The multimode four-photon Hong-Ou-Mandel interference
TThe multimode four-photon Hong-Ou-Mandel interference
A. Ferreri, V. Ansari, C. Silberhorn, and P. R. Sharapova Department of Physics, Paderborn University, Warburger Strasse 100, D-33098 Paderborn, Germany
The two-photon Hong-Ou-Mandel (HOM) interference is a pure quantum effect which indicatesthe degree of indistinguishability of photons. The four-photon HOM interference exhibits richerdynamics in comparison to the two-photon interference and simultaneously is more sensitive to theinput photon states. We demonstrate theoretically and experimentally an explicit dependency ofthe four-photon interference to the number of temporal modes, created in the process of paramet-ric down-conversion. Moreover, we exploit the splitting ratio of the beam splitter to manipulatethe interference between bunching and antibunching. Our results reveal that the temporal modestructure (multimodeness) of the quantum states shapes many-particle interference.
PACS numbers: 42.65.Lm, 42.65.Yj, 42.25.Hz
I. INTRODUCTION
A useful tool to investigate the degree of indistin-guishability of photons in quantum optics is the Hong-Ou-Mandel (HOM) interference [1]. Beyond the intereston the peculiar quantum interference itself, HOM inter-ference is extremely important in quantum information,e.g. for the Bell-state measurement [2], it is used to tes-tify the non-locality of entangled system [3] and in quan-tum lithography [4, 5]. Typically this kind of interfer-ence is studied by involving two photons in free space,though in the last years integrated quantum devices arealso utilized because of their helpful features which allowto manipulate the interference process [6, 7].However, the multiphoton interference which involvesmore than two photons attracts a lot of attention nowa-days [8–11] because it is an essential and indispensabletool for boson sampling [12] and machine learning [13–16]which are the first steps of future quantum computing.Moreover, the multiphoton interference allows to achievea high-dimensional entanglement [17], overcome the stan-dard quantum limit in interferometry [18] and create highdimensional NOON states [19]. In addition, the multi-photon interference can highlight and solve the funda-mental question about the quantum-to-classical transi-tion. In this way, non-monotonic character of the photoninterference with increasing the number of photons wasstudied in [20]. Behaviour of the four-photon interfer-ence with increasing a pump power was investigated in[21]. The multiphoton interference is directly connectedwith the multiparticle indistinguishability and the col-lective phase of photons [22, 23], theoretical descriptionof the multiphoton interference with transition matrix ispresented in [24].In this work we investigate the four-photon interfer-ence in relation to the spectral-temporal properties ofthe photons generated via parametric down-conversion.We show that the number of temporal modes drasticallyinfluences the interference pattern, observing a raising ofHOM dip/peak in the coincidence probability. Our anal-ysis takes into account also different beam splitter param-eters, allowing us to control the interference visibility. We show that the temporal mode structure, particularly theamount of multimodeness, of the photon source plays acrucial role in the multiphoton interference in contrast tothe two-photon interference [6].
II. THEORETICAL MODEL
The type-II parametric down-conversion (PDC) pro-cess produces pairs of photons related by both frequencyand polarization entanglement [25, 26]. The Hamiltonianof the type-II PDC process can be written in terms of thejoint spectral amplitude (JSA) F ( ω s , ω i ) [27]: H = Γ (cid:90) d ω s d ω i F ( ω s , ω i ) a † H ( ω s ) a † V ( ω i ) + H . c ., (1)where the indices "s" and "i" indicate the "signal" and"idler" photons respectively, a s, ( i ) and a † s, ( i ) are the anni-hilation and creation operators of the signal (idler) pho-tons, the coupling constant Γ determines the strength ofinteraction, H and V label the horizontal and vertical po-larization of the photons. In a periodically poled mediumwith the poling period Λ the JSA can be written in theform F ( ω s , ω i ) = e − ( ωs + ωi − ωp )2Ω2 sinc (cid:18) L k (cid:19) e i L ∆ k , (2)where Ω is the pump spectral bandwidth, L is the lengthof the PDC section, ω p is the pump center frequency, ∆ k = k p ( ω p ) − k s ( ω s ) − k i ( ω i )+ π Λ is the phase matchingcondition