Broadband indistinguishability from bright parametric downconversion in a semiconductor waveguide
T. Günthner, B. Pressl, K. Laiho, J. Geßler, S. Höfling, M. Kamp, C. Schneider, G. Weihs
BBroadband indistinguishability from brightparametric downconversion in a semiconductorwaveguide
T. G¨unthner , B. Pressl , K. Laiho , J. Geßler , S.H¨ofling , , M. Kamp , C. Schneider and G. Weihs , Institut f¨ur Experimentalphysik, Universit¨at Innsbruck, Technikerstraße 25,6020 Innsbruck, Austria Technische Physik, Universit¨at W¨urzburg, Am Hubland, 97074 W¨urzburg,Germany School of Physics & Astronomy, University of St Andrews, St Andrews, KY169SS, United Kingdom Institute for Quantum Computing, University of Waterloo, 200 UniversityAvenue W, Waterloo, ON, N2L 3G1, CanadaE-mail: [email protected] and [email protected]
Abstract.
Parametric downconversion (PDC) in semiconductor Bragg-reflection waveguides (BRW) is routinely exploited for photon-pair generationin the telecommunication range. Contrary to many conventional PDC sources,BRWs offer possibilities to create spectrally broadband but nevertheless indis-tinguishable photon pairs in orthogonal polarizations that simultaneously in-corporate high frequency entanglement. We explore the characteristics of co-propagating twin beams created in a type-II ridge BRW. Our PDC source isbright and efficient, which serves as a benchmark of its performance and justi-fies its exploitation for further use in quantum photonics. We then examine thecoalescence of the twin beams and investigate the effect of their inevitable multi-photon contributions on the observed photon bunching. Our results show thatBRWs have a great potential for producing broadband indistinguishable photonpairs as well as multi-photon states.PACS numbers: 42.65.Lm, 42.50.-p, 42.50.Dv, 42.65.Wi
Submitted to:
J. Opt. a r X i v : . [ qu a n t - ph ] J u l
1. Introduction
Versatile quantum light sources are needed for avariety of quantum communication tasks, and thus wewould like to develop them for the telecommunicationwavelengths. Perhaps the best scrutinized processis parametric downconversion (PDC), which producesphotons in pairs, usually denoted as signal and idler.Waveguide realizations have turned PDC sources intoeasy-to-handle and small-scale tools that moreoverprovide higher brightness than their bulk counterparts[1, 2]. Waveguided sources further provide betterintegrability and quantum integrated networks havebeen built both on semiconductor as well as ferro- anddielectric platforms [3–6].Recently, we and others demonstrated PDC inwaveguides being composed of layers of semiconductormaterials that are historically named Bragg-reflectionwaveguides (BRWs) [7–9]. In comparison to ferroelec-tric non-linear optical waveguides the semiconductorstructures benefit from higher nonlinearity and betterintegrability [9]. By embedding the pump laser andthe photon-pair production on the same chip the PDCemission can even be electrically self-pumped [10, 11].BRWs with high signal-idler correlations are suitablefor various quantum optical tasks [9, 12]. Previousexperimental studies include the investigation of thephoton-pair indistinguishability [13] and preparationof polarization entangled states both with co- andcounter-propagating signal and idler schemes [14–16].Furthermore, BRWs offer a lot of flexibility for sourcedesign, which aims at engineering of quantum stateswith desired properties for specific applications [17–22].However, in order to successfully compete withconventional PDC sources, BRWs have to be ableto produce twin beams, in other words signal andidler obeying a strict photon-number correlation, ina bright and efficient manner with a low number ofspurious counts [8, 9]. Still today, their drawbacksare the incompatibility of the utilized spatial modeswith standard single-mode fiber optics, a rather highfacet reflectivity because of the large refractive indexdifference with air, and a high numerical aperture (NA)due to the strong confinement of the spatial modes. Ontop of this, the optical losses both at the pump andthe downconverted wavelengths are significant [9, 23],which limits the useful length of the structures.The phasematching required for the PDC processcan be achieved in semiconductor waveguides byspatial mode matching [24–26] eliminating the need forquasi-phasematching, which is typical for conventionalsources. All in all, the characteristics of signaland idler in their different degrees of freedom aredetermined by the PDC process parameters suchas the strength of the nonlinearity, pump envelopeand the dispersion of the interacting modes in the used geometry. The resulting joint spectrum ofsignal and idler to a large extent dictates for whichquantum optics applications the photonic sourcein question is suitable [27–29]. The state-of-the-art BRW sources provide high entanglement in thespectral degree of freedom [30–32]. Simultaneously,they also offer broadband spectral indistinguishabilityfor signal and idler that are created in orthogonalpolarizations. The former is desired in applicationsrequiring multimode PDC characteristics—or higherdimensional states [33, 34], whereas the latter is abuilding block for many quantum optical networksthat base on photon bunching [35]. The PDC processparameters also govern the photon statistics of the twinbeams, inevitably resulting in higher photon-numbercontributions, which have to be controlled [36].Here, we investigate the characteristics of spec-trally broadband type-II PDC emitted by a BRW ina single-pass configuration. First, we determine theKlyshko efficiency of our BRW source. Thereafter, weutilize correlation functions between signal and idler inorder to investigate the mean photon number in them.We further test the broadband indistinguishability in atwo-photon coalescence experiment. For this purpose,we manipulate the spectral bandwidth of signal andidler by broadband filtering, and via their bunchingat different gains we determine the indistinguishabil-ity governed by the spectral overlap. Our results showthat BRWs are bright and efficient photon sources.
