Characteristics of parametric downconversion from a Bragg-reflection waveguide

T. Gunthner,1, ∗ B. Pressl,1 K. Laiho,1 J. Geßler,2 S. Hofling,2, 3 M. Kamp,2 C. Schneider,2 and G. Weihs1, 4

1Institut fur Experimentalphysik, Universitat Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria2Technische Physik, Universitat Wurzburg, Am Hubland, 97074 Wurzburg, Germany

3School of Physics & Astronomy, University of St Andrews, St Andrews, KY16 9SS, United Kingdom4Institute for Quantum Computing, University of Waterloo,

200 University Avenue W, Waterloo, ON, N2L 3G1, Canada(Dated: December 8, 2014)

We investigate the properties of co-propagating twin beams created by parametric downconversionin a ridge Bragg-reflection waveguide. Our source is bright and efficient although the tightly confinedwaveguide modes are not perfectly compatible with standard single-mode fibre optics. We observethe coalesce of the twin beams and investigate the effect of the multi-photon contributions on theobserved photon bunching. Our source shows a great potential for producing indistinguishablephoton pairs as well as multi-photon states.

I. INTRODUCTION

Versatile quantum light sources are needed for a varietyof quantum communication protocols and thus their de-velopment is pursued by many groups. Perhaps the bestresearched process is parametric downconversion (PDC),which produces photons in pairs, usually denoted as sig-nal and idler. Waveguide realizations have turned PDCsources into easy-to-handle and small-scale tools thatmoreover provide higher brightness than their bulk coun-terparts [1]. Waveguided sources further provide betterintegrability and quantum integrated networks have beenbuilt both on semiconductor and dielectric platforms [2–4].

Recently, the PDC process has been demonstratedin semiconductor waveguides being composed of layersof semiconductor materials that are historically namedBragg-reflection waveguides (BRW) [5, 6]. In comparisonto dielectric non-linear optical waveguides the semicon-ductor structures benefit from higher nonlinearity andbetter integrability [7]. By embedding the pump laserand the photon-pair production on the same chip thePDC emission can even be electrically self pumped [8].BRWs with high signal-idler correlations are suitable formany different quantum optical tasks [7]. Previous ex-perimental studies include demonstration of indistiguish-able photon pairs [9] and preparation of polarization en-tangled states both with co- and counter-propagating sig-nal and idler schemes [10, 11]. Furthermore, BRWs areviable for source design, which aims at engineering ofquantum states with desired properties for specific appli-cations [12, 13].

However, in order to successfully compete with conven-tional PDC sources, BRWs have to be able to producetwin beams, in other words signal and idler obeying astrict photon-number correlation, in a bright and effi-cient manner with a low number of spurious counts. Stilltoday, their drawbacks are the incompatibility of the uti-

∗ thomas.guenthner@uibk.ac.at

lized spatial modes with standard single-mode fiber op-tics, a rather high facet reflectivity because of the largerefractive index difference with air, and a high numericalaperture (NA) due to strong confinement of the spatialmodes. On top of this, the optical losses both at thepump and the downconverted wavelengths are significant[7, 14], which limits the useful length of the structures.

Fortunately, some of these properties can also beexploited—the phasematching required for the PDC pro-cess can be achieved in semiconductor waveguides byspatial mode matching eliminating the need for quasi-phasematching, which is typical for conventional sources[15]. In general, the characteristics of signal and idlerin their different degrees of freedom are determined bythe PDC process parameters such as the strength of thenonlinearity, pump envelope and the dispersion relationin the used geometry. The resulting joint spectrum ofsignal and idler to a large extend dictates for which quan-tum optics applications the photonic source in questionis suitable [16–18]. Additionally, the photon statistics ofthe individual marginal beams that is signal or idler areaffected by the PDC process parameters, therefore, theinevitable higher photon-number contributions have tobe controlled [19].

Here, we investigate the characteristics of spectrallyhighly multimodal PDC emitted by a BRW in a single-pass configuration. Firstly, we determine the Klyshko ef-ficiency of our BRW source. Thereafter, we utilize corre-lation functions between signal and idler in order to inves-tigate the mean photon number in the marginal beams.We further modify the spectral properties by filtering andvia a two-photon coalescence experiment determine theindistinguishability of signal and idler, which is governedby their spectral overlap. Our results show that BRWsare excellent candidates for becoming bright and efficientphotonic sources.