which determines the momentum conservationof the process.The four-photon state generated in the PDC processcan be described by using the second-order of the per-turbation theory. Neglecting the time-ordering effect the a r X i v : . [ phy s i c s . op ti c s ] A p r Figure 1. Schematic setup. The type-II PDC process gen-erates two signal-idler pairs of photons. After PBS, twohorizontally-polarized photons are routed to the upper chan-nel (red line), while two vertically-polarized photons, arerouted to the lower channel (green line). A PC located inthe upper channel converts the horizontally-polarized photonsinto the vertically-polarized. An additional path increment L = l + ∆ l in the lower channel allows to compensate thetime delay between the signal and idler photons. Then fourvertically-polarized photons cross the BS at the same time,the HOM interference occurs. The photons are detected afterthe BS. generated four-photon state is [28–30] | ψ ph (cid:105) = 12 (cid:18) (cid:90) t H ( t (cid:48) ) dt (cid:48) (cid:19) | (cid:105) = ξ (cid:90) + ∞−∞ d ω s d ω i F ( ω s , ω i ) a † H ( ω s ) a † V ( ω i ) × (cid:90) + ∞−∞ d ˜ ω s d ˜ ω i F (˜ ω s , ˜ ω i ) a † H (˜ ω s ) a † V (˜ ω i ) | (cid:105) , (3)where ω s , ω i , ˜ ω s , ˜ ω i are the frequencies of the four gen-erated photons and ξ = Γ t , where t is the time of theinteraction process.The JSA strongly depends on dispersion properties ofthe non-linear material where PDC takes place [31]. Forour investigation we choose a ppKTP waveguide sincethis material finds flexibility to the number of Schmidt modes: a ppKTP waveguide is able to provide a quasi-single-mode PDC state, as well as the strongly multimoderegime by varying the pulse duration only. Such propertyis more difficult to obtain in other materials (LiNbO ,BBO) without filtering.To observe the HOM interference, the incoming pho-tons into a beam splitter (BS) must be fully indistin-guishable in all degrees of freedom: must have the samepolarization and cross the BS without any time delay. Tosatisfy both conditions, we consider the following setupschematically sketched in Fig.1. In Fig.1 the generatedphotons inside the PDC section are separated then by apolarization beam splitter (PBS) in two different spatialchannels. The horizontally-polarized photons is routedin the upper channel, and the vertically-polarized pho-tons in the lower channel. A polarization converter (PC)located in the upper channel converts the horizontally-polarized photons into the vertically-polarized. Due tothe different group velocities of the horizontally- andvertically-polarized photons inside the nonlinear crys-tal, a time delay between the signal and idler photonsis present already after the PDC section. To compen-sate this time delay we create an additional path incre-ment ∆ l in the lower channel. In the end of the setupthe four vertically-polarized photons cross the BS at thesame time and interfere. The interference pattern is mea-sured at the output ports of the BS using photon numberresolving detectors.Mathematically, all these transformations can be ex-pressed as a unitary matrices which act on the initialfour-photon state. The final unitary transformation cantherefore be written as a product of unitary matrices: U tot = BS × F P × P C × F P × P BS × F P , (4)where BS , P C and
P BS are the matrices of the concern-ing elements and the matrices
F P i describe a free propa-gation of the light between optical elements. The outputstate after the all transformations can be obtained afterthe action of the total matrix U tot on the initial four-photon state. In the basis { a † H , a † V , a † H , a † V } , wherethe indexes 1, 2 correspond to the upper and lower chan-nels, the output state can be represented by | ψ out (cid:105) = ξ (cid:90) d ω s d ω i d ˜ ω s d ˜ ω i F ( ω s , ω i ) F (˜ ω s , ˜ ω i ) × U ( ω s ) U † tot ( ω s ) a † H ( ω s ) a † V ( ω s ) a † H ( ω s ) a † V ( ω s ) ⊗ U ( ω i ) U † tot ( ω i ) a † H ( ω i ) a † V ( ω i ) a † H ( ω i ) a † V ( ω i ) ⊗ U (˜ ω s ) U † tot (˜ ω s ) a † H (˜ ω s ) a † V (˜ ω s ) a † H (˜ ω s ) a † V (˜ ω s ) ⊗ U (˜ ω i ) U † tot (˜ ω i ) a † H (˜ ω i ) a † V (˜ ω i ) a † H (˜ ω i ) a † V (˜ ω i ) | (cid:105) , (5)where U ( ω s ) and U ( ω i ) are the initial condition matri- ces [32]. The output state Eq.