PHYSICAL REVIEW A 82, 042104 (2010)

Interference and complementarity for two-photon hybrid entangled states

W. A. T. Nogueira,1,2,* M. Santibanez,1,2 S. Padua,3 A. Delgado,1,2 C. Saavedra,1,2 L. Neves,1,2 and G. Lima1,2

1Center for Optics and Photonics, Universidad de Concepcion, Casilla 4016, Concepcion, Chile2Departamento de Fısica, Universidad de Concepcion, Casilla 160-C, Concepcion, Chile

3Departamento de Fısica, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, MG 30123-970, Brasil(Received 12 August 2010; published 8 October 2010)

In this work we generate two-photon hybrid entangled states (HESs), where the polarization of one photonis entangled with the transverse spatial degree of freedom of the second photon. The photon pair is created byparametric down-conversion in a polarization-entangled state. A birefringent double-slit couples the polarizationand spatial degrees of freedom of these photons, and finally, suitable spatial and polarization projections generatethe HES. We investigate some interesting aspects of the two-photon hybrid interference and present this study inthe context of the complementarity relation that exists between the visibility of the one-photon and that of thetwo-photon interference patterns.

DOI: 10.1103/PhysRevA.82.042104 PACS number(s): 03.65.Ud, 03.67.Mn, 07.60.Ly, 42.50.−p

I. INTRODUCTION

Complementarity of one- and two-particle interference wasfirst proposed by Horne and Zeilinger [1]. They noted thatwhen the two-particle visibility is 1, the one-particle visibilityobserved in either subsystem must be 0, and vice versa. Adetailed investigation of this complementarity relation wasdone by Jaeger et al. [2,3] for the case of a two-particlebeam-splitter-based interferometer. They demonstrated thatthe relation

V 212 + V 2

1 = 1 (1)

holds for any pure bipartite state. Here, V1 is the standardone-particle visibility and V12 is a two-particle visibility thatis obtained from a “corrected” joint detection probability [2].This correction is necessary to interpret the values of two-particle visibility as a scale, from 0, for a product state, to 1,which occurs for a maximally entangled state.

A complementarity relation for the visibilities of the one-and two-photon interference patterns was also derived for atwo-photon double-slit interferometer [4,5]. It consists of asource S, two double-slits, and two detection screens. In thelimit of a small source aperture, the two-photon probability am-plitude reduces to �(1,2) = cos(k θ ′ x1) cos(k θ ′ x2), where θ ′is the angle that is subtended by the slit pairs and the detectingplanes. In the limit of a large source aperture, the two-particleprobability amplitude reduces to �(1,2) = cos[k θ ′ (x1 − x2)]and there is no single-photon interference pattern. In thiscase, the two-photon interference pattern presents “conditionalfringes,” which means that the shape of the fringes dependson the position of both detected photons [6]. When thesource aperture is intermediate in size, both single-photonand conditional two-photon fringes are present. Horne [5]derived a relation for the one- and two-photon fringe visibilitiesthat is equal to Eq. (1). This relation was experimentallyobserved in [7]. Further experimental investigation of thiscomplementarity relation has been presented recently, wherethe authors considered distinct classes of spatially entangledtwo-photon states to test Eq. (1) [8].

*wallon.nogueira@cefop.udec.cl

In fact, Eq. (1) is a special case of the expression whichfully quantifies the complementarity of bipartite two-levelquantum systems [9,10]. It is given by three mutually exclusivequantities that have all the information that can be extractedfrom the quantum state

C2 + V 2k + P 2

k = 1, (2)

where k = 1,2. Vk is the one-particle visibility and Pk isthe path predictability, which, in the context of a double-slitexperiment, gives the knowledge that is available about whichslit the photon has passed by [9–15]. The quantity C is theconcurrence [16], which is a measurement of the entanglementof the composite system. It corresponds to the maximumvalue of the two-photon visibility V12 that can be reached.It also represents the information that cannot be extracted ifa measurement is performed on only a single particle of thecomposed system. Because the quantity C cannot be changedby means of a local unitary operations, the quantity V 2

k + P 2k

is invariant under this kind of operation. A scheme that allowsthe measurement of all quantities that are present in Eq. (2)is presented in Ref. [17]. Complementarity relations involvingmultiparticle states have been also developed [9,11–15,18].

Recently, the experimental investigation of hybrid photonicentanglement (HPE), namely, the entanglement between twodifferent degrees of freedom (DOF) of composite quantumsystems, has been receiving increasing attention [19–26].Correlations of the fields generated in the process of spon-taneous parametric down-conversion (SPDC) [27,28] hasbeen used for the generation of momentum-polarizationentangled states [20–23], angular-momentum-polarization en-tangled states [29,30] and, also, for the generation of time-bin-polarization correlated quantum systems [24,25]. Thegeneration of HPE with continuous variable systems hasalso been demonstrated [26]. The interest in HPE comesfrom the fact that they allow more versatility for quantumcommunication optical networks. Furthermore, momentum-polarization hybrid entangled states (HESs) can be seen, insome cases, as vector-polarization states [23], which havemany practical potentials [31,32].