II. SAMPLE DESIGN AND EXPERIMENT

Our Bragg-reflection waveguide (BRW) sample (de-picted schematically in Fig. 1(a)) is grown on a GaAs

arX

iv:1

412.

2082

v1 [

quan

t-ph

] 5

Dec

201

4

2

ALMO DM

InGaAsAPDs

coincidencediscriminator

BRWsample

(b)

BPF

HWP PBS

Ti:Sapphire

12152 cps

15522 cps

... ...

thuse every 64 trigger pulse

signal & idler

pump

multi-layercore DBR

5.0µm2.0mm

3.4µm

GaAs substrate

(a)

1.2µm

FIG. 1. (Color online) (a) Investigated BRW sample with a core thickness of 1.2 µm, a length of 2.0 mm and a ridge widthand height of 5.0 µm and 3.4 µm, respectively. A commercial-grade simulator eigenmode solver and propagator was used toperform the calculations to find the depicted amplitude distributions of pump, signal and idler modes [20]. The modes areconfined by distributed Bragg-reflectors (DBRs). (b) Setup used to investigate our sample. MO, microscope objective; BRW,Bragg-reflection waveguide; AL, aspheric lens; DM, dichroic mirror; BPF, band-pass filter; PBS, polarizing beam splitter; HWP,half-wave plate; InGaAs APDs, indium gallium arsenide avalanche photo-diodes. For more detais see text.

substrate by molecular beam epitaxy. The sample ismade of AlxGa1−xAs (0 < x < 1) compounds because oftheir inherent high optical second order nonlinearity andthe sophisticated fabrication techniques available. Dueto its zinc blende structure, AlxGa1−xAs has no bire-fringence and, therefore, in order to achieve phasematch-ing the sample is designed to support different spatialmodes that are the Bragg mode for the pump and thetotal internal reflection modes for the twin beams [21].The distributed Bragg reflectors (DBRs) embed a multi-layer core that guarantees good spatial overlap of thepump, signal and idler mode triplets required for an ef-ficient PDC process [22–24]. Finally, the ridge structureis fabricated by electron beam lithography followed byplasma etching. This ensures mode confinement in twodimensions—in the vertical direction by the DBRs andin the horizontal direction by the ridge structure.

In our experiment as shown in Fig. 1(b) we employ apicosecond pulsed Ti:Sapphire laser (76.2 MHz repetitionrate, 772 nm central wavelength) as a pump for the PDCprocess. After power, polarization and spatial mode con-trol, we focus the pump on to the front facet of our BRWwith a 100x microscope objective (MO), which allowsreasonably efficient coupling into the Bragg mode. Atits output facet a high numerical aperture (NA) asphericlens (AL) collects the PDC emission from our BRW, forwhich we numerically determined the NAs of approxi-mately 0.2 and 0.5 in the horizontal and vertical direc-tion, respectively [20]. After removing the residual pumpbeam with a dichroic mirror (DM), we use a band-passfilter (BPF) with either a 12 nm or 40 nm bandwidth tospectrally limit the marginal beams. With an additionalhalf-wave plate (HWP) we change the polarization di-rection of signal and idler, as necessary for the signaland idler coalescence experiment. After separating theorthogonal polarizations using a polarizing beam split-ter (PBS), we couple the marginal beams to single-modefibers for the detection with two commercial time-gatedInGaAs avalanche photo-diodes (APDs) with a variablephoton detection probability up to 25 %. Finally, we em-ploy a time to digital converter to discriminate the sin-

gle and coincidence counts. Due to technical limitations,every 64th laser pulse triggers our InGaAs APDs corre-sponding to a rate R of about 1.19 MHz.

III. SOURCE EFFICIENCY AND BRIGHTNESS

0 100 200 300 400

0.0

2.0

4.0

8.0

10C

oinc

iden

ces/

Sing

les (

%)

6.05.6

(a)

signalidler

0 100 200 300 400Pump power (µW)

0.00.10.20.30.40.5

Mea

n ph

oton

num

ber

(b)

FIG. 2. (Color online) (a) The ratio of coincidences to singlesin both signal and idler and (b) mean photon number 〈n〉 in asingle marginal with respect to the pump power. The verticalerrorbars are smaller than the used symbols.

