Broadband indistinguishability from bright parametric downconversion in a semiconductor waveguide

Parametric downconversion (PDC) in semiconductor Bragg-reflection waveguides (BRW) is routinely exploited for photon-pair generation in the telecommunication range. Contrary to many conventional PDC sources, BRWs offer possibilities to create spectrally broadband but nevertheless indistinguishable photon pairs in orthogonal polarizations that simultaneously incorporate high frequency entanglement. We explore the characteristics of co-propagating twin beams created in a type-II ridge BRW. Our PDC source is bright and efficient, which serves as a benchmark of its performance and justifies its exploitation for further use in quantum photonics. We then examine the coalescence of the twin beams and investigate the effect of their inevitable multi-photon contributions on the observed photon bunching. Our results show that BRWs have a great potential for producing broadband indistinguishable photon pairs as well as multi-photon states.


I. INTRODUCTION
Versatile quantum light sources are needed for a variety of quantum communication protocols and thus their development is pursued by many groups. Perhaps the best researched process is parametric downconversion (PDC), which produces photons in pairs, usually denoted as signal and idler. Waveguide realizations have turned PDC sources into easy-to-handle and small-scale tools that moreover provide higher brightness than their bulk counterparts [1]. Waveguided sources further provide better integrability and quantum integrated networks have been built both on semiconductor and dielectric platforms [2][3][4].
Recently, the PDC process has been demonstrated in semiconductor waveguides being composed of layers of semiconductor materials that are historically named Bragg-reflection waveguides (BRW) [5,6]. In comparison to dielectric non-linear optical waveguides the semiconductor structures benefit from higher nonlinearity and better integrability [7]. By embedding the pump laser and the photon-pair production on the same chip the PDC emission can even be electrically self pumped [8]. BRWs with high signal-idler correlations are suitable for many different quantum optical tasks [7]. Previous experimental studies include demonstration of indistiguishable photon pairs [9] and preparation of polarization entangled states both with co-and counter-propagating signal and idler schemes [10,11]. Furthermore, BRWs are viable for source design, which aims at engineering of quantum states with desired properties for specific applications [12,13].
However, in order to successfully compete with conventional PDC sources, BRWs have to be able to produce twin beams, in other words signal and idler obeying a strict photon-number correlation, in a bright and efficient manner with a low number of spurious counts. Still today, their drawbacks are the incompatibility of the uti- * thomas.guenthner@uibk.ac.at lized spatial modes with standard single-mode fiber optics, a rather high facet reflectivity because of the large refractive index difference with air, and a high numerical aperture (NA) due to strong confinement of the spatial modes. On top of this, the optical losses both at the pump and the downconverted wavelengths are significant [7,14], which limits the useful length of the structures.
Fortunately, some of these properties can also be exploited-the phasematching required for the PDC process can be achieved in semiconductor waveguides by spatial mode matching eliminating the need for quasiphasematching, which is typical for conventional sources [15]. In general, the characteristics of signal and idler in their different degrees of freedom are determined by the PDC process parameters such as the strength of the nonlinearity, pump envelope and the dispersion relation in the used geometry. The resulting joint spectrum of signal and idler to a large extend dictates for which quantum optics applications the photonic source in question is suitable [16][17][18]. Additionally, the photon statistics of the individual marginal beams that is signal or idler are affected by the PDC process parameters, therefore, the inevitable higher photon-number contributions have to be controlled [19].
Here, we investigate the characteristics of spectrally highly multimodal PDC emitted by a BRW in a singlepass configuration. Firstly, we determine the Klyshko efficiency of our BRW source. Thereafter, we utilize correlation functions between signal and idler in order to investigate the mean photon number in the marginal beams. We further modify the spectral properties by filtering and via a two-photon coalescence experiment determine the indistinguishability of signal and idler, which is governed by their spectral overlap. Our results show that BRWs are excellent candidates for becoming bright and efficient photonic sources.