In this work, we investigate interference and complemen-tarity aspects for the source of HPE introduced in Ref. [22]. In

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W. A. T. NOGUEIRA et al. PHYSICAL REVIEW A 82, 042104 (2010)

our case, HESs are prepared in the polarization and transversespatial DOF of a photon pair produced in SPDC. One ofthese photons is sent through a birefringent double-slit (BDS),which discretizes its transverse momentum and couples thephoton-pair polarization and transverse momentum DOF.After a suitable choice of spatial and polarization projections,the HES is generated.

For these states we study properties of the two-photonhybrid interference, which naturally arises when the down-converted photons are let propagate through the free space. Theconditional two-photon interference is shown to be dependentonly on the angle of the polarization projections performedin one photon and on the transverse spatial projectionsimplemented onto the second photon; therefore, we refer toit as two-photon hybrid interference. We present this study inthe context of the complementarity relation that exists betweenthe visibiliy of the one-photon and that of the two-photoninterference patterns. One- and two-photon visibilities areobtained from a set of experimentally produced HESs andtheir values are used to verify Eq. (1).

The article is structured as follows. In Sec. II, we firstdescribe the source of transverse momentum-polarizationHESs used in our experiment. We show how these states canbe obtained, and then we give the theoretical expressions forthe one- and two-photon interferences whose hybrid behavioris analyzed. The complementarity relation for the visibilitiesof these interferences is also discussed, and we show howthis relation can be properly verified in an experiment. Thetheoretical description of this section is fully in accordancewith the experiment performed, and thus, the reader may usethe setup diagram given in Fig. 2 to follow the calculationsdone. The experimental tests of this complementarity relationare presented in Sec. III, the results in Sec. IV, and concludingremarks in Sec. V.

II. THEORY

A. Brief description of the HES source

A detailed description of our HES source is given in [22].Here we briefly describe the main features of this source. Wewould like to stress that we have employed a type II SPDCsource in the geometry of crossed cones [33], instead of the twotype I crystals used in [22]. Therefore, our initial two-photonstate, after spectral filtering and compensation of longitudinaland transverse walk-off effects, is given by

|�〉 = 1√2

(|H 〉s |V 〉i + ejφpol |V 〉s |H 〉i) ⊗ |�spa〉, (3)

where the state |H 〉l (|V 〉l) represents one photon in thepropagation mode l (l = s,i denotes the signal and idlerpropagation modes, respectively) with horizontal (vertical)polarization. We assume here that φpol = 0. |�spa〉 representsthe spatial part of the two-photon state generated. If the idlerphoton is sent through a double-slit, |�spa〉 becomes a discreteentangled state, as shown in Ref. [22]. If we place a spatialfilter in the way of the signal photon, a projection is made ontothe spatial mode |F 〉 defined by the double-slit, which resultsin [22]

|�spa〉 =( |+〉i + ejφspa |−〉i√

2

)⊗ |F 〉s , (4)

where |F 〉s ≡ |+〉s+|−〉s√2

. State |±〉i is a single-photon statedefined, up to a global phase factor, as [34]

|±〉i ≡√

a

π

∫dqi e∓jqid/2 sinc (qia)|qi〉. (5)

State |+〉i (|−〉i) represents the state of the idler photontransmitted by the upper (lower) slit of the double-slit. Thesestates form an orthonormal basis for the Hilbert space of thetransmitted photon [34]. Here, a is the half-width of the slitsand d is the center-to-center separation between the two slits.The phase φspa can be changed by tilting the double-slit, andwe also consider that φspa = 0.

After preparing the two-photon polarization entanglement,the next step to generate our photonic HES is to couple thepolarization and spatial DOF of the idler photon. This can beachieved by quarter-wave plates (QWPs) placed behind eachslit of the double-slit, with their fast axes oriented orthogonally,as shown in Fig. 2(b). This BDS can be seen, up to a phaseshift, as a single-photon two-qubit controlled-NOT gate, wherethe photon polarization is the control qubit and the transversemomentum distribution is the target qubit [22]. The actionof this controlled-NOT can be summarized as |H 〉i |F 〉i ⇒|H 〉i |F 〉i , |H 〉i |A〉i ⇒|H 〉i |A〉i , |V 〉i |F 〉i ⇒j |V 〉i |A〉i , and|V 〉i |A〉i ⇒j |V 〉i |F 〉i , where states |F 〉i and |A〉i are |F 〉i ≡|+〉i+|−〉i√

2and |A〉i ≡ |+〉i−|−〉i√

2[35].

From Eqs. (3) and (4), and taking into account the effectof the BDS, it is straightforward to show that the two-photonstate, after idler transmission through the double-slit, can bewritten as |�〉 = (1/

√2)(|V 〉s |HF 〉i + j |H 〉s |V A〉i), where

we have omitted the factorable spatial state of the signalphoton. This is a two-photon, three-qubit Greenberger-Horne-Zeilinger (GHZ) type state [36], which can be filtered to anHES with a polarization projection on the idler photon. Thismeasurement can be done by placing a polarizer after theBDS, which makes a projective measurement in the idlerpolarization |P 〉i = α|H 〉 + βejφP |V 〉. The HES generatedafter this projection will be given by [22]

|�(P )〉 = α|H 〉s |F 〉i + jβe−jφP |V 〉s |A〉i , (6)

where we have omitted the idler factorable polarization state, αand β are �0, and α2 + β2 = 1. It is important to note that theamount of entanglement of the generated HES can be tuned bychanging the polarization projection of the idler photon [22].The concurrence of the state Eq. (6) is

C = 2αβ, (7)

and it is equal to 0 for a product state and 1 for a hybridmaximally entangled state (HMES).