In conventional dielectric waveguides efficient cou-pling of signal and idler to single-mode fibers has beenachieved at telecommunication wavelengths [25, 26] andalso large mean photon-numbers have been demonstrated[27]. Therefore, we start by investigating the perfor-mance of our BRW source by determining its Klyshkoefficiency [28] and thereafter evaluate the mean photon-number in the marginal beams. For this purpose we usea configuration (Fig. 1(b)), in which the PDC emission

3

is filtered to a spectral width of 40 nm to suppress back-ground light from the waveguide before separating thesignal and idler beams.

To eliminate the effect of the accidental coincidencesproduced by the higher photon-number contributionscreated in the PDC process, we measure the efficiencywith respect to the pump power. In the region ofweak pump powers we can extract the Klyshko efficiency,which is defined only for perfectly photon-number corre-lated photon-pairs, as ηs,i = C/Si,s with C being thecoincidence rate and Ss,i the single count rates of signaland idler, respectively. Our results in Fig. 2(a) show theratio of coincidence counts to single counts for both signaland idler detected with a chosen photon detection prob-ability of 20 % at our APDs. Via extrapolation to zeropump power Klyshko efficiencies of 6.0(1) % and 5.6(2) %are found for signal and idler, respectively. This indicatesthat despite of the high NA and the complex spatial modestructure, the PDC emission can be efficiently collectedwith standard optical components.

By measuring the signal and idler cross-correlation wecan further estimate the mean photon number 〈n〉 ofthe marginal beams in a loss-independent manner viaC/A ≈ 1/ 〈n〉 + 1, in which A = SiSs/R correspondsto the accidental count rate [29]. Since our BRW is ahighly multimodal PDC source (see Appendix A), thisestimation gives the mean photon number in good ap-proximation, being in the worst case the lower bound. InFig. 2(b) we illustrate the mean photon number obtainedfor a single marginal beam with respect to the increasingpump power. Our results show that mean photon num-bers up to 0.5 can be achieved with moderate pump pow-ers. This further indicates that the PDC multi-photoncontributions have to be taken into account especially inthe photon coalescence experiment we investigate next.

IV. COALESCENCE OF SIGNAL AND IDLER

For observing the coalescence of signal and idler pho-tons we follow the experiment in [30] and investigate pho-ton bunching by varying their distinguishability in thepolarization degree of freedom. For this purpose, we de-tect the coincidences between signal and idler while ro-tating the HWP in Fig. 1(b). In case the HWP axes areoriented parallel to the cross-polarized signal and idler,the marginal beams are separated deterministically atthe PBS. Otherwise, signal and idler bunch together ifthey are indistinguishable in all degrees of freedom—notonly in polarization but also spatially and spectrally.

We record the coincidence counts with respect to theHWP angle at several pump powers. Additionally, wechange the photon detection probability of our APDsfrom 20 % to 25 % to increase the count rates. In Fig. 3(a)and (b) we present our results with a 12 nm band-passfilter (BPF) for a low and a high pump power value, re-spectively. From this, we can directly conclude that thehigher the pump power the lower is the visibility V of the

0.0 22.5 45.0 67.5 90.0HWP angle (°)

0

6

12

18

Coi

ncid

ence

s (1/

s) Ppump=22µW

V=0.83(1)

(a)fitdata

0.0 22.5 45.0 67.5 90.0HWP angle (°)

0

30

60

90

120

150

Coi

ncid

ence

s (1/

s) Ppump=170µW

V=0.53(1)

(b)fitdata

0.0 0.1 0.2 0.3 0.4 0.5Mean photon number

0.4

0.6

0.8

1.0

Vis

ibili

ty

(c)(c)12nm BPF40nm BPF

FIG. 3. (Color online) Measured coincidence counts as func-tion of the HWP angle for pump powers of (a) 22 µW and(b) 170 µW for a 12 nm filter bandwidth. (c) The extractedvisibilities decrease with increasing mean photon number.

