa lens LS to meet detector D0, which is placed on theFourier transform plane (focal plane for collimated lightbeam) of the lens. The use of lens LS is to achieve the“far field” condition, but still keep a short distance be-tween the slit and the detector D0. Detector D0 can bescanned along its x-axis by a step motor. The idler pho-ton (photon 2) is sent to an interferometer with equal-path optical arms. The interferometer includes a prismPS, two 50-50 beamsplitters BSA, BSB, two reflectingmirrors MA, MB, and a 50-50 beamsplitter BS. Detec-tors D1 and D2 are placed at the two output ports ofthe BS, respectively, for erasing the which-path infor-mation. The triggering of detectors D3 and D4 providewhich-path information of the idler (photon 2) and inturn provide which-path information of the signal (pho-ton 1). The electronic output pulses of detectors D1, D2,D3, and D4 are sent to coincidence circuits with the out-put pulse of detector D0, respectively, for the countingof “joint detection” rates R01, R02, R03, and R04. Inthis experiment the optical delay (Li − L0) is chosen tobe ≃ 2.5m, where L0 is the optical distance between theoutput surface of BBO and detector D0, and Li is theoptical distance between the output surface of the BBOand detectors D1, D2, D3, and D4, respectively. Thismeans that any information one can learn from photon2 must be at least 8ns later than what one has learnedfrom the registration of photon 1. Compared to the 1nsresponse time of the detectors, 2.5m delay is good enoughfor a “delayed erasure”.Figs.3, 4, and 5 report the experimental results, whichare all consistent with prediction. Figs.3 and 4 show the“joint detection” rates R01 and R02 against the x coor-dinates of detector D0. It is clear we have observed thestandard Young’s double-slit interference pattern. How-ever, there is a pi phase shift between the two interferencefringes. The pi phase shift is explained as follows. Fig.5reports a typical R03 (R04), “joint detection” countingrate between D0 and “which-path” D3 (D4), against thex coordinates of detector D0. An absence of interferenceis clearly demonstrated. There is no significant differencebetween the curves of R03 and R04 except the small shiftof the center.To explain the experimental results, a standard quan-tum mechanical calculation is presented in the following.The “joint detection” counting rate, R0i, of detector D0and detector Dj , on the time interval T , is given by theGlauber formula [8]:R0j ∝1T∫ T0∫ T0dT0dTj〈Ψ|E(−)0 E(−)j E(+)j E(+)0 |Ψ〉=1T∫ T0∫ T0dT0dTj |〈0|E(+)j E(+)0 |Ψ〉|2, (1)where T0 is the detection time of D0, Tj is the detec-tion time of Dj( j = 1, 2, 3, 4) and E(±)0,j are positive andnegative-frequency components of the field at detectorsD0 and Dj, respectively. |Ψ〉 is the entangled state ofSPDC,|Ψ〉 =∑s,iC(ks,ki) a†s(ω(ks)) a†i (ω(ki))|0〉, (2)where C(ks,ki) = δ(ωs + ωi − ωp)δ(ks + ki − kp), forthe SPDC in which ωj and kj (j = s, i, p) are the fre-quency and wavevectors of the signal (s), idler (i), andpump (p), respectively, ωp and kp can be considered asconstants, a single mode laser line is used for pump anda†s and a†i are creation operators for signal and idler pho-tons, respectively. For the case of two scattering atoms,see ref. [3], and in the case of cascade radiation, see ref.[9], C(ks,ki) has a similar structure but without the mo-mentum delta function. The δ functions in eq.