B. Two-photon hybrid interference

Next we consider the situation where the signal photon istransmitted through a polarization analyzer and then is detectedby a “bucket” detector—an opened avalanche photodiodewhich registers a photon, but not its transverse position. Wealso assume that the idler photon is let to propagate freelyafter its transmission through the BDS and the polarizer and,then, is detected at a distant plane (where the Fraunhoffer ap-proximation is valid) by a “point-like” detector: an avalanche

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INTERFERENCE AND COMPLEMENTARITY FOR TWO- . . . PHYSICAL REVIEW A 82, 042104 (2010)

photodiode whose aperture is small compared to the transversediffraction pattern formed by the idler down-converted beam.

By assuming also that the photoionization of the idlerdetector is independent of the idler polarization chosen forthe generation of the HES, the probability distribution of jointdetection will depend only on the signal polarization analyzerorientation (θs) and the idler detector transverse position (xi).This distribution, Pcc(θs,xi), is therefore given by

Pcc(θs,xi) ∝ tr[ρ�E(−)i (xi)E

(+)i (xi) ⊗ Ps(θs)], (8)

where ρ� = |�(P )〉〈�(P )| is the hybrid two-photon densityoperator [see Eq. (6)]. The terms E

(−)i (xi) and E

(+)i (xi) are

the negative- and positive-frequency parts of the electric fieldoperator that represents the spatial evolution of the idler photonfrom the BDS to the detection plane and at the transverseposition xi . The positive-frequency part is given, up to anirrelevant phase, by [28,37]

E(+)i (xi) ∝

∫dq a(q) exp

{j

[qxi − q2 z

2k

]}, (9)

where k represents the wave number of the idler down-converted beam, z is the distance from the double-slit to thedetector plane, and a(q) is the destruction operator inthe transverse spatial mode q. The operator Ps(θs) representsthe action of the signal polarization analyzer, and it is definedin the usual way, by the projector Ps(θs) = |θs〉〈θs |, where|θs〉 ≡ cos(θs)|H 〉s + ejφs sin(θs)|V 〉s .

Taking into account the completeness relation for the Fockstates of light, and the definition of the spatial states |F 〉iand |A〉i , it is straightforward to show that the joint detectionprobability distribution Pcc(θs,xi) may be written as

Pcc(θs,xi) ∝ |〈θs |〈vac|E(+)i (xi)|�〉|2

=∣∣∣∣jβe−jφP cos(θs)√

2(〈vac|E(+)

i (xi)|+〉i−〈vac|E(+)i (xi)|−〉i)

+α sin(θs)√2

(〈vac|E(+)i (xi)|+〉i + 〈vac|E(+)

i (xi)|−〉i)∣∣∣∣2

,

(10)

where |vac〉 represents the vacuum state. The detectors’s quan-tum efficiencies η are assumed to be η = 1. It is straightforwardto show that the matrix elements 〈vac|E(+)

i (xi)|±〉 are givenby

〈vac|E(+)i (xi)|±〉 ∝ exp

[jk(xi ∓ d)2

2z

]sinc

[ka(xi ∓ d)

z

].

(11)

Considering usual values for the parameters in Eq. (11), suchas those used in our experiment: a = 40 µm, d = 250 µm,k = 702 nm, and z = 42 cm, it is reasonable to assume thatsinc[ ka(xi∓d)

z] ∼ sinc[ kaxi

z] and, also, that exp [ jkx2

i

2z] ∼ 1. By

using Eq. (5) and Eq. (11) to calculate Eq. (10), and consideringfor simplicity that the signal polarization projections areimplemented with φs = 0, we get the following two-photon

0 100 200 3000

0.5

1

θs (degree)

Ccc

(θs,x

i)

0 100 200 3000

0.5

1

θs (degree)

Ccc

(θs,x

i)

-2 -1 0 1 2 0

0.5

1

xi (mm)

Ccc

(θs,x

i)

-2 -1 0 1 2 0

0.5

1

xi (mm)

Ccc

(θs,x

i)

(a) (b)

(c) (d)

Pc

cP

cc

Pc

cP

cc

FIG. 1. (Color online) (a, b) Non-normalized two-photon polar-ization curves, with the idler detector fixed at xi = πz

kdand xi = 0,

respectively, and the signal polarizer analyzer rotated. (c, d) Non-normalized two-photon Young’s interferences for cases where thesignal polarizer analyzer is set to the vertical and horizontal directions,respectively, and the detector idler scanned transversally. Graphicswere created using the values a = 40 µm, d = 250 µm, k = 702 nm,and z = 42 cm.

conditional probability distribution:

Pcc(θs,xi) =(

2ka

π2z

)sinc 2(Axi)[β

2 cos2(θs) sin2(Bxi)

+α2 sin2(θs) cos2(Bxi)

+ 1

2αβ sin(2θs) sin(2Bxi) cos(φP )], (12)

where A = (ka/z) and B = (kd/2z). This equation presentsmany interesting aspects that we are going to analyze.