2. Sample design and experiment
Our BRW sample depicted schematically in figure 1(a)is grown on a GaAs substrate by molecular beamepitaxy. The sample having the same structuraldesign as in [9, 15] is made of Al x Ga − x As (0 1) compounds because of their inherent highoptical second order nonlinearity and the sophisticatedfabrication techniques available. Due to its zincblendestructure, Al x Ga − x As has no birefringence and,therefore, in order to achieve phasematching thesample is designed to support different spatial modesthat are the Bragg mode for the pump and thetotal internal reflection (TIR) modes for the twinbeams [38]. The distributed Bragg reflectors (DBRs)embed a multi-layer core that guarantees good spatialoverlap of the pump, signal and idler mode tripletsrequired for an efficient PDC process [39–41]. Finally,the ridge structure is fabricated by electron beamlithography followed by plasma etching. This ensuresmode confinement in two dimensions—in the verticaldirection by the DBRs and in the horizontal directionby the ridge structure.In our experiment as shown in figure 1(b)we employ a picosecond pulsed Ti:Sapphire laser ALMO DM InGaAsAPDscoincidencediscriminatorBRWsample (b) BPF HWP PBSTi:Sapphire ... ... th use every 64 trigger pulse s i gna l & i d l e r pumpmulti-layercore DBR . µ m . m m GaAs substrate (a) Figure 1. (Color online) (a) Investigated BRW sample with a core thickness of 1 . µ m, a length of 2 . . µ m and 3 . µ m, respectively. The depicted amplitude distributions of pump, signal and idler mode were inferred witha commercial-grade simulator eigenmode solver and propagator [37]. (b) Experimental setup for investigating the source efficiencyand brightness as well as the signal and idler coalescence. For abbreviations and more details see text. (76 . . . − and 200 s − , correspondingly. Finally, we employ atime to digital converter to discriminate the singleand coincidence counts. Due to technical limitations,only every 64 th laser pulse gates our InGaAs APDscorresponding to a rate R of about 1 . 19 MHz. 3. Source efficiency and brightness With conventional PDC sources efficient single-modefiber couplings and large mean photon numbers can beachieved at telecommunication wavelengths [34,42–44].Therefore, we start by investigating the performance ofour BRW source by determining its Klyshko efficiency[45] and thereafter evaluate the mean photon numberin the individual twin beams. For this purpose weuse the configuration shown in figure 1(b), in which C o i n c i d e n ce s / S i ng l e s ( % ) (a) signalidler Pump power ( µ W) M ea n pho t on nu m b e r (b) Figure 2. (Color online) (a) The ratio of coincidences to singlesin both signal and idler and (b) mean photon number (cid:104) n (cid:105) in oneof the generated twin beams with respect to the pump power.The vertical errorbars are smaller than the used symbols. the PDC emission is filtered to a spectral width of40 nm to suppress background light from the waveguidebefore the signal and idler beams are deterministicallyseparated at PBS.To eliminate the effect of the accidental coinci-dences produced by the higher photon-number contri-butions created in PDC, we measure the process ef-ficiency with respect to the pump power. In the re-gion of weak pump powers we can extract the Klyshkoefficiency, which is defined only for perfectly photon-number correlated photon-pairs, as η s,i = C/S i,s with C being the coincidence rate and S s,i the single countrates of signal and idler, respectively. Our results in fig-ure 2(a) show the ratio of coincidence counts to singlecounts for both signal and idler detected