(2) are theresults of approximations for an infinite size SPDC crys-tal and for infinite interaction time. We introduce thetwo-dimensional function Ψ(t0, tj) as in eq.(1),2Ψ(t0, tj) ≡ 〈0|E(+)j E(+)0 |Ψ〉. (3)Ψ(t0, tj) is the joint count probability amplitude (“wave-function” for short), where t0 ≡ T0 − L0/c, tj ≡ Tj −Lj/c, j = 1, 2, 3, 4, L0 (Lj) is the optical distance be-tween the output point on the BBO crystal and D0 (Dj).It is straightforward to see that the four “wavefunctions”Ψ(t0, tj), correspond to four different “joint detection”measurements, having the following different forms:Ψ(t0, t1) = A(t0, tA1 ) +A(t0, tB1 ),Ψ(t0, t2) = A(t0, tA2 )−A(t0, tB2 ), (4)Ψ(t0, t3) = A(t0, tA3 ), Ψ(t0, t4) = A(t0, tB4 ), (5)where as in Fig.1 the upper index of t (A or B) labels thescattering crystal (A or B region) and the lower index of tindicates different detectors. The different sign betweenthe two amplitudes Ψ(t0, t1) and Ψ(t0, t2) is caused bythe transmission-reflection unitary transformation of thebeamsplitter BS, see Fig.1 and Fig.2. It is also straight-forward to calculate each of the A(ti, tj) [10]. To sim-plify the calculations, we consider the longitudinal inte-gral only and write the two-photon state in terms of theintegral of ke and ko:|Ψ〉 = A′0∫dke∫dko δ(ωe + ωo − ωp)×Φ(∆kL)a†kea†ko |0〉, (6)where a type-II phase matching crystal with finite lengthof L is assumed. Φ(∆kL) is a sinc-like function,Φ(∆kL) = (ei(∆kL) − 1)/i(∆kL). Using eqs. (3) and(6) we find,A(ti, tj) = A0∫dke∫dkoδ(ωe + ωo − ωp)×Φ(∆kL)fi(ωe)fj(ωo)e−i(ωete1+ωoto2), (7)where fi,j(ω), is the spectral transmission function of anassumed filter placed in front of the kth detector and isassumed Gaussian to simplify the calculation. To com-plete the integral, we define ωe = Ωe+ν and ωo = Ωo−ν,where Ωe and Ωo are the center frequencies of the SPDC,Ωe +Ωo = Ωp and ν is a small tuning frequency, so thatωe + ωo = Ωp still holds. Consequently, we can expandke and ko around Ke(Ωe) and Ko(Ωo) to first order in ν:ke = Ke + νdωedke∣∣∣∣Ωe= Ke +νue,ko = Ko − νdωodko∣∣∣∣Ωo= Ko −νuo, (8)where ue and uo are recognized as the group velocitiesof the e-ray and o-ray at frequencies Ωe and Ωo, respec-tively. Completing the integral, the biphoton wavepacketof type-II SPDC is thus:A(ti, tj) = A0Π(ti − tj)e−iΩitie−iΩjtj , (9)where we have dropped the e, o indices. The shape ofΠ(t1−t2) is determined by the bandwidth of the spectralfilters and the parameter DL of the SPDC crystal, whereD ≡ 1/uo− 1/ue. If the filters are removed or have largeenough bandwidth, we have a rectangular pulse functionΠ(t1 − t2).Π(t0 − tj) ={1 if 0 ≤ t0 − tj ≤ DL,0 otherwise.It is easy to find that the two amplitudes in Ψ(t0, t1) andΨ(t0, t2) are indistinguishable (overlap in both t0−tj andt0 + tj), respectively, so that interference is expected inboth the coincidence counting rates, R01 and R02; how-ever, with a pi phase shift due to the different sign,R01 ∝ cos2(xpid/λf), and R02 ∝ sin2(xpid/λf).If we consider “slit” A and B both have finite width (notinfinitely narrow), an integral is necessary to sum all pos-sible amplitudes along slit A and slit B. We will have astandard interference-diffraction pattern forR01 andR02,R01 ∝ sinc2(xpia/λf) cos2(xpid/λf),R02 ∝ sinc2(xpia/λf) sin2(xpid/λf), (10)where a is the width of the slit A and B (equal width),d is the distance between the center of slit A and B,λ = λs = λi is the wavelength of the signal and idler, andf is the focal length of lens LS. We have also applied the“far field approximation” for the signal and equal opticaldistance of the interferometer for the idler. After consid-ering the finite size of the detectors and the divergenceof the pump beam for further integrals, the interferencevisibility is reduced to the level close to the observation.For the “joint detection” R03 and R04, it is seenthat the “wavefunction” in eq.