1. Hybrid behavior

Two-photon hybrid interference can be measured withcompletely distinct types of measurements. Whenever the idlerdetector is fixed and the polarization analyzer is rotated, wewill have interference curves that are typical of a polarization-entangled two-photon state [33] (see Fig. 1). In contrast, whenthe polarization analyzer is fixed and the idler detector istransversally displaced, there will be a conditional fourth-orderYoung’s interference formed [6], which has been observed onlywhen spatially correlated photons were transmitted throughdouble-slits [8,37,38].

2. Conditionality of the hybrid interference

In Figs. 1(a) and 1(b), two-photon polarization curvesare plotted in terms of the signal polarizer analyzer angle,when the idler detector position is fixed at xi = πz

kdand

xi = 0, respectively. When xi = πzkd

, the interference fringeswill be governed by the first term in Eq. (12), while whenxi = 0, they will be governed by the second term. One canclearly see the conditionality of the two-photon polarizationinterference. Two-photon conditional Young’s interferencesare shown in Figs. 1(c) and 1(d), for cases where the signal

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W. A. T. NOGUEIRA et al. PHYSICAL REVIEW A 82, 042104 (2010)

polarization analyzer is set to the vertical and horizontaldirection, respectively, and the idler detector is scannedtransversally. For plotting these curves, we considered thevalues of k, z, a, and d of the experimental setup used.

3. Conditionality of V12

The visibility of two-photon hybrid interference can bedefined as

V12 ≡ [Pcc(θs,xi)]max − [Pcc(θs,xi)]min

[Pcc(θs,xi)]max + [Pcc(θs,xi)]min. (13)

As is also the case with polarization-only or spatial-onlyentangled photons, hybrid two-photon visibility is dependenton the measurement bases considered, even for an HMES.For example, in the case of an HMES, two-photon Young’sinterference will disappear when the idler detector is scannedwith the signal polarization analyzer oriented at 45◦ [39]. Thiseffect is related to the wave-particle duality of the photontransmitted by the BDS.

4. Applications

As well summarized in [40], when there is some priorknowledge about the quantum state, it is possible to use two-photon interference to determine the amount of entanglement[16]. Several works have used this technique to measuredistinct types of quantum correlations [8,37,41,42]. Hybridtwo-photon interference can be used in this way to quantify thehybrid entanglement. As mentioned before, the concurrencefor the HES in Eq. (6) is given in terms of α and β,which can be determined from the measurements describedin Fig. 1. The advantage is that one can choose which typeof measurement, the two-photon polarization curve or thetwo-photon Young’s interference, is more suitable for theexperimental entanglement determination.

C. One-photon interference

There are two distinct types of one-photon interferencethat can be observed, one for each DOF involved in thephotonic entanglement. When the counts of the idler detectorare registered as a function of its transverse displacement, aYoung’s second-order interference will be formed. When thesignal detection is registered as a function of the orientation ofits polarization analyzer, a sinusoidal curve will be formed.

1. One-photon spatial interference

The probability of detecting the idler photon is proportionalto the spatial second-order correlation function, defined astr[ρiE

(−)i (xi)E

(+)i (xi)]. ρi is the density operator representing

the idler photon and it can be obtained by tracing out thepolarization of the signal photon from the HES given byEq. (6). Considering the approximations for the elementsof matrix 〈vac|E(+)

i (xi)|±〉 exposed before, the probabilitydistribution of detection of a single photon by the idler detectoris given by

P1s(xi) ∝ |α|2|〈vac|E(+)i (xi)|F 〉i |2 + |β|2|〈vac|E(+)

i (xi)|A〉i |2

=(

2ka

πz

)1

2sinc 2 (Axi) [1 + (α2 − β2) cos(Bxi)].

(14)

We can see that one-photon spatial interference has a visibility

V1s = |α2 − β2|, (15)

which is 0 when down-converted photons are in a hybridmaximally entangled state and 1 for a product state. Notethat V1s may also be written as V1s = √

1 − C2.Experimental measurement of P1s(xi) can be easily per-

formed. The trace over the signal polarization is done byremoving the polarization analyzer from its propagation path,and the coincidence counts, in this case, will map P1s(xi) whilethe idler detector is scanned. As explained in [8] and [37], it isimportant to understand that correct measurement of P1s(xi)must be done using the coincidence counts recorded and notthe single counts, since the latter also include photons that donot belong to the two-photon HES of Eq. (6).

2. One-photon polarization interference

The probability of detecting a signal photon is proportionalto tr[ρsPs(θs)]. ρs is the reduced density operator that representsthe signal photon and it is obtained after tracing out thespatial content of the idler photon from the two-photon HES ofEq. (6). So, we have a probability distribution given by

P1p(θs) ∝ |α|2|〈θs |H 〉|2 + |β|2|〈θs |V 〉|2

=(

1

2π

)[1 + (β2 − α2) cos(2θs)], (16)

which has the same visibility as the one-photon spatialinterference.