(5) can be modified andwritten in the basis | m, n (cid:105) with the probability ampli-tudes C m,n (∆ l, τ, L ) , where m indicates the number ofphotons in the upper channel, n - the number of photonsin the lower channel: | ψ out (cid:105) = (cid:90) d ω s d ω i d ˜ ω s d ˜ ω i ( C (∆ l, τ, L ) | , (cid:105) + C (∆ l, τ, L )( | , (cid:105) + | , (cid:105) ) + C (∆ l, τ, L )( | , (cid:105) + | , (cid:105) )) . (6)By using the final state Eq. (6), we can calculate theexpectation values of the simultaneous positive-operatorvalued measures (POVM) which corresponds to the mea-suring the coincidence probability at the detectors : P (∆ l, τ, L ) = (cid:90) d ω b d ω c d ˜ ω b d ˜ ω c |(cid:104) | √ √ d ( ω b ) d ( ω c ) d (˜ ω b ) d (˜ ω c ) | ˜ ψ out (cid:105)| (7) P (∆ l, τ, L ) = (cid:90) d ω b d ω c d ˜ ω b d ˜ ω c |(cid:104) | √ d ( ω b ) d ( ω c ) d (˜ ω b ) d (˜ ω c ) | ˜ ψ out (cid:105)| (8) P (∆ l, τ, L ) = (cid:90) d ω b d ω c d ˜ ω b d ˜ ω c |(cid:104) | √ d ( ω b ) d ( ω c ) d (˜ ω b ) d (˜ ω c ) | ˜ ψ out (cid:105)| , (9)where d and d are the annihilation operators of thedetectors placed in the upper and the lower channels re-spectively, ω b , ω c , ˜ ω b , ˜ ω c are the frequencies of the detec-tors, | ˜ ψ out (cid:105) = | ψ out (cid:105) / (cid:104) ψ out | ψ out (cid:105) is the normalized outputstate. The Eq. (7,8,9) present the probability P mn to de-tect m photons in the upper channel and n photons inthe lower channel respectively. III. RESULTS AND DISCUSSIONA. Experiment
To study the impact of time-frequency correlations ofthe state on the four-photon interference, we employ anengineered, programmable PDC source. Our source isan 8 mm long waveguided channel in KTP, engineered tothe symmetric group velocity matching condition whichallows us to flexibly control the frequency correlation be-tween signal and idler photons by modulating the pumppulses only [33, 34]. We pump the source with ultra-short pulses out of an Ti:Sa oscillator, followed by a pulseshaper. The pulse shaper is based on a spatial light mod-ulator in a 4f setup which allows us to shape the spectralamplitude and phase of pump pulses. With this con-figuration we can have pulses with a time bandwidthsranging from 0.3 to 40 picoseconds. Further details ofthe experimental setup are given in Fig. 2 [33, 35].
SMFCOUPLERPBS SPECTRALSHAPING 765 nmPUMP Ti:SaPPKTP WAVEGUIDE SNSPDsBPFHWP1 1234HWP2INTERFEROMETER DETECTION50:50
Figure 2. Experimental setup. A femtosecond tita-nium:sapphire (Ti:Sa) oscillator with repetition rate of 80MHz is used to pump a PPKTP waveguide designed for type-II PDC. For spectral shaping of the pump, we use a spatiallight modulator (SLM) in a folded 4f setup to shape the de-sired spectral amplitude and phase. An 8 nm wide bandpassfilter (BPF) centered at 1532 nm was used to block the pumpand phasematching side-lobes. The orthogonally polarizedPDC photons were sent to the interferometer setup wherewe used a polarising beamsplitter (PBS), a half waveplate(HWP), and an adjustable time delay stage ∆ τ to control theinterference. Then the photons were sent to a single-modefibre coupler with an adjustable coupling ratio where inter-ference happens. Each output port of the fibre coupler isthen connected to a balanced fibre splitter followed by super-conducting nanowire single photon detectors (SNSPD). In this work we consider three PDC states (A, B, C),generated by pump pulses with bandwidths of 0.14, 1.29,and 6.62 picoseconds, as summarized in Table I. To shapethe temporal profile of the pump field in the case of statesA and B we simply increase the pulse duration and carveout the corresponding spectral amplitude with a constantspectral phase. This approach works pretty well for shortpulse duration but for longer pulse duration results in asignificantly small pulse energies which in turn reducesthe probability of generating PDC photons. For this rea-son, in the case of state C we take an alternative method;we use the same spectral amplitude as in the case Abut with a quadratic spectral phase with the constant D = 1 . . This additional phase modifies the JSA asfollows ¯ F ( ω s , ω i ) = F ( ω s , ω i ) e iD ( ω s + ω i − ω p ) . (10)To characterize the joint spectral intensity (JSI), | F ( ω s , ω i ) | , of the PDC states, we employ a time-of-flight spectrometer with a resolution of 0.1 nm [36]. Inthe first column of Fig. 3 we plot measured JSIs of threePDC states. Since the JSI does not contain any informa-tion about the spectral phase, the JSI of the state A andC are essentially identical. State B, however, features astrong frequency anti-correlation. The second column ofFig. 3 presents theoretically calculated spectral ampli-tude and phase of considered JSAs; it is clearly seen that S T A T E A S T A T E B S T A T E C Experimental JSI Theoretical JSA Theoretical JTA HOMI