II. SAMPLE DESIGN AND EXPERIMENT
Our Bragg-reflection waveguide (BRW) sample (depicted schematically in Fig. 1(a) substrate by molecular beam epitaxy. The sample is made of Al x Ga 1−x As (0 < x < 1) compounds because of their inherent high optical second order nonlinearity and the sophisticated fabrication techniques available. Due to its zinc blende structure, Al x Ga 1−x As has no birefringence and, therefore, in order to achieve phasematching the sample is designed to support different spatial modes that are the Bragg mode for the pump and the total internal reflection modes for the twin beams [21]. The distributed Bragg reflectors (DBRs) embed a multilayer core that guarantees good spatial overlap of the pump, signal and idler mode triplets required for an efficient PDC process [22][23][24]. Finally, the ridge structure is fabricated by electron beam lithography followed by plasma etching. This ensures mode confinement in two dimensions-in the vertical direction by the DBRs and in the horizontal direction by the ridge structure.
In our experiment as shown in Fig. 1(b) we employ a picosecond pulsed Ti:Sapphire laser (76.2 MHz repetition rate, 772 nm central wavelength) as a pump for the PDC process. After power, polarization and spatial mode control, we focus the pump on to the front facet of our BRW with a 100x microscope objective (MO), which allows reasonably efficient coupling into the Bragg mode. At its output facet a high numerical aperture (NA) aspheric lens (AL) collects the PDC emission from our BRW, for which we numerically determined the NAs of approximately 0.2 and 0.5 in the horizontal and vertical direction, respectively [20]. After removing the residual pump beam with a dichroic mirror (DM), we use a band-pass filter (BPF) with either a 12 nm or 40 nm bandwidth to spectrally limit the marginal beams. With an additional half-wave plate (HWP) we change the polarization direction of signal and idler, as necessary for the signal and idler coalescence experiment. After separating the orthogonal polarizations using a polarizing beam splitter (PBS), we couple the marginal beams to single-mode fibers for the detection with two commercial time-gated InGaAs avalanche photo-diodes (APDs) with a variable photon detection probability up to 25 %. Finally, we employ a time to digital converter to discriminate the sin-gle and coincidence counts. Due to technical limitations, every 64 th laser pulse triggers our InGaAs APDs corresponding to a rate R of about 1.19 MHz. In conventional dielectric waveguides efficient coupling of signal and idler to single-mode fibers has been achieved at telecommunication wavelengths [25,26] and also large mean photon-numbers have been demonstrated [27]. Therefore, we start by investigating the performance of our BRW source by determining its Klyshko efficiency [28] and thereafter evaluate the mean photonnumber in the marginal beams. For this purpose we use a configuration ( Fig. 1(b)), in which the PDC emission is filtered to a spectral width of 40 nm to suppress background light from the waveguide before separating the signal and idler beams.

III. SOURCE EFFICIENCY AND BRIGHTNESS
To eliminate the effect of the accidental coincidences produced by the higher photon-number contributions created in the PDC process, we measure the efficiency with respect to the pump power. In the region of weak pump powers we can extract the Klyshko efficiency, 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 count rates of signal and idler, respectively. Our results in Fig. 2(a) show the ratio of coincidence counts to single counts for both signal and idler detected with a chosen photon detection probability of 20 % at our APDs. Via extrapolation to zero pump power Klyshko efficiencies of 6.0(1) % and 5.6(2) % are found for signal and idler, respectively. This indicates that despite of the high NA and the complex spatial mode structure, the PDC emission can be efficiently collected with standard optical components.
By measuring the signal and idler cross-correlation we can further estimate the mean photon number n of the marginal beams in a loss-independent manner via C/A ≈ 1/ n + 1, in which A = S i S s /R corresponds to the accidental count rate [29]. Since our BRW is a highly multimodal PDC source (see Appendix A), this estimation gives the mean photon number in good approximation, being in the worst case the lower bound. In Fig. 2(b) we illustrate the mean photon number obtained for a single marginal beam with respect to the increasing pump power. Our results show that mean photon numbers up to 0.5 can be achieved with moderate pump powers. This further indicates that the PDC multi-photon contributions have to be taken into account especially in the photon coalescence experiment we investigate next.