We also note that P1p(θs) can easily be measured. Theoperation of tracing out the information on the momentumdistribution of the idler photon is done by opening the idlerdetector, which thus becomes a “bucket” detector, in the sensediscussed before. The coincidence counts will then map theP1p(θs) curve, while the polarization analyzer before the signaldetector is rotated.

D. Testing the complementarity relation

As discussed in [2] and [3], the definition of the visibilityof the two-photon interference pattern given by Eq. (13) failsto capture the intended sense of two-particle interference,since it yields V12 = 1 even if the two-photon HES is aproduct state. So, to properly study the one- and two-photoncomplementarity relation, we adopt the correction proposed byJaeger et al. [2,3], where the presence of a correction factor forthe joint probability allows one to have a corrected two-photonvisibility equal to 1 for a maximally entangled state and 0for a product state. However, before continuing we wouldlike to emphasize that consideration of the joint probabilitydistribution in the form given in Eq. (12) is completely relevantsince it is the only two-photon distribution that indeed can bemeasured directly in the laboratory.

The corrected joint detection probability distribution,Pcc(θs,xi), is, in accordance with the idea of a correction factorexposed in references [2] and [3], given by

Pcc(θs,xi) = Pcc(θs,xi)−P1s(xi)P1p(θs)

+(

2ka

π2z

)1

4sinc 2(Axi), (17)

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INTERFERENCE AND COMPLEMENTARITY FOR TWO- . . . PHYSICAL REVIEW A 82, 042104 (2010)

where the extra factor, sinc 2(Axi), accounts for propagationand diffraction effects that appear after the idler photon istransmitted through the BDS. Even though this probabilitycannot be measured directly, it can be calculated from theexperimental results of Pcc(θs,xi), P1s(xi), and P1p(θs), as donein [7], for spatially correlated photons.

By substituting Eqs. (12), (14), and (16) into Eq. (17), oneobtains that

Pcc(θs,xi) =(

ka

2π2z

)sinc 2 (Axi)

×{1 + C sin(2θs) sin (2Bxi) cos(φP )

+C2 cos(2θs) cos(2Bxi)}, (18)

which has the same structure as Eq. (82b) in Ref. [3].Thus, the corrected two-photon visibility becomes

V12 ≡ [Pcc(θs,xi)]max − [Pcc(θs,xi)]min

[Pcc(θs,xi)]max + [Pcc(θs,xi)]min, (19)

and it is interesting to note is that the complementarity relationgiven by Eq. (1) can be tested considering four distincttypes of measurements. One can choose (i) to measure theone-photon spatial interference and compare its visibility withthe two-photon Young’s interference-corrected visibility or(ii) to compare it with the corrected visibility of the two-photon polarization curve. One can also choose to comparethe visibility of the one-photon polarization curve with thecorrected visibilities of the (iii) two-photon polarization curveor (iv) the two-photon spatial curve.

As mentioned before, the visibility of two-photon interfer-ence depends on the measurement basis chosen. In fact, itsvalue can vary from C2 � V12 � C [3]. Here, we considerthe measurement of the conditional polarization curve whilethe idler detector is fixed at xi = 0 and the measurementof the conditional Young’s interference when the signalpolarizer analyzer is fixed in the vertical direction θs = π/2.For these measurements, it is possible to derive simpleexpressions for the corrected two-photon interferences, andthey allow one to clearly see the relation between V12 andthe concurrence C of the HES. When the signal polarizationanalyzer is fixed in the vertical direction, the corrected two-photon spatial interference of Eq. (17) may be written as

Pcc(π/2,xi) =(

ka

2π2z

)sinc 2 (Axi)

× [1 + C2 cos (2Bxi)], (20)

which has visibility V12 = C2. When the idler detector is fixedat xi = 0, we will have Pcc(θs,xi) in terms of the angle of thesignal polarization analyzer given by

Pcc(θs,0) =(

ka

2π2z

)[1 − C2 cos(2θs)], (21)

which, of course, has the same visibility, V12 = C2.

III. EXPERIMENT

A. Experimental setup

The experimental setup considered is outlined inFig. 2(a). A single-mode collimated Ar+-ion laser operating

at 351.1 nm, with an average power of 71 mW and in theTEM00 mode with a transverse profile of ∼2-mm FWHM,is sent through a 5-mm-long β-barium borate (BBO) crystalcut for type II SPDC. Degenerated down-converted photonsof 702 nm are selected using interference filters that havesmall bandwidths, 1-nm FWHM) and are mounted in frontof the idler and signal detectors, which are referred to asDi and Ds, respectively, henceforth. To prepare the |�+

pol〉polarization state, a half-wave plate (HWP) and BBO crystals2.5 mm thick are placed on each down-converted arm tocompensate longitudinal and transverse walk-off effects [33].In the idler path, a BDS is placed at a distance of 40 cmfrom the crystal, followed by a polarization analyzer, whichis composed of a QWP, an HWP plate, and a PBS. The BDSis sketched in Fig. 2(b). The slit width is 2a = 80; µm andtheir center-to-center separation is d = 250; µm. Upon itstransmission through this system, the idler photon propagatesfreely through 42 cm, until it reaches the single-photon detectorDi, which is mounted on a translation stage that allows itsdisplacement in the x transverse direction.

