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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 114008 T Jennewein et alphoton pairs and obtain the overall two-photon coincidenceprobability asP b,c ′ (τ) = 1 ∫ d/2 ∫ d/2Pd 2 b,c (t a,b − t c,d − τ)dt a,b dt c,d , (3)−d/2−d/2where τ is the offset of the two arms of the photons to the beamsplitter. We further assume that the photon wave-packets area Gaussian shape with the width equal to the coherence timeτ cohr of the photons, which interfere with time delay τ andthe offset t a,b − t c,d . Therefore the coincidence probability forthe interference of two photons isP b,c (t a,b − t c,d − τ)= 1 2(1− 2V VI1+ V VIexp (t a,b − t c,d − τ) 2τ 2 cohr).In our experiment the crystal had a thickness of d = 2 mm,hence the photon pairs are undefined within the range of|t a,b | 180 fs (large compared to the photon coherenceτ cohr = 775 fs for 1 nm bandwidth filters). The HOM visibilityis defined as V VII = (P ′ (inf) − P ′ (0))/P ′ (inf). The explicitnumerical calculation of the integral (3) for filter band widthsof λ FWHM of 1 nm, 2 nm, 3 nm leads to the HOM visibilitiesof V VII,1nm = 0.9641,V VII,2nm = 0.8817,V VII,3nm = 0.7778respectively. A further advanced calculation should alsoinclude the synchronization jitter of two pump lasers, andis given by Kaltenbaek [39].The standard approach for minimizing or even avoidingthe effects of this group velocity mismatch is to use narrowbandwidth filters and thereby extend their coherence timeof the SPDC photons beyond the group velocity dispersion.Unfortunately this leads to a huge loss of the generatedphotons. This timing effect is finally resolved by utilizingparticular crystals with well chosen or even specificallytailored group velocity properties, that do not require anyfilters for the SPDC and still achieve high-quality two-photoninterference [40–43].4. Experimental resultsThe expected contributions of all the above discussed effectsare summarized in table 1. The total visibility is calculatedby taking the product of all contributing visibilities. Inour experiment we performed a range of test measurementsin order to compare the experimentally attainable HOMvisibilities with the estimated visibilities, these discussedabove. The measured results are summarized in figure 2and compared with the respective estimates. Note the verygood agreement of the experiment with the calculation.Now it is possible to understand the various effects thatreduce the experimental quality in our entanglement swappingexperiment. More importantly we are able the adequatelychoose the filter bandwidth as 1 nm FWHM, such thatall further measurements on the entanglement swapping areperformed at the required quality of the experiment.The correlations of the entangled photons are quantifiedby measuring the correlations between photons a and d withthe expectation value for joint polarization measurementsE(φ a ,φ d ), where φ a and φ d are the polarizer setting for photons(4)a and d, respectively [44]. The correlation coefficients aredefined asE(φ a ,φ d ) = (N ++ (φ a ,φ d ) − N +− (φ a ,φ d )∑− N −+ (φ a ,φ d ) + N −− (φ a ,φ d ))/Nij , (5)where as usual the N ij (φ a ,φ d ) are the coincidence countsbetween the i-channel of the polarizer of photon a set at angleφ a , and the j-channel of the polarizer of photon d set at angleφ d . Each measurement run of the correlation coefficientsE(φ a ,φ d ) for photons a and d at settings φ a = φ d = 45 ◦involved a scan through the relative time delay betweenthe photons b and c by moving a mirror in the path ofthe pump laser for one of the SPDC sources, see figure 3.Clearly the dependence of the photon interference on the delaybetween the two photons can be observed and highlights theoperation of the interferometric Bell-state analysis used inthis entanglement swapping experiment. This measurementactually shows the very high correlation of the entangledphotons after the entanglement swapping of E = 0.91 forthe | − 〉 ad and E = 0.86 for the | + 〉 ad .Several such expectation values E were used to obtainmeasures for the state fidelity, defined as F =〈|ρ〉. Tosee how this was done, first take the general definition of atwo-qubit density matrix ρ( 3∑)ρ = 1 T mk σ m ⊗ σ k (6)40where T mk are the expectation values Tr{ρσ m σ k } for aparticular spin measurement, σ 0 is the identity and σ 1,2,3are the Pauli matrices. We also consider the projectorsof the two Bell states used in our experiment, which are| ∓ 〉〈 ∓ |=0.25(1 − σ 1 σ 1 ∓ σ 2 σ 2 ∓ σ 3 σ 3 ) and obtain thefidelity via the trace F | ∓ 〉 = Tr{ρ| ∓ 〉〈 ∓ |}, leading toF | ∓ 〉 = 1 4 [1 − T 11 ∓ T 22 ∓ T 33 ]. (7)This expression is experimentally easily accessible, since itonly requires three different measurements and not the setof the nine measurements (combinations of the three basissettings on either photon) needed for performing a tomographyof the state – which made a huge practical difference giventhe limited count rates of our experiment. In our case theaxis of the Pauli matrices are identified with the polarization,where σ 1,2,3 correspond to analyser setting in the H, 45 andL polarization basis. Therefore the coefficients in the aboveexpression for the fidelity are T 11 = E(H, H), and likewise forthe other polarization bases.We have performed the correlation measurements on theswapped entangled photons which end up in the | − 〉 ad or| + 〉 ad depending on the outcome of the bell-state analyser, seetable 2. These results lead to overlap fidelities of the swappedentangled photons with the ideal Bell states of F | − 〉 = 0.892and F | + 〉 = 0.879 for states | − 〉 ad and | + 〉 ad , respectively.As a sufficient indication of the presence of entanglementwe tested a Bell inequality, where the suitable version forexperiments is the Clauser–Horne–Shimony–Holt (CHSH)variant [29], which has the following form:S =|E(φ a ′ ,φ′ d ) − E(φ′ a ,φ′′ d )| + |E(φ′′ a ,φ′ d ) + E(φ′′a ,φ′′ d)| 2.(8)5