IV. COALESCENCE OF SIGNAL AND IDLER
For observing the coalescence of signal and idler photons we follow the experiment in [30] and investigate photon bunching by varying their distinguishability in the polarization degree of freedom. For this purpose, we detect the coincidences between signal and idler while rotating the HWP in Fig. 1(b). In case the HWP axes are oriented parallel to the cross-polarized signal and idler, the marginal beams are separated deterministically at the PBS. Otherwise, signal and idler bunch together if they are indistinguishable in all degrees of freedom-not only in polarization but also spatially and spectrally.
We record the coincidence counts with respect to the HWP angle at several pump powers. Additionally, we change the photon detection probability of our APDs from 20 % to 25 % to increase the count rates. In Fig. 3(a) and (b) we present our results with a 12 nm band-pass filter (BPF) for a low and a high pump power value, respectively. From this, we can directly conclude that the higher the pump power the lower is the visibility V of the measured fringes given by V = (C max − C min )/(C max + C min ). We nevertheless achieved a maximum visibility of 0.83(1), which lies significantly above the classical limit of 1/3 for a completely distinguishable photon pair and delivers us a measure of the signal and idler indistinguishability. In order to understand the effect of the multi-photon contributions to the signal and idler coalescence, we further examine the visibility in terms of the mean photon number in the marginal beams, which is also provided by the measured data. In Fig. 3(c) we depict the measured visibilities with respect to the estimated mean photon number in the marginal beams for both 12 nm and 40 nm BPFs. In both cases the visibility clearly decreases with increasing mean photon number. However, not only the increasing multi-photon contributions but also the spectral overlap of signal and idler affect the visibility in the coalescence experiment. From our results for the visibility we can infer the spectral overlap O of the downconverted photon pairs that is given by (see Appendix B) By fitting our results in Fig. 3(c) against Eq. (1) we can retrieve spectral overlap values of 95.0(6) % and 81.6(9) % for 12 nm and 40 nm filter bandwidth, respectively. The large difference in the spectral overlaps with the two investigated filters are caused by the the different group velocities of the signal and idler wavepackets due to which signal and idler pulses are temporally shifted with respect to each other. Theoretically the joint spectral distribution of signal and idler govern the spectral overlap (see Appendix A). Our results are in good accordance with numerical simulations, which predict spectral overlaps of 98.0 % and 83.8 %, respectively.

V. CONCLUSION
Integratable and easy-to-handle sources of parametric downconversion are highly desired in many quantum optics applications. Bragg-reflection waveguides based on semiconductor compounds provide a platform that can meet these demands. We investigated the properties of parametric downconversion from a BRW sample and showed that Klyshko efficiencies on the order of a few percent can be achieved for signal and idler. Moreover, our source is bright and capable of producing higher photon numbers as desired for multiphoton production. Finally, we examined the coalescence between signal and idler in order to assess their indistinguishability. Although the visibility of the measured fringes is diminished by the multi-photon contributions of signal and idler, the created photon pairs show a high degree of indistinguishability that can be quantitfied with the spectral overlap. We extended our model to take into account both these process parameters and showed that our results are in good agreement with numerical simulations. We believe our work gives a detailed insight of the PDC process in our BRW both in the spectral and photon-number degrees of freedom. This will become important when optimizing and adapting this photonic source in to a quantum optical network. of signal and idler In this appendix we investigate the joint spectral