In the signal path, the spatial mode of the photon is definedby a spatial filter placed after the compensating crystal. It iscomposed of two lenses and a horizontal slit of 50-µm width,which is placed in the focal plane of both lenses. The firstand second lenses have focal distances of 150 and 200 mm,respectively. After the spatial filter, a polarization analyzer(QWP, HWP, and PBS) allows for polarization projection inany basis. The detector Ds is located 25 cm from the secondlens of the spatial filter. The detectors Di and Ds are connectedto a circuit used to record single and coincidence counts.

B. State preparation

The first step in generating different HESs is to ensure thatthe polarization entangled state generated [see Sec. II A] hasa high degree of entanglement and purity. To verify this pointwe have measured interference curves in the +45◦/ − 45◦polarization basis, as described in [33], and observed visibilitythat reached 0.95. This was done before placing the BDS onthe idler arm and the spatial filter on the signal arm. Besides,we made a tomographic reconstruction of the two-photonpolarization state [43] and observed a purity of 0.92 ± 0.02[tr(ρ2)] and a fidelity [44] of 0.95 ± 0.03 with state |�+

pol〉 =(1/

√2)(|H 〉i |V 〉s + |V 〉i |H 〉s).

For testing the complementarity relation, we generatedHESs with different degrees of entanglement by means ofa suitable choice of the polarization projection implementedin the idler photon, as described in Sec. II A. The first statewas an HMES, generated with α = β = 1√

2and φP = π/2,

which corresponds to projection of the idler photon onto theleft-circular polarization. This state is given by

|�(P )〉 = 1√2

(|V 〉s |F 〉i + |H 〉s |A〉i). (22)

The other three prepared states are generated by consideringdistinct linear polarization projections in the idler arm. Theyhave the form

|�(P )〉 = cos(2ξ )|V 〉s |F 〉i +√

cos2(2ξ ) − 1|H 〉s |A〉i , (23)

and in our case, we adopted ξ = 0◦, ξ = 5◦, and ξ = 10◦.

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Mi, Ms

Di

Ds

Electronic System

20cm15cm

BBO Crystal Type II

L2P1/2

P1/4

P1/2

Ci

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idler

signal

Fast axis

Fast axis

QWP

QWP

F1L1

Mi

Ms

(a)

(b) (c)

BDS

FIG. 2. (Color online) Experimental setup. (a) A single-mode Ar+-ion laser with a wavelength of 351.1 nm passes through a β-bariumborate (BBO) type II crystal 5 mm wide. In both arms there are half-wave plates (P1/2) that rotate the polarization by 90◦, followed by a BBOcompensating crystal 2.5 mm wide (Ci and Cs). Also, half-wave plates, quarter-wave plates, and polarizing beam splitters are placed beforethe detectors to effect projection in the desired polarization basis. The birefringent double-slit (BDS) is placed in the idler arm. In the signalarm there is a spatial filter made by two lenses (L1 and L2) with 15- and 20-cm focal distances, respectively) and a horizontal slit, F1, 50 µmwide. Di and Ds are the idler and signal single-photon detectors, respectively. (b) Sketch of the BDS: two quarter-wave plates with their fastaxes in perpendicular directions—one in front of the upper slit and the other in front of the bottom slit. (c) Mi and Ms represent the spatial andfrequency filters placed before Di and Ds, respectively. They are composed of an iris of controllable diameter followed by an interference filtercentered at 702 nm, with 1-nm FWHM.

IV. RESULTS

We measured four types of interference curves for eachstate—one- and two-photon polarization curves and one- andtwo-photon spatial interferences—from which we obtain thevisibilities V1p, V12p, V1s , and V12s , respectively. For thesemeasurements, we used a 1.3-mm-diameter iris in front of Ds.The aperture in front of detector Di was modified dependingon the measurement performed.

For measurement of two-photon polarization curves, asingle slit of 100 µm × 5 mm (xy), was placed in frontof detector Di, which was fixed at xi = 0, corresponding tospatial projection of the idler photon onto |F 〉i [22,35]. In thecase of one-photon polarization curves, a 2.0-mm-diameteriris was placed before detector Di. This action turns Di into a“bucket” detector, which detects a photon without registeringits transverse position. This is necessary to trace over thespatial DOF, which means that Di must be unable to distinguishbetween |F 〉i and |A〉i [37]. One- and two-photon polarizationcurves were obtained by measuring the coincidence rate interms of the HWP angle at the signal arm. Coincidence curveswere recorded for each of the four prepared HESs.

For measurements of one- and two-photon spatial patterns,a 100 µm × 5-mm slit was placed in front of Di. In the case ofone-photon spatial patterns, the signal polarization analyzer infront of detector Ds was removed to perform a trace over thepolarization DOF. In the case of two-photon spatial patterns,

the polarization analyzer was fixed in the vertical direction(θs = π/2), determining the conditional measurement thatwe desired. All spatial curves were obtained by recordingthe coincidence counts as a function of the Di transverseposition (x).