with a cho-sen photon detection probability of 20 % at our APDs.Thence, we extrapolate Klyshko efficiencies of 6 . . C/A , in which A = S i S s /R corresponds to theaccidental count rate, we can further estimate the meanphoton number (cid:104) n (cid:105) created in one of the twin beamsin a loss-independent manner via C/A ≈ / (cid:104) n (cid:105) + 1[46]. Since our BRW is a highly multimodal PDCsource (see Appendix A), this estimate gives themean photon number in good approximation, beingin the worst case the lower bound. In figure 2(b) weillustrate the obtained mean photon number growinglinearly with respect to the increasing pump power asexpected for weakly excited PDC. Our results showthat mean photon numbers up to 0 . 4. Coalescence of signal and idler For observing the coalescence of signal and idler pho-tons we follow the experiment in [31] and investigatephoton bunching by varying their distinguishability inthe polarization degree of freedom. For this purpose,we detect the coincidences between signal and idlerwhile rotating the HWP in figure 1(b). In case theHWP axes are oriented parallel to the cross-polarizedsignal and idler, they are separated deterministicallyat the PBS. In any other case, signal and idler bunchtogether if they are indistinguishable in all degrees offreedom—not only in polarization but also spatiallyand spectrally.We record the coincidence counts with respect tothe HWP angle at several pump powers. Additionally,we change the photon detection probability of ourAPDs from 20 % to 25 % to increase the count rates.Figure 3(a) and (b) show our results with a 12 nmband-pass filter (BPF) for a low and a high pumppower value, respectively. From this, we can directlyconclude that the higher the pump power the lower isthe visibility of the measured fringes given by V =( C max . − C min . ) / ( C max . + C min . ). We neverthelessachieved a maximum visibility of 0 . / C o i n c i d e n ce s ( / s ) P pump =22 µ W V =0.83(1) (a) fitdata ° )0306090120150 C o i n c i d e n ce s ( / s ) P pump =170 µ W V =0.53(1) (b) fitdata V i s i b ilit y (c)(c) Figure 3. (Color online) Measured coincidence counts asfunction of the HWP angle for pump powers of (a) 22 µ W and(b) 170 µ W for a 12 nm filter bandwidth. (c) The extractedvisibilities decrease with increasing mean photon number. cence, we further examine the visibility in terms ofthe mean photon number, which is also directly pro-vided by the measured data, when signal and idler aredeterministically split at HWP. In figure 3(c) we de-pict the measured visibilities with respect to the esti-mated mean photon numbers for both 12 nm and 40 nmBPFs. In both cases the visibility clearly decreaseswith increasing mean photon number. However, notonly the increasing multi-photon contributions but alsothe spectral mismatch between signal and idler affectthe visibility in the coalescence experiment. From themeasured visibilities we can infer the spectral overlapof the downconverted photon pairs via (see AppendixB) V ≈ O − O + 4 (cid:104) n (cid:105) . (1)By fitting our results in figure 3(c) against (1) weretrieve for the spectral overlap with 12 nm and 40 nmfilter bandwidths the values of 95 . . joint spectral distribution of signal and idler governsthe spectral overlap (see Appendix A). Our resultsare in good accordance with numerical simulations,which predict spectral overlaps of about 98 % and 83 %,respectively for the two filters. 