(5) (which clearly provides“which-path” information) has only one amplitude andno interference is expected.In conclusion, we have realized a quantum eraser ex-periment of the type proposed in ref. [3]. The experimen-tal results demonstrate the possibility of observing bothparticle-like and wave-like behavior of a light quantumvia quantum mechanical entanglement. The which-pathor both-path information of a quantum can be erased ormarked by its entangled twin even after the registrationof the quantum.This work was supported, in part, by the U.S. Office ofNaval Research, the Army Research Office - the NationalSecurity Agency, the National Science Foundation, andthe Welch Foundation. MOS wishes to thank RolandHagen for helpful and stimulating discussions.[1] N. Bohr, Naturwissenschaften, 16, 245 (1928).3[2] R. Feynman, R. Leighton, and M. Sands, The FeynmanLectures on Physics, Vol. III, Addison Wesley, Reading(1965).[3] M.O. Scully and K. Dru¨hl, Phys. Rev. A 25, 2208 (1982).[4] See Wheeler’s “delayed choice”, in Quantum Theory andMeasurement, edited by J.A. Wheeler and W.H. Zurek,Princeton Univ. Press (1983).[5] A.G. Zajonc et al., Nature, 353, 507 (1991); P.G. Kwiatet al., Phys. Rev. A 49, 61 (1994); T.J. Herzog et al.,Phys. Rev. Lett., 75, 3034 (1995); T.B. Pittman et al.,Phys. Rev. Lett., 77, 1917 (1996).[6] D.N. Klyshko, Photon and Nonlinear Optics, Gordon andBreach Science, New York (1988); A. Yariv, QuantumElectronics, John Wiley and Sons, New York (1989).[7] In type-I SPDC, signal and idler are both ordinary raysof the crystal; however, in type-II SPDC the signal andidler are orthogonal polarized, i.e., one is ordinary rayand the other is extraordinary ray of the crystal.[8] R.J. Glauber, Phys. Rev. 130, 2529 (1963); 131, 2766(1963).[9] M.O. Scully and M.S. Zubairy, Quantum Optics, Cam-bridge Univ. Press, Cambridge, UK (1997).[10] M.H. Rubin, D.N. Klyshko, and Y.H. Shih, Phys. Rev.A 50, 5122 (1994).D0D3D1D2D4ABBSABSBBSxLSFIG. 1. A proposed quantum eraser experiment. A pairof entangled photons is emitted from either atom A or atomB by atomic cascade decay. “Clicks” at D3 or D4 providewhich-path information and “clicks” at D1 or D2 erase thewhich-path information.PumpfD0 D2D1D3xABCoincidenceCircuitBBOMAMBBSBSALSFIG. 2. Schematic of the experimental setup. The pumplaser beam of SPDC is divided by a double-slit and incidentonto a BBO crystal at two regions A and B. A pair of sig-nal-idler photons is generated either from A or B region. Thedetection time of the signal photon is 8ns earlier than that ofthe idler.0.0 0.5 1.0 1.5 2.0 2.5 3.0D0 Position(mm)020406080100120140CoincidenceR01FIG. 3. R01 (“joint detection” rate between detectors D0and D1) against the x coordinates of detector D0. A standardYoung’s double-slit interference pattern is observed.0.0 0.5 1.0 1.5 2.0 2.5 3.0D0 Position(mm)020406080100120140CoincidenceR02FIG. 4. R02 (“joint detection” rate between detectors D0and D2) Note, there is a pi phase shift compare to R01 shownin Fig.30.0 0.5 1.0 1.5 2.0 2.5 3.0D0 Position(mm)020406080100120140CoincidenceR03FIG. 5. R03 (“joint detection” rate between detectors D0and D3). An absence of interference is clearly demonstrated.4