amplitude (JSA) of signal and idler and numerically estimate the spectral overlap O, which determines their indistinguishability in the low gain regime. Following [16,17,31] we find that the joint spectral characteristics in a collinear single-pass PDC source are given by in which N accounts for the normalization dω s dω i |f (ω s , ω i )| 2 = 1, α(ω p = ω s + ω i ) describes the 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. In a Gaussian approximation we can describe the pump amplitude as with σ p describing the bandwidth of the pump. We use a simple PDC model for uniform waveguides [18,24] with constant-valued non-linearity over the whole length of the waveguide and the overlap of spatial modes effectively affects only its strength. Thus, in the single-pass configuration we can write the PM function as [32] φ(ω s , ω i ) = sinc in the final form of which we have used a Gaussian approximation for the sinc-function. In Eq. (A3) L denotes the BRW length, ∆k(ω s , in which the detunings are defined as is determined by the group velocity mismatch of the downconverted photons and the pump photon, while Λ s,i = 1 2 k s,i (ω 0 s,i ) − 1 2 k p (ω 0 p ) and Λ p = k p (ω 0 p ) are related to the group velocity dispersions. For our simulation we substitute Eqs. (A2)-(A4) into Eq. (A1) and evaluate the JSA. From a commercially available solver (Mode Solutions) we obtain the parameters of the PDC process in our BRW ( Fig. 1(a) Fig. 4(c) the amplitude of the simulated JSA as a function of the signal and idler frequencies f s,i = ω s,i /2π that we evaluated at the extracted degeneracy point at f 0 s =f 0 i =193.3 THz corresponding to 1551.1 nm. The simulated degeneracy point is very close to the measured one found at 1544.1(8) nm. We believe this slight discrepancy is due to the experimental limitations in the BRW growth process concerning the accuracy at which the refractive index and the thickness of the layers can be controlled.
While the pump spectrum is responsible for energy conservation, i.e. ω p = ω s + ω i , the PM function represents frequencies, at which the condition ∆k ≈ 0 is fulfilled. Due to the different group velocities of the signal and idler photon in the vicinity of the degeneracy point the tilt of the PM function θ ≈ arctan(κ s /κ i ) deviates from that of the pump spectrum only by approximately 0.5 • . In combination with the curved PM-function, the resulting JSA is asymmetric around the degeneracy point leading to different marginal spectral properties of signal and idler.
The spectral overlap determined by [30 with * denoting complex conjugate depends remarkably on the group velocity mismatch. As in our case signal and idler travel with different group velocities, their wavepackets are slightly temporally shifted, which is evident in the phase term of the JSA. For the unfiltered JSA we determine a spectral overlap of only about 32 %, while 78 % could be achieved if the temporal mismatch was corrected. We restrict the influence of the group velocity mismatch by spectral filtering close to the JSA degeneracy point and, therefore, we expect a spectral overlap of 98 % and 84 % when filtering the JSA with a 12 nm (1.5 THz) Gaussian and a 40 nm (5.0 THz) super-Gaussian BPF, respectively. These values are in good accordance with our experimentally determined results in Sec. IV.

Appendix B: Quantum interference experiment with a twin beam state
We utilize the description of parametric downconversion as multimode squeezer in order to estimate the effect of the higher photon-number contributions on a quantum interference between signal and idler twin beams [29,33]. We start by considering the configuration in which expresses the photon annihilatorsĉ(t µ ) andd(t µ ) (µ = 1, 2) at the beam splitter output ports in terms of those at its input portsâ(t µ ) andb(t µ ). Further, we assume that our detectors have a spectrally broadband response and their detection windows are much longer than the duration of the generated pulsed wavepackets. Thus, the coincidence rate can be evaluated via [16] the integrand in which describes the probability of a coincidence at times t 1 and t 2 between the two detectors.