5. Conclusion Integratable and easy-to-handle sources of parametricdownconversion are highly desired in many quantumoptics applications. Bragg-reflection waveguides basedon semiconductor compounds provide a platform thatcan meet these demands. Our BRW sample showsa good performance, and we can reach Klyshkoefficiencies up to a few percent with avalanchephotodetection, regardless of the non-standard modeprofile of the signal and idler beams. Moreover, oursource provides a high brightness and is capable ofproducing higher photon numbers as is desired formultiphoton production. We further examined thecoalescence between the twin beams filtered to a fewtens of nanometers bandwidth in order to assess theirindistinguishability. The visibility of the measuredfringes is diminished by the multi-photon contributionsof signal and idler, but we can nevertheless extract ahigh degree of indistinguishability, which is quantifiedby their spectral overlap. We extended our modelto take into account both these process parametersand showed that our results are in good agreementwith numerical simulations. Thus, being characteristicfor BRWs, our source provides signal and idler inorthogonal polarizations that are over a broad spectralband highly indistinguishable in frequency. We believeour work gives a detailed insight of the PDC processin our BRW both in the spectral and photon-numberdegrees of freedom. This will become important whenoptimizing and adapting BRW sources into quantumoptical networks. Acknowledgements We thank Matthias Covi for assistance with theexperimental setup. This work was supported in partby the ERC, project EnSeNa (257531) and by theFWF through project no. I-2065-N27. Appendix A. Modelling the joint spectralproperties of signal and idler In this appendix we investigate the joint spectralamplitude (JSA) of signal and idler and numericallyestimate the spectral overlap O , which determines theirindistinguishability in the low gain regime. Following[27,28,47] we find that the joint spectral characteristicsin a collinear single-pass PDC source are given by f ( ω s , ω i ) = 1 N α ( ω s + ω i ) φ ( ω s , ω i ), (A.1)in which N accounts for the normalization of the JSAvia (cid:82) dω s dω i | f ( ω s , ω i ) | = 1, α ( ω p = ω s + ω i ) describesthe pump spectrum in terms of the frequencies ω µ ( µ = p, s, i ) for pump, signal and idler, respectively,and φ ( ω s , ω i ) is the phasematching (PM) function. Ina Gaussian approximation we can describe the pumpamplitude as α ( ω s + ω i ) = e − σ p ( ω s + ω i ) (A.2)with σ p being the bandwidth of the pump. We use asimple PDC model for uniform waveguides [29,40] withconstant-valued non-linearity over the whole length ofthe waveguide and assume that the overlap of spatialmodes effectively affects only its strength. Thus, in thesingle-pass configuration we can write the PM functionas [48] φ ( ω s , ω i ) = sinc (cid:18) L k ( ω s , ω i ) (cid:19) e i L ∆ k ( ω s ,ω i ) ≈ e − γ L ∆ k ( ω s ,ω i ) e i L ∆ k ( ω s ,ω i ) , (A.3)in the final form of which we have used a Gaussianapproximation for the sinc-function. In (A.3) L denotes the BRW length, ∆ k ( ω s , ω i ) = k p ( ω s + ω i ) − k s ( ω s ) − k i ( ω i ) describes the phase mismatchin terms of k µ ( ω µ ) = n µ ω µ /c with n µ being theeffective refractive index and c the speed of light, while γ ≈ . 