Delayed ���Choice��� Quantum Eraser

Crossref

Delayed “Choice” Quantum Eraser

Texas A&M University

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A Delayed Choice Quantum Eraser
Yoon-Ho Kim, R. Yu, S.P. Kulik∗, and Y.H. Shih
Department of Physics, University of Maryland, Baltimore County,
Baltimore, MD 21250
Marlan O. Scully
Department of Physics, Texas A & M University, College Station, TX 77842
and Max-Planck Institut fu¨r Quantenoptik, Mu¨nchen, Germany
(submitted to PRL)
This paper reports a “delayed choice quantum eraser” experiment proposed by Scully and Dru¨hl in 1982. The experimental
results demonstrated the possibility of simultaneously observing both particle-like and wave-like behavior of a quantum via
quantum entanglement. The which-path or both-path information of a quantum can be erased or marked by its entangled twin
even after the registration of the quantum.
PACS Number: 03.65.Bz, 42.50.Dv
Complementarity, perhaps the most basic principle of
quantum mechanics, distinguishes the world of quantum
phenomena from the realm of classical physics. Quantum
mechanically, one can never expect to measure both pre-
cise position and momentum of a quantum at the same
time. It is prohibited. We say that the quantum ob-
servables “position” and “momentum” are “complemen-
tary” because the precise knowledge of the position (mo-
mentum) implies that all possible outcomes of measuring
the momentum (position) are equally probable. In 1927,
Niels Bohr illustrated complementarity with “wave-like”
and “particle-like” attributes of a quantum mechanical
object [1]. Since then, complementarity is often super-
ficially identified with “wave-particle duality of matter”.
Over the years the two-slit interference experiment has
been emphasized as a good example of the enforcement
of complementarity. Feynman, discussing the two-slit
experiment, noted that this wave-particle dual behav-
ior contains the basic mystery of quantum mechanics
[2]. The actual mechanisms that enforce complementar-
ity vary from one experimental situation to another. In
the two-slit experiment, the common “wisdom” is that
the position-momentum uncertainty relation δxδp ≥ h¯2
makes it impossible to determine which slit the photon
(or electron) passes through without at the same time
disturbing the photon (or electron) enough to destroy
the interference pattern. However, it has been proven [3]
that under certain circumstances this common interpre-
tation may not be true. In 1982, Scully and Dru¨hl found
a way around this position-momentum uncertainty obsta-
cle and proposed a quantum eraser to obtain which-path
or particle-like information without scattering or
∗Permanent Address: Department of Physics, Moscow State
University, Moscow, Russia
otherwise introducing large uncontrolled phase factors
to disturb the interference. To be sure the interference
pattern disappears when which-path information is ob-
tained. But it reappears when we erase (quantum era-
sure) the which-path information [3,4]. Since 1982, quan-
tum eraser behavior has been reported in several experi-
ments [5]; however, the original scheme has not been fully
demonstrated.