Our goal is to rewrite Eq. (B3) in terms of the beam splitter input operations and then evaluate the expectation values regarding the desired input state. By plugging the beam splitter transformations in Eqs (B1) and (B2) together with their conjugates to Eq. (B3) and utilizing the Fourier transformations given byâ(t) = dωb(ω)e −iωt with ω being the optical angular frequency, only few terms survive and we can write down the coincidence rate in form in which we have carried out the integration over time by extending the limits to infinity (δ(ω) = 1/(2π) ∞ −∞ dt µ e itµω ). Now, we re-express Eq. (B4) in terms of broadband detection modes that correspond to those of our downconverter. The multimode squeezed state send to the input arms a and b of the beam splitter is defined via the unitary squeezing operatorŜ a,b as [29] |Ψ =Ŝ a,b |0 = e k r kÂ † kB † in which the real valued squeezing strength r k = Bλ k is related to the gain of the PDC process B and to the Schmidt modes λ k ( k λ 2 k = 1) of the joint spectral correlation function of signal and idler given by f (ω s , ω i ) = k λ k ϕ k (ω s )φ k (ω i ). With help of the two sets of orthonormal basis functions {ϕ k } and {φ k } for signal and idler, respectively, we define the k-th mode sent to the input arm a aŝ for which it applies Similarly, the k-th mode sent to the input arm b can be written asB † the basis functions in which obey conditions similar to those in Eq. (B7). The broadband mode transformations of the k-th input modes can be presented in the form [29] S † In the following we consider only the case of weak squeezing and approximate sinh(r k ) ≈ r k = Bλ k and cosh(r k ) ≈ 1. Further, we estimate the mean photon number in the both input arms as n = k sinh 2 (r k ) ≈ B 2 . In order to transform Eq. (B4) to the broadband-mode picture, we require the identitieŝ Thence, we re-express the coincidence rate in Eq. (B4) as We directly recognize that several terms in Eq. (B12) correspond to Glauber correlation functions G(w, υ) = : ( qÂ † qÂq ) w ( q B † q B q ) υ : with indices w and υ describing the order of the correlation for the twin beam modes a and b, respectively [29]. Thence, the expectation values in the first and last terms deliver G(2, 0) = G(0, 2) = n 2 [1 + 1 K ], and in the fourth term G(1, 1) = n 2 [1 + 1 K ] + n , where K corresponds to the effective number of the excited modes (K = 1/ k λ 4 k ) [29]. The rest of the mean values can be evaluated by plugging in the transformations from Eqs (B9) and (B10) together with their hermitian conjugates. While the second and fifth terms vanish, the third term delivers k,n,l,mÂ † nB † kÂlBm dωϕ n (ω)φ m (ω) dωφ k (ω)ϕ l (ω) in which O = dω dω f (ω,ω)f (ω, ω) = k,n λ n λ k dωϕ n (ω)φ k (ω) dωφ n (ω)ϕ k (ω) (B14) describes the spectral overlap between signal and idler and A = dω dω g s (ω,ω)g i (ω, ω) = k,n λ 2 n λ 2 k dωϕ n (ω)φ k (ω) dωφ k (ω)ϕ n (ω) (B15) determines the overlap of signal and idler spectral densities that are given by g s (ω,ω) = dω i f (ω, ω i )f (ω, ω i ) and (B16) g i (ω, ω) = dω s f (ω s ,ω)f (ω s , ω).
Note that if the spectral densities of signal and idler are equal this term will end up giving the purity of the photon wavepacket, 1/K.
Finally, we determine an expression for the visibility of our quantum interference experiment in Sec. IV. When signal and idler are expected to bunch, we can estimate the rate of the coincidences according to Eq. (B12) as This rate is compared with the one obtained when the signal and idler beams are separated deterministically. By using the same model as above but assuming a beam splitter with T = 1 in Fig. 5, we gain R max ∝ G(1, 1) = n + n 2 1 + 1 K . (B19) The visibility is then given by For a highly multimode PDC process such as ours we estimate that the visibility can be evaluated in terms of the mean photon number as that contains in addition to the spectral overlap known for a true photon-pair state [30] a degradation of the visibility due to the higher photon-number contributions.