measured fringes given by V = (Cmax − Cmin)/(Cmax +Cmin). We nevertheless achieved a maximum visibility of0.83(1), which lies significantly above the classical limitof 1/3 for a completely distinguishable photon pair anddelivers us a measure of the signal and idler indistin-guishability. In order to understand the effect of themulti-photon contributions to the signal and idler coa-lescence, we further examine the visibility in terms ofthe mean photon number in the marginal beams, whichis also provided by the measured data. In Fig. 3(c) wedepict the measured visibilities with respect to the esti-mated mean photon number in the marginal beams forboth 12 nm and 40 nm BPFs. In both cases the visibilityclearly decreases with increasing mean photon number.However, not only the increasing multi-photon contri-butions but also the spectral overlap of signal and idleraffect the visibility in the coalescence experiment. Fromour results for the visibility we can infer the spectral over-lap O of the downconverted photon pairs that is given by(see Appendix B)

V =1 +O

3−O + 4〈n〉. (1)

4

By fitting our results in Fig. 3(c) against Eq. (1) we canretrieve spectral overlap values of 95.0(6) % and 81.6(9) %for 12 nm and 40 nm filter bandwidth, respectively. Thelarge difference in the spectral overlaps with the two in-vestigated filters are caused by the the different group ve-locities of the signal and idler wavepackets due to whichsignal and idler pulses are temporally shifted with respectto each other. Theoretically the joint spectral distribu-tion of signal and idler govern the spectral overlap (seeAppendix A). Our results are in good accordance withnumerical simulations, which predict spectral overlaps of98.0 % and 83.8 %, respectively.

V. CONCLUSION

Integratable and easy-to-handle sources of parametricdownconversion are highly desired in many quantum op-tics applications. Bragg-reflection waveguides based onsemiconductor compounds provide a platform that canmeet these demands. We investigated the propertiesof parametric downconversion from a BRW sample andshowed that Klyshko efficiencies on the order of a few per-cent can be achieved for signal and idler. Moreover, oursource is bright and capable of producing higher photonnumbers as desired for multiphoton production. Finally,we examined the coalescence between signal and idler inorder to assess their indistinguishability. Although thevisibility of the measured fringes is diminished by themulti-photon contributions of signal and idler, the cre-ated photon pairs show a high degree of indistinguishabil-ity that can be quantitfied with the spectral overlap. Weextended our model to take into account both these pro-cess parameters and showed that our results are in goodagreement with numerical simulations. We believe ourwork gives a detailed insight of the PDC process in ourBRW both in the spectral and photon-number degreesof freedom. This will become important when optimiz-ing and adapting this photonic source in to a quantumoptical network.

ACKNOWLEDGEMENTS

We thank Matthias Covi for assistance with the exper-imental setup. This work was supported in part by theERC, project EnSeNa (257531) and by the FWF throughproject nos. P-22979-N16 and I-2065-N27 (DACH).

Appendix A: Modelling the joint spectral propertiesof signal and idler

In this appendix we investigate the joint spectral am-plitude (JSA) of signal and idler and numerically es-timate the spectral overlap O, which determines theirindistinguishability in the low gain regime. Following

[16, 17, 31] we find that the joint spectral characteristicsin a collinear single-pass PDC source are given by

f(ωs, ωi) =1

Nα(ωs + ωi)φ(ωs, ωi), (A1)

in which N accounts for the normalization∫dωsdωi|f(ωs, ωi)|2 = 1, α(ωp = ωs + ωi) describes

the pump spectrum in terms of the frequencies ωµ(µ = p, s, i) for pump, signal and idler, respectively,and φ(ωs, ωi) is the phasematching (PM) function. Ina Gaussian approximation we can describe the pumpamplitude as

α(ωs + ωi) = e1σp

(ωs+ωi)2

(A2)

with σp describing the bandwidth of the pump. We use asimple PDC model for uniform waveguides [18, 24] withconstant-valued non-linearity over the whole length ofthe waveguide and the overlap of spatial modes effec-tively affects only its strength. Thus, in the single-passconfiguration we can write the PM function as [32]

φ(ωs, ωi) = sinc

(L

2∆k(ωs, ωi)

)eiL2 ∆k(ωs,ωi)

≈ e−γ L2

4 ∆k2(ωs,ωi)eiL2 ∆k(ωs,ωi), (A3)

in the final form of which we have used a Gaussian ap-proximation for the sinc-function. In Eq. (A3) L de-notes the BRW length, ∆k(ωs, ωi) = kp(ωs + ωi) −ks(ωs)−ki(ωi) describes the phase mismatch in terms ofkµ(ωµ) = nµωµ/c with nµ being the effective refractiveindex and c the speed of light, while γ ≈ 0.193 adjuststhe width of the approximated PM function. Performinga Taylor series of ∆k to the second order at a chosenpoint ω0

p = ω0s +ω0

i we can write the phase mismatch as

∆k(ωs, ωi) ≈ κsνs + κiνi + Λsν2s + Λiν

2i −Λpνsνi, (A4)

in which the detunings are defined as νµ = ωµ − ω0µ.