One proposed quantum eraser experiment very close
to the 1982 proposal is illustrated in Fig.1. Two atoms
labeled by A and B are excited by a laser pulse. A pair of
entangled photons, photon 1 and photon 2, is then emit-
ted from either atom A or atom B by atomic cascade
decay. Photon 1, propagating to the right, is registered
by a photon counting detector D0, which can be scanned
by a step motor along its x-axis for the observation of
interference fringes. Photon 2, propagating to the left, is
injected into a beamsplitter. If the pair is generated in
atom A, photon 2 will follow the A path meeting BSA
with 50% chance of being reflected or transmitted. If the
pair is generated in atom B, photon 2 will follow the B
path meeting BSB with 50% chance of being reflected or
transmitted. Under the 50% chance of being transmitted
by either BSA or BSB, photon 2 is detected by either
detectorD3 orD4. The registration ofD3 orD4 provides
which-path information (path A or path B) of photon 2
and in turn provides which-path information of photon
1 because of the entanglement nature of the two-photon
state of atomic cascade decay. Given a reflection at either
BSA or BSB photon 2 will continue to follow its A path
or B path to meet another 50-50 beamsplitter BS and
then be detected by either detector D1 or D2, which are
placed at the output ports of the beamsplitter BS. The
triggering of detectorsD1 orD2 erases the which-path in-
formation. So that either the absence of the interference
or the restoration of the interference can be arranged via
an appropriately contrived photon correlation study. The
1
experiment is designed in such a way that L0, the optical
distance between atoms A, B and detector D0, is much
shorter than Li, which is the optical distance between
atoms A, B and detectors D1, D2, D3, and D4, respec-
tively. So that D0 will be triggered much earlier by pho-
ton 1. After the registration of photon 1, we look at these
“delayed” detection events of D1, D2, D3, and D4 which
have constant time delays, τi ≃ (Li − L0)/c, relative to
the triggering time ofD0. It is easy to see these “joint de-
tection” events must have resulted from the same photon
pair. It was predicted that the “joint detection” counting
rate R01 (joint detection rate between D0 and D1) and
R02 will show interference pattern when detector D0 is
scanned along its x-axis. This reflects the wave property
(both-path) of photon 1. However, no interference will be
observed in the “joint detection” counting rate R03 and
R04 when detector D0 is scanned along its x-axis. This
is clearly expected because we now have indicated the
particle property (which-path) of photon 1. It is impor-
tant to emphasize that all four “joint detection” rates
R01, R02, R03, and R04 are recorded at the same time
during one scanning of D0 along its y-axis. That is, in
the present experiment we “see” both wave (interference)
and which-path (particle-like) with the same apparatus.
We wish to report a realization of the above quantum
eraser experiment. The schematic diagram of the experi-
mental setup is shown in Fig.2. Instead of atomic cascade
decay, spontaneous parametric down conversion (SPDC)
is used to prepare the entangled two-photon state. SPDC
is a spontaneous nonlinear optical process from which a
pair of signal-idler photons is generated when a pump
laser beam is incident onto a nonlinear optical crystal
[6]. In this experiment, the 351.1nm Argon ion pump
laser beam is divided by a double-slit and incident onto a
type-II phase matching [7] nonlinear optical crystal BBO
(β−BaB2O4) at two regions A and B. A pair of 702.2nm
orthogonally polarized signal-idler photon is generated ei-
ther from A or B region. The width of the SPDC region
is about 0.3mm and the distance between the center of
A and B is about 0.7mm. A Glen-Thompson prism is
used to split the orthogonally polarized signal and idler.
The signal photon (photon 1, either from A or B) passes
a lens LS to meet detector D0, which is placed on the
Fourier transform plane (focal plane for collimated light
beam) of the lens. The use of lens LS is to achieve the
“far field” condition, but still keep a short distance be-
tween the slit and the detector D0. Detector D0 can be
scanned along its x-axis by a step motor. The idler pho-
ton (photon 2) is sent to an interferometer with equal-
path optical arms. The interferometer includes a prism
PS, two 50-50 beamsplitters BSA, BSB, two reflecting
mirrors MA, MB, and a 50-50 beamsplitter BS. Detec-
tors D1 and D2 are placed at the two output ports of
the BS, respectively, for erasing the which-path infor-
mation. The triggering of detectors D3 and D4 provide
which-path information of the idler (photon 2) and in
turn provide which-path information of the signal (pho-
ton 1). The electronic output pulses of detectors D1, D2,
D3, and D4 are sent to coincidence circuits with the out-
put pulse of detector D0, respectively, for the counting
of “joint detection” rates R01, R02, R03, and R04. In
this experiment the optical delay (Li − L0) is chosen to
be ≃ 2.5m, where L0 is the optical distance between the
output surface of BBO and detector D0, and Li is the
optical distance between the output surface of the BBO
and detectors D1, D2, D3, and D4, respectively. This
means that any information one can learn from photon
2 must be at least 8ns later than what one has learned
from the registration of photon 1. Compared to the 1ns
response time of the detectors, 2.5m delay is good enough
for a “delayed erasure”.