Figure 3 shows measurements based on the proceduredescribed for the case of an HMES. Figures 3(a) and 3(b) showthe observed one- and two-photon spatial interference patterns,respectively. The one-photon curve was fitted according toEq. (14). From this curve, we obtained the one-photon spatialvisibility V1s , which was, together with the amplitude, a freeparameter in the fit. The conditional spatial curve was fittedwith Eq. (12) by choosing θs = π/2 and leaving the amplitudeas a free parameter. In both fits, the vertical error bars areproportional to the square roots of the measured coincidences.

These curves, properly normalized with the factors presentin Eqs. (12) and (14), were used in the calculation of two-photon corrected curves. The same procedure was used tomake the fits in Figs. 3(c) and 3(d) for polarization DOF,using the respective equation, that is, Eq. (16) for Fig. 3(c) andEq. (12), with the choice xi = 0, for Fig. 3(d).

Based on the curves obtained for one- and two-photonpatterns, it was possible to obtain corrected coincidencecurves for all four prepared HESs. Figure 4 summarizes theexperimental results obtained for these states; the two-photonvs one-photon visibilities are plotted. From left to right, the

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0

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60

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160

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0

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120

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s

Di vertical position (mm)

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100

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250

300

350

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ence

sin

500s

HWP angle (degrees) for Ds

0

20

40

60

80

100

120

50 100 150 200 250

Coi

ncid

ence

sin

1000

s

HWP angle (degrees) for Ds

(d)(c)

(b)(a)

FIG. 3. (Color online) Experimental results for HMESs. (a) One-photon and (b) two-photon spatial interference patterns. (c) One-photonand (d) two-photon polarization interference patterns. In all plots, vertical error bars represent the square root of the measurements. One cansee that there is good agreement between theoretical predictions (solid lines) and experimental results (circles).

experimental points (red circles) correspond to the HMES andnonmaximally HESs with ξ equal to 10◦, 5◦, and 0◦ (productstate) given in Eq. (23).

The agreement of experimental data and the ideal com-plementarity relation (solid curve) given in Eq. (1) is good.Discrepancies between theory and experiment can be at-tributed mainly to the nonperfect polarization entangled stateinitially prepared, to the nonperfect coupling in the BDS dueto the misalignment of the QWP, and, also, to the polarizationprojection to generate the HES. Finally, there is another sourceof error in the process of spatial tracing in our measurementsof one-photon polarization curves. As already noted, the iris

in front of detector Di must be large enough so as not todistinguish between |F 〉i and |A〉i .

V. CONCLUSION

In this work we have extensively analyzed the propertiesof two-photon hybrid interference patterns, which naturallyarise when HESs are let propagate through free space, andpresented them in the context of the complementarity relationthat exists between one- and two-photon visibilities. Wetheoretically obtained expressions for the one- and two-photon interference curves that we expected to measure and

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

V1p

V12

s

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

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V1s

V12

p

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

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V1p

V12

p

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0 (c)(b) (d)

V1s

V12

s

(a)

FIG. 4. (Color online) Two-photon versus one-photon visibilities obtained. Each circle corresponds to a distinct HES. (a) Two-photon versusone-photon spatial visibilities. (b) Two-photon versus one-photon polarization visibilities. (c) Two-photon polarization versus one-photon spatialvisibilities. (d) Two-photon versus one-photon polarization visibilities. Two-photon visibilities for HMESs are the greatest (circles at the left).Two-photon visibilities for product states are the smallest (circles at the right). The solid line corresponds to the complementarity relation givenby Eq. (1).

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analyzed the hybrid behavior presented in the two-photon jointprobability. The complementarity relation for the visibilitiesof these interferences has also been theoretically discussed,and an experiment based on a type II SPDC source of HEShas been performed to verify this relation. Our experimentcorresponds to 12 tests for this complementarity relation.

As a direct application, we mention that the hybridtwo-photon interference can be used to quantify the hybridentanglement. In particular, it is possible to choose whichtype of measurement, the two-photon polarization curve or thetwo-photon Young’s interference, is best suited for determi-nation of experimental entanglement. Another possibility isthe investigation of complementarity in higher dimensionalquantum systems [9,12,14] by generating multiqubit and

qubit-qudit HESs. Generation of the latter class of states isdiscussed in Ref. [22] and requires the use of a birefringentmultislit.

ACKNOWLEDGMENTS

We would like to thank C. H. Monken for lending usthe sanded quarter-wave plates used in the double-slit. Thiswork was supported by Grants No. Milenio ICM P06-067F,CONICYT PFB-0824, PBCT Red21, FONDECYT 11085055,and FONDECYT 11085057. S. Padua acknowledges thesupport of the CNPq, FAPEMIG, and National Institute ofScience and Technology in Quantum Information, Brazil.M. Santibanez thanks CONICYT for support.

[1] M. A. Horne and A. Zeilinger, in Microphysical Reality andQuantum Formalism, Proceedings of the Conference at Urbino,Italy, Sept. 25–Oct. 3, 1985, edited by A. van der Merwe,F. Selleri and G. Tarozzi (Kluwer Academic, Dordrecht, 1988),Vol. 2, p. 401.

[2] G. Jaeger, M. A. Horne, and A. Shimony, Phys. Rev. A 48, 1023(1993).