Figure 3. Spectral-temporal properties of considered PDC states: state A is a nearly decorrelated PDC, state B is a standardfrequency anti-correlated PDC, and state C is a PDC state with spectral phase anti-correlations from a strongly chirped pump.The first column presents measured joint spectral intensity (JSI) which contains no information about the spectral phase. Thesecond column depicts the absolute value and the phase of theoretical joint spectral amplitudes (JSAs). The third column isthe absolute value of theoretical joint temporal amplitudes (JTAs). The fourth column shows the calculated (red solid line)and measured (blue dots) two-photon Hong-Ou-Mandel intereference (HOMI), with error bars smaller than the dots.Table I. Overview of studied PDC states. All bandwidths ∆ are referring to the standard deviation of the amplitude theoptical field.PDC state A B C ∆ λ pump (nm) 1.8 0.2 1.8 ∆ ω pump (THz) 3.479 0.386 3.44 D ( ps ) 0 0 1.9 ∆ t pump (ps) 0.14 1.29 6.62 g (2) ( τ = 0) 1 . ± .
011 1 . ± .
010 1 . ± . K = g (2) − states A and C are completely different due to the phase.To find a better understanding of our experiment, itwould be helpful to translate the spectral representationto temporal description by means of Fourier transform,since the HOM interference shows an indistinguishabilityof photons with the same polarization and mode struc- ture in time F ( t s , t i ) = (cid:90) d ω s d ω i F ( ω s , ω i ) e i ( ω s t s + ω i t i ) . (11)The absolute value of the joint time amplitude (JTA),Eq. (11), is depicted in the third column of Fig. 3. Inthese plots, it is clearly seen that the temporal correla-tions between signal and idler photons, the multimode-ness in the state, is increasing from state A to B and toC. Despite distinctively different pump pulse bandwidths,the width of the two-photon HOM interference is mainlydetermined by the fixed crystal length and thus nearlyidentical for all three states, as experimentally shownin fourth column of Fig. 3. This independence of thetwo-photon HOM interference on pump pulse duration isa consequence of special dispersion characteristic of ourPDC source which has been reported in the past [28, 37].The visibility of the HOM dip, on the other hand, isgoverned by the symmetry of JSA around the main di-agonal λ s = λ i : for the non-symmetrical JSA even for a Figure 4. Theoretical coincidence probabilities to detect: (a) two photons per channel, (b) three photons in one channel andone photon in the other channel (the green and blue curves are dotted in order to show the overlapping with the black and redcurves respectively), (c) four photons in one channel for different pump pulses.Figure 5. Comparison between theory and experiment of the P probability in cases a) A, b) B and c) C. zero time delay the probability to observe two photonsin two channels is not zero.To characterize the Schmidt number K = g (2) − ofPDC states, we measure the unheralded second-ordercorrelation function g (2) ( τ = 0) [38]. The g (2) measure-ment probes the photon number statistics of signal oridler photons, where a single-mode PDC state shows a g (2) = 2 and a highly multimode state gives g (2) = 1 .As shown in Table I, state A is nearly single-mode whilestates B and C have an increasing amount of multimod-eness. B. Balanced BS
In this section, we investigate the four-photon interfer-ence using the aforementioned three PDC states. In thefirst part of our analysis, we use a balanced BS, i.e. hav-ing the same values for the reflection and transmissioncoefficients. The probabilities P mn to detect m photonsin the upper channel and n photons in the lower channeldescribed by Eq. (7,8,9) are plotted in Fig.4.We can clearly observe that by varying the pump pulseduration the four-photon HOM interference is modified considerably, unlike what was expected in the case oftwo-photon interference, seen in Fig. 3. Such behaviourreflects the complexity of the multiphoton interferenceand can be explained by the number of Schmidt modes[39] in the PDC state. Indeed, the black line in Fig.4corresponds to the quasi-single-mode case, state A, withthe Schmidt number K = 1 . and near circular JTA.Increasing the pump temporal duration, states B and C,results in a higher number of modes which leads to astronger anti-bunching