Figs.3, 4, and 5 report the experimental results, which
are all consistent with prediction. Figs.3 and 4 show the
“joint detection” rates R01 and R02 against the x coor-
dinates of detector D0. It is clear we have observed the
standard Young’s double-slit interference pattern. How-
ever, there is a pi phase shift between the two interference
fringes. The pi phase shift is explained as follows. Fig.5
reports a typical R03 (R04), “joint detection” counting
rate between D0 and “which-path” D3 (D4), against the
x coordinates of detector D0. An absence of interference
is clearly demonstrated. There is no significant difference
between the curves of R03 and R04 except the small shift
of the center.
To explain the experimental results, a standard quan-
tum mechanical calculation is presented in the following.
The “joint detection” counting rate, R0i, of detector D0
and detector Dj , on the time interval T , is given by the
Glauber formula [8]:
R0j ∝
1
T
∫ T
0
∫ T
0
dT0dTj〈Ψ|E
(−)
0 E
(−)
j E
(+)
j E
(+)
0 |Ψ〉
=
1
T
∫ T
0
∫ T
0
dT0dTj |〈0|E
(+)
j E
(+)
0 |Ψ〉|
2, (1)
where T0 is the detection time of D0, Tj is the detec-
tion time of Dj( j = 1, 2, 3, 4) and E
(±)
0,j are positive and
negative-frequency components of the field at detectors
D0 and Dj, respectively. |Ψ〉 is the entangled state of
SPDC,
|Ψ〉 =
∑
s,i
C(ks,ki) a
†
s(ω(ks)) a
†
i (ω(ki))|0〉, (2)
where C(ks,ki) = δ(ωs + ωi − ωp)δ(ks + ki − kp), for
the SPDC in which ωj and kj (j = s, i, p) are the fre-
quency and wavevectors of the signal (s), idler (i), and
pump (p), respectively, ωp and kp can be considered as
constants, a single mode laser line is used for pump and
a†s and a
†
i are creation operators for signal and idler pho-
tons, respectively. For the case of two scattering atoms,
see ref. [3], and in the case of cascade radiation, see ref.
[9], C(ks,ki) has a similar structure but without the mo-
mentum delta function. The δ functions in eq.(2) are the
results of approximations for an infinite size SPDC crys-
tal and for infinite interaction time. We introduce the
two-dimensional function Ψ(t0, tj) as in eq.(1),
2
Ψ(t0, tj) ≡ 〈0|E
(+)
j E
(+)
0 |Ψ〉. (3)
Ψ(t0, tj) is the joint count probability amplitude (“wave-
function” for short), where t0 ≡ T0 − L0/c, tj ≡ Tj −
Lj/c, j = 1, 2, 3, 4, L0 (Lj) is the optical distance be-
tween the output point on the BBO crystal and D0 (Dj).