[3] G. Jaeger, A. Shimony, and L. Vaidman, Phys. Rev. A 51, 54(1995).

[4] M. A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett.62, 2209 (1989); D. M. Greenberger et al., Phys. Today 46,22 (1993); in Quantum Coherence and Reality, Proceedingsof the International Conference on Fundamental Aspects ofQuantum Theory at Columbia, United States, 10–12 December1992, edited by J. S. Anandan and J. L. Safko (World Scientific,Singapore, 1995), p. 233.

[5] M. Horne, in Experimental Metaphysics, edited by R. S. Cohen,M. Horne, and J. Stachel (Kluwer, Boston, 1997), Vol. 1, p. 109.

[6] R. Ghosh and L. Mandel, Phys. Rev. Lett. 59, 1903 (1987).[7] A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko,

and M. C. Teich, Phys. Rev. A 63, 063803 (2001).[8] W. H. Peeters, J. J. Renema, and M. P. van Exter, Phys. Rev. A

79, 043817 (2009).[9] M. Jakob and J. A. Bergou, Phys. Rev. A 76, 052107 (2007).

[10] M. Jakob and J. A. Bergou, Opt. Commun. 283, 827 (2010).[11] T. E. Tessier, Found. Phys. Lett. 18, 107 (2005).[12] X. Peng, X. Zhu, D. Suter, J. Du, M. Liu, and K. Gao, Phys.

Rev. A 72, 052109 (2005).[13] A. Hosoya, A. Carlini, and S. Okano, Int. J. Mod. Phys. C 17,

493 (2006).[14] X. Peng, J. Zhang, J. Du, and D. Suter, Phys. Rev. A 77, 052107

(2008).[15] B. C. Hiesmayr and M. Huber, Phys. Rev. A 78, 012342

(2008).[16] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).[17] F. de Melo, S. P. Walborn, J. A. Bergou, and L. Davidovich,

Phys. Rev. Lett. 98, 250501 (2007).[18] M. Jakob and J. Bergou, Phys. Rev. A 66, 062107 (2002).[19] M. Zukowsky and A. Zeilinger, Phys. Lett. A 155, 69 (1991).[20] X.-s. Ma, A. Qarry, J. Kofler, T. Jennewein, and A. Zeilinger,

Phys. Rev. A 79, 042101 (2009).[21] L. Neves, G. Lima, J. Aguirre, F. A. Torres-Ruiz, C. Saavedra,

and A. Delgado, New J. Phys. 11, 073035 (2009).

[22] L. Neves, G. Lima, A. Delgado, and C. Saavedra, Phys. Rev. A80, 042322 (2009).

[23] J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, Phys. Rev. Lett. 105,030407 (2010).

[24] M. Fujiwara, M. Toyoshima, M. Sasaki, K. Yoshino, Y. Nambu,and A. Tomita, Appl. Phys. Lett. 95, 261103 (2009).

[25] F. Bussieres, J. A. Slater, J. Jin, N. Godbout, and W. Tittel, Phys.Rev. A 81, 052106 (2010).

[26] C. Gabriel et al., e-print arXiv:1007.1322v1 [quant-ph].[27] D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84

(1970).[28] C. K. Hong and L. Mandel, Phys. Rev. A 31, 2409 (1985).[29] E. Nagali and F. Sciarrino, Opt. Express 18, 18243 (2010).[30] E. Karimi et al., Phys. Rev. A 82, 022115 (2010).[31] Q. Zhan, Adv. Opt. Photon. 1, 1 (2009).[32] R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901

(2003).[33] P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V.

Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337 (1995).[34] L. Neves, G. Lima, J. G. Aguirre Gomez, C. H. Monken,

C. Saavedra, and S. Padua, Phys. Rev. Lett. 94, 100501 (2005).[35] G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra,

and S. Padua, J. Phys. B 41, 185501 (2008).[36] D. M. Greenberger, M. A. Horne, and A. Zeilinger, Bell’s

Theorem, Quantum Theory, and Conceptions of the Universe,edited by M. Kafatos (Kluwer Academic, Dordrecht, 1989),pp. 73–76; D. M. Greenberger et al., Am. J. Phys. 58, 1131(1990).

[37] L. Neves, G. Lima, E. J. S. Fonseca, L. Davidovich, andS. Padua, Phys. Rev. A 76, 032314 (2007).

[38] E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, andS. Padua, Phys. Rev. A 61, 023801 (2000).

[39] Similarly, it happens when the idler detector is fixed at xi = πz

2kd,

and the signal analyzer is rotated. There is no interference at all.[40] J.-M. Cai, Z.-W. Zhou, and G.-C. Guo, Phys. Rev. A 73, 024301

(2006).[41] M. Franca Santos, P. Milman, A. Z. Khoury, and P. H. Souto

Ribeiro, Phys. Rev. A 64, 023804 (2001).[42] S. M. Lee, S.-W. Ji, H.-W. Lee, and M. Suhail Zubairy, Phys.

Rev. A 77, 040301(R) (2008).[43] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys.

Rev. A 64, 052312 (2001).[44] R. Jozsa, J. Mod. Opt. 41, 2315 (1994).

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