In Eq. (A4) κµ = k′µ(ω0µ) − k′p(ω0

p) = 1/vg(µ) − 1/vg(p)is determined by the group velocity mismatch of thedownconverted photons and the pump photon, whileΛs,i = 1

2k′′s,i(ω

0s,i) − 1

2k′′p (ω0

p) and Λp = k′′p (ω0p) are re-

lated to the group velocity dispersions.

TABLE I. Parameters of the pump (p), signal (s) and idler(i) mode extracted from numerical simulations [20].

vg(µ) (µm/ps) κµ (10−3ps/µm) Λµ (10−6ps2/µm)p s i s i p s i

74.0 90.1 90.4 -2.40 -2.44 5.74 -2.16 -2.17

For our simulation we substitute Eqs. (A2)-(A4) intoEq. (A1) and evaluate the JSA. From a commerciallyavailable solver (Mode Solutions) we obtain the parame-ters of the PDC process in our BRW (Fig. 1(a)) listed inTab. I. Together with the pump spectrum and the PM

5

x =

(a) (b) (c)

185 190 195 200

185

190

195

200

Frequency signal (THz)

Fre

quen

cyid

ler

(TH

z)

α(f +f )s i

185 190 195 200

185

190

195

200

Frequency signal (THz)

Fre

quen

cyid

ler

(TH

z)

|ϕ(f ,f )|s i

185 190 195 200

185

190

195

200

Frequency signal (THz)

Fre

quen

cyid

ler

(TH

z)

| f (f ,f )|s i

1.581.62 1.54 1.50

1.58

1.62

1.54

1.50

Wavelength signal (µm)W

avelength idler (µ

m)

FIG. 4. (Color online) (a) Pump spectrum with a linewidth of 0.176 THz (≡ 0.35 nm), (b) PM function and (c) JSA versusthe frequency of signal and idler, respectively. A commercial-grade simulator eigenmode solver and propagator was used tosimulate the dispersion of pump, signal and idler in our BRW sample [20]. The solid line and the dashed line represent thediagonal and anti-diagonal, respectively.

function in Figs 4(a) and (b), we show in Fig. 4(c) theamplitude of the simulated JSA as a function of the sig-nal and idler frequencies fs,i = ωs,i/2π that we evaluatedat the extracted degeneracy point at f0

s=f0i =193.3 THz

corresponding to 1551.1 nm. The simulated degener-acy point is very close to the measured one found at1544.1(8) nm. We believe this slight discrepancy is dueto the experimental limitations in the BRW growth pro-cess concerning the accuracy at which the refractive indexand the thickness of the layers can be controlled.

While the pump spectrum is responsible for energyconservation, i.e. ωp = ωs + ωi, the PM function rep-resents frequencies, at which the condition ∆k ≈ 0 is ful-filled. Due to the different group velocities of the signaland idler photon in the vicinity of the degeneracy pointthe tilt of the PM function θ ≈ arctan(κs/κi) deviatesfrom that of the pump spectrum only by approximately0.5◦. In combination with the curved PM-function, theresulting JSA is asymmetric around the degeneracy pointleading to different marginal spectral properties of signaland idler.

The spectral overlap determined by [30] O =∫ ∫dωsdωif(ωs, ωi)f

∗(ωi, ωs) with ∗ denoting complexconjugate depends remarkably on the group velocity mis-match. As in our case signal and idler travel with differ-ent group velocities, their wavepackets are slightly tem-porally shifted, which is evident in the phase term of theJSA. For the unfiltered JSA we determine a spectral over-lap of only about 32 %, while 78 % could be achieved if thetemporal mismatch was corrected. We restrict the influ-ence of the group velocity mismatch by spectral filteringclose to the JSA degeneracy point and, therefore, we ex-pect a spectral overlap of 98 % and 84 % when filteringthe JSA with a 12 nm (1.5 THz) Gaussian and a 40 nm(5.0 THz) super-Gaussian BPF, respectively. These val-ues are in good accordance with our experimentally de-termined results in Sec. IV.