193 adjusts the width of the approximated PMfunction. Performing a Taylor expansion of ∆ k to thesecond order at a phasematched point ω p = ω s + ω i we can write the phase mismatch as∆ k ( ω s , ω i ) ≈ κ s ν s + κ i ν i +Λ s ν s +Λ i ν i − Λ p ν s ν i , (A.4)in which the detunings are defined as ν µ = ω µ − ω µ .In (A.4) κ µ = k (cid:48) µ ( ω µ ) − k (cid:48) p ( ω p ) = 1 /v g ( µ ) − /v g ( p ) is determined by the group velocity mismatch of thedownconverted photons and the pump photon, whileΛ s,i = k (cid:48)(cid:48) s,i ( ω s,i ) − k (cid:48)(cid:48) p ( ω p ) and Λ p = k (cid:48)(cid:48) p ( ω p ) arerelated to the group velocity dispersions.For our simulation we substitute (A.2)-(A.4) into(A.1) and evaluate the JSA. From a commerciallyavailable solver (Mode Solutions [37]) we obtain forthe PDC process in our BRW (in figure 1(a)) thedispersion properties listed in table A1. In figure A1we show the joint spectral intensity (JSI), | f ( ω s , ω i ) | ,as a function of the signal and idler frequencies f s,i = ω s,i / π . We evaluated JSI at the extracteddegeneracy point of f s = f i =193 . . . 25 nm broadspectrum. The simulated degeneracy point is very closeto the measured one found at 1544 . Table A1. Parameters of the pump (p), signal (s) and idler (i)mode extracted from numerical simulations [37]. v g ( µ ) ( µ m/ps) κ µ (10 − ps/ µ m) Λ µ (10 − ps / µ m)p s i s i p s i74.0 90.1 90.4 -2.40 -2.44 5.74 -2.16 -2.17Due to small difference in the group velocitiesof the signal and idler photons in the vicinity ofthe degeneracy point, the tilt of the PM function θ ≈ arctan( κ s /κ i ) deviates from that of perfect anti-correlation by about 0 . ◦ . Altogether, our simulatedJSI is slightly asymmetric around the degeneracypoint, leading to different spectral properties ofsignal and idler centered at 1567 nm and 1535 nm,respectively. Both marginal spectra are originallyapproximately 90 nm wide. Thus, they are two ordersof magnitude broader than the linewidth of the JSI atits degeneracy point, being about 0 . O = (cid:82) (cid:82) dω s dω i f ( ω s , ω i ) f ∗ ( ω i , ω s ) and depends remarkablyon the group velocity mismatch. As in our casesignal and idler travel with slightly different groupvelocities, their wavepackets are temporally shifted,which is evident from the phase term of the JSA. Forthe unfiltered JSA we determine a spectral overlapof only about 26 %, while 76 % could be achieved ifthe temporal mismatch was corrected. We restrictthe influence of the group velocity mismatch byspectral filtering close to the JSA degeneracy pointand, therefore, we expect a spectral overlap of about98 % and 83 % when filtering with a 12 nm (1 . . Frequency signal (THz) F r e qu e n c y i d l e r( T H z ) | f (f ,f )|² s i W a v e l e ng t h i d l e r ( µ m ) 180 190 200 210180190200210 Intensity90 nm n m . . . - . . + . . n m D e t un i ng ( n m ) I n t e n s it y I n t e n s it y Figure A1. (Color online) A contour plot of JSI with respectto the signal and idler frequencies/wavelengths. The insetillustrates the width of the JSI in terms of detuning from thedegeneracy point. The red and green solid curve represent themarginal spectra. The diagonal (blue solid line) and the anti-diagonal (brown dashed line) as well as the dotted lines provideguides for the eye. Appendix B. Quantum interferenceexperiment with a twin beam state We utilize the description of parametric downconver-sion as multimode squeezer in order to estimate theeffect of the higher photon-number contributions on aquantum interference between signal and idler [46, 49].We start by considering the configuration in figure B1,in which the bunching takes place at a symmetric beamsplitter having a transmission of T = 1 / a and b , and output arms, c and d , in time t asˆ c ( t ) = 1 / √ (cid:104) √ η ˆ a ( t ) + (cid:112) − η ˆ v ( t )+ √ η ˆ b ( t ) + (cid:112) − η ˆ v ( t ) (cid:105) (B.1)ˆ d ( t ) = 1 / √ (cid:104) √ η ˆ a ( t ) + (cid:112) − η ˆ v ( t ) − √ η ˆ b ( t ) − (cid:112) − η ˆ v ( t ) (cid:105) , (B.2)which expresses the photon annihilators ˆ c ( t ς ) and ˆ d ( t ς )( ς = 1 , 2) at the beam