It is straightforward to see that the four “wavefunctions”
Ψ(t0, tj), correspond to four different “joint detection”
measurements, having the following different forms:
Ψ(t0, t1) = A(t0, t
A
1 ) +A(t0, t
B
1 ),
Ψ(t0, t2) = A(t0, t
A
2 )−A(t0, t
B
2 ), (4)
Ψ(t0, t3) = A(t0, t
A
3 ), Ψ(t0, t4) = A(t0, t
B
4 ), (5)
where as in Fig.1 the upper index of t (A or B) labels the
scattering crystal (A or B region) and the lower index of t
indicates different detectors. The different sign between
the two amplitudes Ψ(t0, t1) and Ψ(t0, t2) is caused by
the transmission-reflection unitary transformation of the
beamsplitter BS, see Fig.1 and Fig.2. It is also straight-
forward to calculate each of the A(ti, tj) [10]. To sim-
plify the calculations, we consider the longitudinal inte-
gral only and write the two-photon state in terms of the
integral of ke and ko:
|Ψ〉 = A
′
0
∫
dke
∫
dko δ(ωe + ωo − ωp)×
Φ(∆kL)a
†
ke
a†ko |0〉, (6)
where a type-II phase matching crystal with finite length
of L is assumed. Φ(∆kL) is a sinc-like function,
Φ(∆kL) = (e
i(∆kL) − 1)/i(∆kL). Using eqs. (3) and
(6) we find,
A(ti, tj) = A0
∫
dke
∫
dkoδ(ωe + ωo − ωp)×
Φ(∆kL)fi(ωe)fj(ωo)e
−i(ωet
e
1
+ωot
o
2
), (7)
where fi,j(ω), is the spectral transmission function of an
assumed filter placed in front of the kth detector and is
assumed Gaussian to simplify the calculation. To com-
plete the integral, we define ωe = Ωe+ν and ωo = Ωo−ν,
where Ωe and Ωo are the center frequencies of the SPDC,
Ωe +Ωo = Ωp and ν is a small tuning frequency, so that
ωe + ωo = Ωp still holds. Consequently, we can expand
ke and ko around Ke(Ωe) and Ko(Ωo) to first order in ν:
ke = Ke + ν
dωe
dke
∣∣∣∣
Ωe
= Ke +
ν
ue
,
ko = Ko − ν
dωo
dko
∣∣∣∣
Ωo
= Ko −
ν
uo
, (8)
where ue and uo are recognized as the group velocities
of the e-ray and o-ray at frequencies Ωe and Ωo, respec-
tively. Completing the integral, the biphoton wavepacket
of type-II SPDC is thus:
A(ti, tj) = A0Π(ti − tj)e
−iΩitie−iΩjtj , (9)
where we have dropped the e, o indices. The shape of
Π(t1−t2) is determined by the bandwidth of the spectral
filters and the parameter DL of the SPDC crystal, where
D ≡ 1/uo− 1/ue. If the filters are removed or have large
enough bandwidth, we have a rectangular pulse function
Π(t1 − t2).
Π(t0 − tj) =
{
1 if 0 ≤ t0 − tj ≤ DL,
0 otherwise.
It is easy to find that the two amplitudes in Ψ(t0, t1) and
Ψ(t0, t2) are indistinguishable (overlap in both t0−tj and
t0 + tj), respectively, so that interference is expected in
both the coincidence counting rates, R01 and R02; how-
ever, with a pi phase shift due to the different sign,
R01 ∝ cos
2(xpid/λf), and R02 ∝ sin
2(xpid/λf).
If we consider “slit” A and B both have finite width (not
infinitely narrow), an integral is necessary to sum all pos-
sible amplitudes along slit A and slit B. We will have a
standard interference-diffraction pattern forR01 andR02,
R01 ∝ sinc
2(xpia/λf) cos2(xpid/λf),
R02 ∝ sinc
2(xpia/λf) sin2(xpid/λf), (10)
where a is the width of the slit A and B (equal width),
d is the distance between the center of slit A and B,
λ = λs = λi is the wavelength of the signal and idler, and
f is the focal length of lens LS. We have also applied the
“far field approximation” for the signal and equal optical
distance of the interferometer for the idler. After consid-
ering the finite size of the detectors and the divergence
of the pump beam for further integrals, the interference
visibility is reduced to the level close to the observation.
For the “joint detection” R03 and R04, it is seen
that the “wavefunction” in eq.(5) (which clearly provides
“which-path” information) has only one amplitude and
no interference is expected.