Appendix B: Quantum interference experiment witha twin beam state

We utilize the description of parametric downconver-sion as multimode squeezer in order to estimate the effectof the higher photon-number contributions on a quan-tum interference between signal and idler twin beams[29, 33]. We start by considering the configuration in

a

b

c

d

T

&

FIG. 5. The twin beam state is send to the input arms aand b of beam splitter with transmission c and coincidencesare counted between the two output arms c and d.

Fig. 5 with a symmetric beam splitter having a transmis-sion of T = 1/2 and define the transformation betweenits input arms, a and b, and output arms, c and d, intime t as

c(t1) = 1/√

2[a(t1) + b(t1)

](B1)

d(t2) = 1/√

2[a(t2)− b(t2)

], (B2)

which expresses the photon annihilators c(tµ) and d(tµ)(µ = 1, 2) at the beam splitter output ports in terms of

those at its input ports a(tµ) and b(tµ). Further, weassume that our detectors have a spectrally broadband

6

response and their detection windows are much longerthan the duration of the generated pulsed wavepackets.Thus, the coincidence rate can be evaluated via [16]

R ∝∫dt1

∫dt2 〈c†(t1)d

†(t2)d(t2)c(t1)〉 , (B3)

the integrand in which describes the probability of a co-incidence at times t1 and t2 between the two detectors.

Our goal is to rewrite Eq. (B3) in terms of the beamsplitter input operations and then evaluate the expecta-tion values regarding the desired input state. By plug-ging the beam splitter transformations in Eqs (B1) and(B2) together with their conjugates to Eq. (B3) andutilizing the Fourier transformations given by a(t) =

1√2π

∫dω a(ω)e−iωt and b(t) = 1√

2π

∫dω b(ω)e−iωt with

ω being the optical angular frequency, only few termssurvive and we can write down the coincidence rate inform

R ∝ 1

4

⟨ ∫dω

∫dω[a†(ω)a†(ω)a(ω)a(ω)− a†(ω)a†(ω)b(ω)b(ω)− 2a†(ω)b

†(ω)a(ω)b(ω)

+2a†(ω)b†(ω)b(ω)a(ω)− b

†(ω)b

†(ω)a(ω)a(ω) + b

†(ω)b

†(ω)b(ω)b(ω)

] ⟩(B4)

in which we have carried out the integration overtime by extending the limits to infinity (δ(ω) =1/(2π)

∫∞−∞ dtµe

itµω ).

Now, we re-express Eq. (B4) in terms of broadbanddetection modes that correspond to those of our down-converter. The multimode squeezed state send to theinput arms a and b of the beam splitter is defined via theunitary squeezing operator Sa,b as [29]

|Ψ〉 = Sa,b |0〉 = e∑k rkA

†kB

†k−h.c. |0〉 , (B5)

in which the real valued squeezing strength rk = Bλkis related to the gain of the PDC process B and tothe Schmidt modes λk (

∑k λ

2k = 1) of the joint spec-

tral correlation function of signal and idler given byf(ωs, ωi) =

∑k λkϕk(ωs)φk(ωi). With help of the two

sets of orthonormal basis functions {ϕk} and {φk} forsignal and idler, respectively, we define the k-th modesent to the input arm a as

A†k =

∫dω ϕk(ω) a†(ω), (B6)

for which it applies∫dω ϕ?k(ω)ϕk′(ω) =

{1 if k = k′,0 otherwise

and∑k

ϕk(ω)ϕ?k(ω) = δ(ω − ω). (B7)

Similarly, the k-th mode sent to the input arm b can bewritten as

B†k =

∫dω φk(ω) b

†(ω), (B8)

the basis functions in which obey conditions similar tothose in Eq. (B7). The broadband mode transformations

of the k-th input modes can be presented in the form [29]

S†a,bAkSa,b = cosh(rk)Ak + sinh(rk)B

†k (B9)