splitter output ports in termsof those of the inputs ˆ a ( t ς ) and ˆ b ( t ς ) and the vacuummodes ˆ v , ( t ς ). The detection efficiencies correspondto the transmission coefficients η , . Further, we a b cd T & v v η η Figure B1. Model of bunching experiment with a twin beamstate. The twin beams send to the input arms a and b aredegraded due to losses at the beam splitters with transmissions η and η , to which the optical vacuum modes v and v arecoupled, respectively. Bunching occurs at the beam splitter withtransmission T and coincidences are counted between the twooutput arms c and d . For more details see text. assume that our detectors have a spectrally broadresponse and their detection windows are much longerthan the duration of the generated pulsed wavepackets.Thus, the coincidence rate can be evaluated via [27] R ∝ (cid:90) dt (cid:90) dt (cid:104) ˆ c † ( t )ˆ d † ( t )ˆ d ( t )ˆ c ( t ) (cid:105) , (B.3)the integrand in which describes the probability ofa coincidence at times t and t between the twodetectors.Our goal is to rewrite (B.3) in terms of thebeam splitter input operators and then evaluate theexpectation values regarding the desired input state.By plugging the beam splitter transformations in (B.1)and (B.2) together with their conjugates to (B.3) andutilizing the Fourier transformations given by ˆ a ( t ) = √ π (cid:82) dω ˆ a ( ω ) e − iωt and ˆ b ( t ) = √ π (cid:82) dω ˆ b ( ω ) e − iωt with ω being the optical angular frequency, only fewterms survive and we can write down the coincidencerate in the form R ∝ (cid:10) (cid:90) dω (cid:90) d ˜ ω (cid:2) η ˆ a † ( ω )ˆ a † (˜ ω )ˆ a (˜ ω )ˆ a ( ω ) − η η ˆ a † ( ω )ˆ a † (˜ ω )ˆ b (˜ ω )ˆ b ( ω ) − η η ˆ a † ( ω )ˆ b † (˜ ω )ˆ a (˜ ω )ˆ b ( ω )+2 η η ˆ a † ( ω )ˆ b † (˜ ω )ˆ b (˜ ω )ˆ a ( ω ) − η η ˆ b † ( ω )ˆ b † (˜ ω )ˆ a (˜ ω )ˆ a ( ω )+ η ˆ b † ( ω )ˆ b † (˜ ω )ˆ b (˜ ω )ˆ b ( ω ) (cid:3) (cid:11) (B.4)in which we have carried out the integration overtime by extending the limits to infinity ( δ ( ω ) =1 / (2 π ) (cid:82) ∞−∞ dt µ e it µ ω ).Now, we re-express (B.4) in terms of broadbanddetection modes that correspond to those of ourdownconverter. The multimode squeezed state sent to the input arms a and b of the beam splitter is definedvia the unitary squeezing operator ˆ S a,b as [46] | Ψ (cid:105) = ˆ S a,b | (cid:105) = e (cid:80) k r k ˆ A † k ˆ B † k − h.c. | (cid:105) , (B.5)in which the real valued squeezing strength r k = B λ k is related to the gain of the PDC process B andto the Schmidt modes λ k ( (cid:80) k λ k = 1) of the jointspectral correlation function of signal and idler givenby f ( ω s , ω i ) = (cid:80) k λ k ϕ k ( ω s ) φ k ( ω i ). With the help ofthe two sets of orthonormal basis functions { ϕ k } and { φ k } for signal and idler, respectively, we define thek-th mode sent to the input arm a asˆ A † k = (cid:90) dω ϕ k ( ω ) ˆ a † ( ω ) , (B.6)for which the following relations hold: (cid:90) dω ϕ (cid:63)k ( ω ) ϕ k (cid:48) ( ω ) = (cid:26) k = k (cid:48) , (cid:88) k ϕ k ( ω ) ϕ (cid:63)k (˜ ω ) = δ ( ω − ˜ ω ) . (B.7)Similarly, the k-th mode sent to the input arm b canbe written as ˆ B † k = (cid:90) dω φ k ( ω ) ˆ b † ( ω ) , (B.8)the basis functions in which obey conditions similar tothose in (B.7). The broadband mode transformationsof the k-th input modes can be presented in the form[46] ˆ S † a,b ˆ A k ˆ S a,b = cosh( r k ) ˆ A k + sinh( r k ) ˆ B † k (B.9)ˆ S † a,b ˆ B k ˆ S a,b = cosh( r k ) ˆ B k + sinh( r k ) ˆ A † k . (B.10)In the following we consider only the case of weaksqueezing and approximate sinh( r k ) ≈ r k = B λ k and cosh( r k ) ≈ 1. Further, we estimate