effect at zero time delay which isreflected in growing of the P probability.Simultaneously, for the zero time delay under the as-sumption of symmetrical JSA, i.e. F ( ω s , ω i ) = F ( ω i , ω s ) ,the P and P = 1 − P probabilities can be calculatedanalytically with using of the Schmidt decomposition ofJSA : F ( ω s , ω i ) = (cid:80) n √ Λ n u n ( ω s ) v n ( ω i ) , where Λ n areeigenvalues, u n and v n are eigenfunctions of the Schmidtdecomposition [39] (see Supplementary): P = 12 + 2 (cid:80) n Λ n . (12)It means that P can be varied from P = 1 / in thesingle mode regime to the P = 1 / in the strongly mul-timode case. This relation between the degree of multi- Figure 6. Theoretical coincidence probabilities to detect: (a) two photons per channel, (b) three photons in one channel andone photon in the other channel, (c) four photons in one channel for different pump pulses in the unbalanced BS case.Figure 7. Comparison between theory and experiment of the P probability for cases a) A, b) B and c) C in the unbalancedBS configuration. modeness, which is characterized by the Schmidt number K = 1 / (cid:80) n Λ n , and antibunching properties of light canbe observed only in the case of multiphoton interference,which is highly relevant for any quantum networks deal-ing with multimode many-particle states like boson sam-pling. It is important to note that there are no analogousvariations in the case of the two-photon interference.In the case of state C the JSI is the same as in thequasi-single-mode case A but due to the quadratic phaseof pump pulse, their JTAs are completely different, seeFig. 3, the last leads to different interference patterns.However, for states of light with the same JTAs thecoincidence probabilities are identical since behavior ofthe multiphoton interference is determined by temporalproperties of light. For example, the interference pat-terns in the case C and in the case of the small pumpspectral bandwidth . THz are close to each other (Fig.4b, green and blue lines) and show the multimodeness oflight.Moreover, in the four-photon case the symmetry ofJSA is very important because it is strongly affects the P probability. In particular, in cases A and C the JSAis not fully symmetrical around the main diagonal, thisleads to the non-zero P probability even for zero time delay, Fig.4b, black and green curves. Nevertheless, bydecreasing the pump spectral bandwidth the JSA be-comes more symmetrical respect to the signal-idler vari-ables exchange, case B, and as a consequence, P van-ishes when ∆ l = 0 (red line in Fig. 4b).To justify our theoretical analysis we built the experi-mental setup is sketched in Fig. 2. Four detectors in theend of the setup allow to analyze the four-photon inter-ference. The comparison between measured coincidenceprobabilities and theoretical calculations are illustratedin Fig.5 by black and red curves respectively. C. Unbalanced BS
A balanced beam splitter, i.e. a beam splitter havingthe same values for both transmission and refection coef-ficients, is a fundamental tool in the two-photon interfer-ence scenario, since it allows to inhibit the probability tomeasure two photons in different channels. In the four-photon scenario, a balanced BS is able to annul the P probability but cannot erase the P probability. Thatis why it is also interesting to take into account anothervalues of the BS parameters, which allow to inhibit theoutput state with two photons in both channels and pro-vide an analogy with the two-photon interference. Thevalues of the transmission and reflection coefficients ofthe BS which vanish the P probability for the zero timedelay in the single-mode (plane wave) case are (3 ±√ / [40].In Fig.6 one can observe the behaviour of the P mn probabilities for the different JSAs depicted in Fig.3. Asit clearly seen, with using the unbalanced BS we caninhibit drastically the P by imposing a circular JSA,black curve in Fig. 6a. Nevertheless, it is not possible toannul the P totally since the JSA is not perfectly cir-cular. With increasing the number of modes the P isgrowing. It is worth to observe, that the P probability,which shows a dip using a balanced BS, transforms tothe broad peak in the unbalanced case, which is conse-quence of the unbalanced BS. The comparison with theexperimental data is shown in Fig.7. Figure 8. The two-photon HOM dip with a balanced BS andthe four-photons HOM dip with an unbalanced BS in thesingle-mode regime: the pulse duration is 0.29 ps.