In conclusion, we have realized a quantum eraser ex-
periment of the type proposed in ref. [3]. The experimen-
tal results demonstrate the possibility of observing both
particle-like and wave-like behavior of a light quantum
via quantum mechanical entanglement. The which-path
or both-path information of a quantum can be erased or
marked by its entangled twin even after the registration
of the quantum.
This work was supported, in part, by the U.S. Office of
Naval Research, the Army Research Office - the National
Security Agency, the National Science Foundation, and
the Welch Foundation. MOS wishes to thank Roland
Hagen for helpful and stimulating discussions.
[1] N. Bohr, Naturwissenschaften, 16, 245 (1928).
3
[2] R. Feynman, R. Leighton, and M. Sands, The Feynman
Lectures on Physics, Vol. III, Addison Wesley, Reading
(1965).
[3] M.O. Scully and K. Dru¨hl, Phys. Rev. A 25, 2208 (1982).
[4] See Wheeler’s “delayed choice”, in Quantum Theory and
Measurement, edited by J.A. Wheeler and W.H. Zurek,
Princeton Univ. Press (1983).
[5] A.G. Zajonc et al., Nature, 353, 507 (1991); P.G. Kwiat
et al., Phys. Rev. A 49, 61 (1994); T.J. Herzog et al.,
Phys. Rev. Lett., 75, 3034 (1995); T.B. Pittman et al.,
Phys. Rev. Lett., 77, 1917 (1996).
[6] D.N. Klyshko, Photon and Nonlinear Optics, Gordon and
Breach Science, New York (1988); A. Yariv, Quantum
Electronics, John Wiley and Sons, New York (1989).
[7] In type-I SPDC, signal and idler are both ordinary rays
of the crystal; however, in type-II SPDC the signal and
idler are orthogonal polarized, i.e., one is ordinary ray
and the other is extraordinary ray of the crystal.
[8] R.J. Glauber, Phys. Rev. 130, 2529 (1963); 131, 2766
(1963).
[9] M.O. Scully and M.S. Zubairy, Quantum Optics, Cam-
bridge Univ. Press, Cambridge, UK (1997).
[10] M.H. Rubin, D.N. Klyshko, and Y.H. Shih, Phys. Rev.
A 50, 5122 (1994).
D0
D
3
D
1
D2
D4
A
B
BSA
BSB
BS
x
LS
FIG. 1. A proposed quantum eraser experiment. A pair
of entangled photons is emitted from either atom A or atom
B by atomic cascade decay. “Clicks” at D3 or D4 provide
which-path information and “clicks” at D1 or D2 erase the
which-path information.
Pump
f
D0 D2
D1
D3
x
A
B
Coincidence
Circuit
BBO
MA
MB
BSBSA
LS
FIG. 2. Schematic of the experimental setup. The pump
laser beam of SPDC is divided by a double-slit and incident
onto a BBO crystal at two regions A and B. A pair of sig-
nal-idler photons is generated either from A or B region. The
detection time of the signal photon is 8ns earlier than that of
the idler.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
D0 Position(mm)
0
20
40
60
80
100
120
140
Co
in
cid
en
ce
R01
FIG. 3. R01 (“joint detection” rate between detectors D0
and D1) against the x coordinates of detector D0. A standard
Young’s double-slit interference pattern is observed.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
D0 Position(mm)
0
20
40
60
80
100
120
140
Co
in
cid
en
ce
R02
FIG. 4. R02 (“joint detection” rate between detectors D0
and D2) Note, there is a pi phase shift compare to R01 shown
in Fig.3
0.0 0.5 1.0 1.5 2.0 2.5 3.0
D0 Position(mm)
0
20
40
60
80
100
120
140
Co
in
cid
en
ce
R03
FIG. 5. R03 (“joint detection” rate between detectors D0
and D3). An absence of interference is clearly demonstrated.
4


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