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 114008 T Jennewein et alTable 2. Measured correlation values for photons a and d forpolarizer settings θ a ; θ d , after the entanglement swapping protocol.The outcome of the Bell-state measurement on photons b and cdetermines the type of correlation to be a − or + . The first threemeasurements show the high quality of the entangled state in thethree bases. These measurements give us the very high state fidelityof F = 0.892 and F = 0.879. The following four measurementsallow us to test the CHSH inequality. As the local realistic upperlimit for S is 2, above results for both Bell states clearly violate theCHSH inequality. Note that the correlation coefficients for the twoBell states were measured within the same measurement run for aparticular polarizer setting by sorting the data into correspondingsubsets. The given errors are the statistical uncertainties due tophoton counting. The differences in the correlation coefficientscome from the higher correlation fidelity for analyser settings closerto 0 ◦ and 90 ◦ .E(θ a ; θ d ) | − 〉 ad | − 〉 bc | + 〉 ad | + 〉 bcE(0 ◦ ; 0 ◦ ) −0.972 ± 0.0 −0.957 ± 0.0E(45 ◦ ; 45 ◦ ) −0.832 ± 0.0 0.771 ± 0.0E(L; L) −0.765 ± 0.0 0.789 ± 0.0E(0 ◦ ; 22.5 ◦ ) −0.61 ± 0.04 −0.55 ± 0.04E(0 ◦ ; 67.5 ◦ ) 0.80 ± 0.04 0, 60 ± 0, 04E(45 ◦ ; 67.5 ◦ ) −0.64 ± 0.05 0.48 ± 0.05E(45 ◦ ; 22.5 ◦ ) −0.55 ± 0.05 0.67 ± 0.05Bell-parameter S 2.60 ± 0.09 2.30 ± 0.09It is straightforward to show that the prediction of quantummechanics leads to E QM (φ a ,φ d ) =−cos(2(φ a − φ d )). Withthe settings chosen as (φ a ′,φ′ d ,φ′′ a ,φ′′ d ) = (0◦ , 22.5 ◦ , 45 ◦ ,67.5 ◦ ) the maximal value of S QM = 2 √ 2, which clearlyviolates the limit of 2 imposed by inequality (8). Thenecessary series of correlation coefficients of photons a andd were measured, and are given in table 2. Note that thecorrelation coefficients for the two different Bell states weremeasured within the same measurement run for a particularpolarizer setting by sorting the data into corresponding subsetsdepending upon the observed Bell state. Our results show clearviolations of the Bell inequality with S = 2.60 ± 0.09 for the| − 〉 ad and S = 2.30 ± 0.09 for the | + 〉 ad Bell state. Thelower value of S for | + 〉 ad is due to the difficult polarizationalignment of the output fibres of the beam splitter.5. ConclusionWe demonstrate an entanglement swapping experimentusing photons produced by spontaneous parametric downconversion(SPDC) and an optimal Bell-state analyser with50% efficiency, i.e. capable of detecting two of the four Bellstates perfectly. Via a thorough analysis and optimization ofthe experimental errors in our system, we achieved a very highexperimental entanglement correlation and tested the violationof two Bell inequalities on the swapped photons in the sameexperiment run. In addition, we showed that the final entangledBell states had an observed state fidelity F = 0.892 andF = 0.879 with the respective ideal states | − 〉 bc and | + 〉 bc .In particular, we studied the various deteriorating effects in thespontaneous parametric down conversion which are crucialfor performing high-quality multi-photon experiments, andgave a simple model that allowed us to estimate the finalstate fidelity. We consequently improved the experimentalperformance by carefully optimizing the several sources oferrors. As the main experimental obstacle for SPDC systemswe also estimated the group velocity mismatch between thepump and the down-conversion photons in the SPDC crystal ina simple mathematical model, and compared the calculationswith several test experiments. These calculations also allowedus to understand why in our case the required bandpass filtersfor the interfering photons must be the narrow value of 1 nmFWHM. We note that finally the problem of the groupvelocity mismatch will be solved by cancelling the dispersioneffects with advanced phase-matching methods [40–43]. Thevarious effects of errors must be considered in most types ofmulti-photon experiments that rely on two-photon interferenceeffects and are based on parametric down-conversion.From the fundamental side it is interesting to note that theobserved violations of the Bell inequalities in our entanglementswapping experiment drastically highlights the non-classicalnature of entanglement, since the outcome of the Bellmeasurementof the two inner photons b and c (| − 〉 bc and| + 〉 bc ) even determines the type of entanglement of the twoouter photons a and d (| − 〉 ad and | + 〉 ad )[45].Since teleportation using a two-outcome Bell-statemeasurement is also one of the building blocks of linearopticalquantum gates [46], this experiment clearly representan important step for linear-optical quantum informationprocessing, the creation of multi-photon entangled states andquantum repeater protocols.AcknowledgmentsWe thank M Zukowski, D Greenberger, C Brukner, T Rudolph,A White, M Barbieri and R Kaltenbaek for fruitful discussionson experimental and theoretical questions. This work has beensupported by the Austrian Science Foundation (FWF) ProjectNo. F1506 and F1520, and by the European Commission underthe Integrated Project Qubit Applications (QAP) funded bythe IST directorate and the DTO-funded U.S. Army ResearchOffice.References[1] Bennett C H, Brassard G, Crépeau C, Jozsa R, Peres Aand Wootters W K 1993 Phys. Rev. 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