In the case of the unbalanced BS and taking the single-mode state, the P probability is similar to the typicaltwo-photon HOM-dip and we can therefore compare suchcase with the two-photon interference (the P probabil-ity) by using the same parameters but the balanced BS.The comparison is presented in Fig.8, one can observethat the width of the dip in P and P cases are closeto each other. Hovewer, the minimum point in the P case is deeper respect to the P , which means that P is more stable against the non-symmetrical JSA. Simul-taneously the P is broader in comparison to the P (FWHM ≈ . mm for two photons, FWHM ≈ . mmfor four photons) that indicates larger range of indistin-guishability of four photons due to the unbalanced BS. IV. CONCLUSION
The four-photon Hong-Ou-Mandel interference was in-vestigated by using different mode content of the PDCsource. It was observed that the HOM profiles dependstrictly on the parameters of the source in terms of boththe number of Schmidt modes and the symmetry of theJSA. An antibunching behavior in the interference pat-tern of four-photon interference is directly connected withthe number of temporal modes (multimodeness) in thesystem and becomes more pronounced with increasingthe Schmidt parameter. Such behaviour can be observedin the case of multiphotom interference only.The number of modes was modified by varying thepump spectral bandwidth and for the first time withusing of a chirped pump pulse. The last case with ar-tificial creating of multimodenes by adding a quadraticspectral phase allows to increase significantly the num-ber of modes without decreasing the pulse energy andmaintaining the same signal-idler spectrum.Also we demonstrated that it is possible to changedrastically the shape of the HOM curves by varying thetransmission and reflection parameters of the beam split-ter. It was shown that with a specific choice of suchparameters the four-photon interference can be similarto the two-photon interference but with larger range ofphotons indistinguishability and better stability to theasymmetry of JSA.Presented results illuminate features of multiphoton in-terference with using a real multimode photon source andopen an avenue for further investigation of multiphotoninterfernece and its implementation into quantum net-works and quantum computing algorithms based on pho-tonic structures.
V. ACKNOWLEDGMENTS
We acknowledge the financial support of the DeutscheForschungsgemeinschaft (DFG) via TRR 142/1, projectC02. P. R. Sh. thanks the state of North Rhine-Westphalia for support by the
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Eberly, Physical ReviewLetters , 5304 (2000).[40] B. Liu, F. Sun, Y. Gong, Y. Huang, G. Guo, and Z. Ou,Optics letters , 1320 (2007). VI. APPENDIXA. Relation between P and the number of modes In this Appendix we show how the peak in P at the zero time delay depends on the number of Schmidt modes ofthe source. At the zero time delay, the expression for P probability can be drastically reduced due to the symmetryof the JSA respect to the main diagonal: P = (cid:82) d ω b d ω c d ˜ ω b d ˜ ω c F ( ω c , ω d ) F (˜ ω c , ˜ ω d ) F ∗ ( ω d , ω c ) F ∗ (˜ ω d , ˜ ω c )2 + 2 (cid:82) d ω b d ω c d ˜ ω b d ˜ ω c F ( ω c , ω d ) F (˜ ω c , ˜ ω d ) F ∗ ( ω d , ˜ ω c ) F ∗ (˜ ω d , ω c ) . (13)The integration in the numerator tends to the unity due to the normalization of the JSA. To calculate the expressionin the denominator it is helpful to perform the Schmidt decomposition of the JSAs: (cid:90) d ω b d ω c d ˜ ω b d ˜ ω c F ( ω c , ω d ) F (˜ ω c , ˜ ω d ) F ∗ ( ω d , ˜ ω c ) F ∗ (˜ ω d , ω c ) = (cid:88) αβγδ (cid:112) Λ α Λ β Λ γ Λ δ (cid:90) d ω b d ω c d ˜ ω b d ˜ ω c u α ( ω c ) u β (˜ ω c ) u ∗ γ ( ω d ) u ∗ δ (˜ ω d ) v α ( ω c ) v β (˜ ω c ) v ∗ γ ( ω d ) v ∗ δ (˜ ω d ) , (14)where parameters Λ n and functions u n ( ω ) and v n ( ω ) are Schmidt eigenvalues and eigenfunctions respectively. Dueto the symmetry of JSA we can assume u ≡ v . With using of the orthonormalization of the Schmidt-mode basis, theintegral in the denominator can be taken and the final P probability at the zero time delay can be obtained: P = 12(1 + k ) , (15)where K = 1 / ( (cid:80) α Λ α ))