the meanphoton number in the both input arms as (cid:104) n (cid:105) = (cid:80) k sinh ( r k ) ≈ B .In order to transform (B.4) to the broadband-mode picture, we require the identitiesˆ a † ( ω ) = (cid:88) k ˆ A † k ϕ (cid:63)k ( ω ) andˆ b † ( ω ) = (cid:88) k ˆ B † k φ (cid:63)k ( ω ) . (B.11)Thence, we re-express the coincidence rate in (B.4) as R ∝ (cid:28)(cid:20) η (cid:88) k,n ˆ A † n ˆ A † k ˆ A k ˆ A n − η η (cid:88) k,n,l,m ˆ A † n ˆ A † k ˆ B l ˆ B m (cid:90) dωϕ (cid:63)n ( ω ) φ m ( ω ) (cid:90) d ˜ ωϕ (cid:63)k (˜ ω ) φ l (˜ ω ) − η η (cid:88) k,n,l,m ˆ A † n ˆ B † k ˆ A l ˆ B m (cid:90) dωϕ (cid:63)n ( ω ) φ m ( ω ) (cid:90) d ˜ ωφ (cid:63)k (˜ ω ) ϕ l (˜ ω )+ 2 η η (cid:88) k,n ˆ A † n ˆ B † k ˆ B k ˆ A n − η η (cid:88) k,n,l,m ˆ B † n ˆ B † k ˆ A k ˆ A n (cid:90) dωφ (cid:63)n ( ω ) ϕ m ( ω ) (cid:90) d ˜ ωφ (cid:63)k (˜ ω ) ϕ l (˜ ω )+ η (cid:88) k,n ˆ B † n ˆ B † k ˆ B k ˆ B n (cid:21)(cid:29) . (B.12)We directly recognize that several terms in (B.12)correspond to Glauber correlation functions G ( w, υ ) = (cid:104) : ( (cid:80) q ˆ A † q ˆ A q ) w ( (cid:80) q (cid:48) ˆ B † q (cid:48) ˆ B q (cid:48) ) υ : (cid:105) with indices w and υ describing the order of the correlation for the twinbeam modes a and b , respectively [46]. Thence, whendisregarding the losses, the expectation values in thefirst and last terms deliver G (2 , 0) = G (0 , 2) = (cid:104) n (cid:105) [1+ K ], and in the fourth term G (1 , 1) = (cid:104) n (cid:105) [1 + K ] + (cid:104) n (cid:105) , where K corresponds to the effective number ofexcited modes ( K = 1 / (cid:80) k λ k ) [46]. The rest ofthe mean values can be evaluated by plugging in thetransformations from (B.9) and (B.10) together withtheir hermitian conjugates. While the second and fifthterms vanish, the third term delivers (cid:10) (cid:88) k,n,l,m ˆ A † n ˆ B † k ˆ A l ˆ B m (cid:90) dωϕ (cid:63)n ( ω ) φ m ( ω ) (cid:90) d ˜ ωφ (cid:63)k (˜ ω ) ϕ l (˜ ω ) (cid:11) = (cid:104) n (cid:105) O + (cid:104) n (cid:105) A , (B.13)in which O = (cid:90) dω (cid:90) d ˜ ω f (cid:63) ( ω, ˜ ω ) f (˜ ω, ω ) (B.14)= (cid:88) k,n λ n λ k (cid:90) dωϕ (cid:63)n ( ω ) φ k ( ω ) (cid:90) d ˜ ωφ (cid:63)n (˜ ω ) ϕ k (˜ ω )describes the spectral overlap between signal and idlerand A = (cid:90) dω (cid:90) d ˜ ω g s ( ω, ˜ ω ) g i (˜ ω, ω ) (B.15)= (cid:88) k,n λ n λ k (cid:90) dωϕ (cid:63)n ( ω ) φ k ( ω ) (cid:90) d ˜ ωφ (cid:63)k (˜ ω ) ϕ n (˜ ω )determines the overlap of signal and idler spectral densities that are given by g s ( ω, ˜ ω ) = (cid:90) dω i f (cid:63) ( ω, ω i ) f (˜ ω, ω i ) and (B.16) g i (˜ ω, ω ) = (cid:90) dω s f (cid:63) ( ω s , ˜ ω ) f ( ω s , ω ) . (B.17)We note that if the spectral densities of signal and idlerare equal this term will end up giving the purity of thephoton wavepacket 1 /K .Finally, we determine an expression for thevisibility of our quantum interference experiment inSec. 4. When signal and idler are expected to bunch,we can estimate the rate of the coincidences accordingto (B.12) as R min. ∝ (cid:104) n (cid:105) (1 + 1 K )( η + η )+ 12 (cid:16) (cid:104) n (cid:105) + (cid:104) n (cid:105) (1 + 1 K ) (cid:17) η η − (cid:16) (cid:104) n (cid:105) O + (cid:104) n (cid:105) A (cid:17) η η . (B.18)This rate is compared with the one obtained when thesignal and idler beams are separated deterministically.By using the same model as above but assuming abeam splitter with T = 1 in figure B1, we gain R max. ∝ (cid:18) (cid:104) n (cid:105) + (cid:104) n (cid:105) (cid:16) K (cid:17)(cid:19) η η . (B.19)The visibility is then given by V = R max. − R min. 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