S†a,bBkSa,b = cosh(rk)Bk + sinh(rk)A

†k. (B10)

In the following we consider only the case of weaksqueezing and approximate sinh(rk) ≈ rk = Bλk andcosh(rk) ≈ 1. Further, we estimate the mean photonnumber in the both input arms as 〈n〉 =

∑k sinh2(rk) ≈

B2.In order to transform Eq. (B4) to the broadband-mode

picture, we require the identities

a†(ω) =∑k

A†kϕ

?k(ω) and

b†(ω) =

∑k

B†kφ

?k(ω). (B11)

Thence, we re-express the coincidence rate in Eq. (B4)as

R ∝ 1

4

⟨[∑k,n

A†nA†kAkAn

−∑

k,n,l,m

A†nA†kBlBm

∫dωϕ?n(ω)φm(ω)

∫dωϕ?k(ω)φl(ω)

− 2∑

k,n,l,m

A†nB†kAlBm

∫dωϕ?n(ω)φm(ω)

∫dωφ?k(ω)ϕl(ω)

+ 2∑k,n

A†nB†kBkAn

−∑

k,n,l,m

B†nB†kAkAn

∫dωφ?n(ω)ϕm(ω)

∫dωφ?k(ω)ϕl(ω)

+∑k,n

B†nB†kBkBn

]⟩. (B12)

7

We directly recognize that several terms in Eq. (B12)correspond to Glauber correlation functions G(w, υ) =

〈: (∑q A†qAq)

w(∑q′ B

†q′Bq′)

υ :〉 with indices w and υ de-scribing the order of the correlation for the twin beammodes a and b, respectively [29]. Thence, the expecta-tion values in the first and last terms deliver G(2, 0) =

G(0, 2) = 〈n〉2 [1 + 1K ], and in the fourth term G(1, 1) =

〈n〉2 [1 + 1K ] + 〈n〉, where K corresponds to the effective

number of the excited modes (K = 1/∑k λ

4k) [29]. The

rest of the mean values can be evaluated by plugging inthe transformations from Eqs (B9) and (B10) togetherwith their hermitian conjugates. While the second andfifth terms vanish, the third term delivers⟨ ∑k,n,l,m

A†nB†kAlBm

∫dωϕ?n(ω)φm(ω)

∫dωφ?k(ω)ϕl(ω)

⟩= 〈n〉O + 〈n〉2A, (B13)

in which

O =

∫dω

∫dω f?(ω, ω)f(ω, ω)

=∑k,n

λnλk

∫dωϕ?n(ω)φk(ω)

∫dωφ?n(ω)ϕk(ω) (B14)

describes the spectral overlap between signal and idlerand

A =

∫dω

∫dω gs(ω, ω)gi(ω, ω)

=∑k,n

λ2nλ

2k

∫dωϕ?n(ω)φk(ω)

∫dωφ?k(ω)ϕn(ω) (B15)

determines the overlap of signal and idler spectral densi-ties that are given by

gs(ω, ω) =

∫dωi f

?(ω, ωi)f(ω, ωi) and (B16)

gi(ω, ω) =

∫dωs f

?(ωs, ω)f(ωs, ω). (B17)

Note that if the spectral densities of signal and idler areequal this term will end up giving the purity of the photonwavepacket, 1/K.

Finally, we determine an expression for the visibility ofour quantum interference experiment in Sec. IV. Whensignal and idler are expected to bunch, we can estimatethe rate of the coincidences according to Eq. (B12) as

Rmin ∝1

2〈n〉(1−O

)+ 〈n〉2

(1 +

1

K− A

2

). (B18)

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 a beamsplitter with T = 1 in Fig. 5, we gain

Rmax ∝ G(1, 1) = 〈n〉+ 〈n〉2(

1 +1

K

). (B19)

The visibility is then given by

V =Rmax −Rmin

Rmax +Rmin

=[1 +O] +A〈n〉

[3−O] + 4 〈n〉 (1 + 1K )−A〈n〉

. (B20)

For a highly multimode PDC process such as ours weestimate that the visibility can be evaluated in terms ofthe mean photon number as

V ≈ 1 +O3−O + 4 〈n〉

(B21)

that contains in addition to the spectral overlap knownfor a true photon-pair state [30] a degradation of thevisibility due to the higher photon-number contributions.

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