In quantum computing, it is important to distinguish between physical and logical qubits. The first is the two-level physical systems that are processed in the quantum hardware, while the second is the clusters of redundant physical qubits which store the information [1–5]. On photonic platforms, the number of physical qubits is typically associated with the number of photons whose quality is intrinsically quantified by the purity of the photon states and extrinsically quanitified by their indistinguishability [6,7]. The demand for reliable physical qubits translates into engineering high-quality sources of photons. Currently, there are two main types of integrated sources: deterministic and probabilistic. In the case of deterministic sources, the integration in a silicon photonic-integrated circuit (SiPIC) is still very demanding because of the high coupling loss and the complex fabrication process. On the other hand, there have been many reports of probabilistic photon-pair sources integrated in SiPIC over the last 10 years [8,9]. Then, the correlation between the photons is exploited to produce single photon sources through the heralding mechanism, where one photon of the pair is detected to herald the presence of its twin photon [10]. In the case of deterministic sources, the quality of a generated single photon state is quantified by how close to zero the second-order coherence function g(2)(0) is or, in the case of probabilistic ones, how close to zero the heralded second-order coherence function gh(2)(0) is.

In this paper, we aim to address the following question: what type of source performs better among the probabilistic sources? This is not a trivial question. The limited amount of physical qubits in the so-called noisy intermediate scale quantum (NISQ) computing era prevents the use of error-correction techniques in universal quantum computing hardware and brings in people to work with quantum simulators, i.e., devices that perform non-universal and problem-focused algorithms [11,12]. Therefore, the most suitable source of photons should be determined by the requirements of the specific implemented algorithm. The ideal short-term aim would be the realization of a sources’ library, which can be consulted as needed and updated with the improvements of the sources present in the list.

Among probabilistic sources, the nonlinearity of the photonic material is typically used to create pairs of photons through nonlinear spontaneous parametric processes. Once the generation is achieved, the generated photons are led to reconfigurable networks of Mach–Zehnder interferometers (MZIs) to produce interference patterns. These are used to qualify the quality of the generated photons [10]. Photon-pair generation and interference between the photons within a SiPIC have gained more and more attention with the growing anticipation toward using a fault-tolerant quantum computer based on linear optical quantum computation [13–17]. In silicon photonics, photon pairs can be generated by spontaneous four-wave mixing (SFWM) in a long waveguide [18,19] or a compact microring resonator [20,21]. Quantum interference of photons in a SiPIC has been demonstrated using heralded photons from non-degenerate photon pairs [20,21] or using degenerate photon pairs composed of two photons that are hard to distinguish from each other [18]. The photon pairs that degenerate in the signal-idler energies can be generated by pumping the SFWM process with two laser wavelengths in, for example, spiraled waveguides, which form the arms of an integrated MZI. The propagation path of the photons can be controlled by adjusting the phase of the MZI, and on-chip quantum interference such as the Hong–Ou–Mandel (HOM) effect can be observed [18,22]. On the other hand, using integrated microring resonators as photon-pair sources, the HOM effect [23–25] of the heralded photons has also been reported more recently [20,21], and a degenerate photon-pair generation from a single microring resonator has been demonstrated [26]. However, on-chip quantum interference of degenerate photon pairs from two microring resonators has not yet been shown and neither has its direct comparison with an experiment using waveguide spirals.

Here, we aim at comparing probabilistic degenerate photon-pair sources made of microring resonators or waveguides. In order to make a fair comparison, we used sources integrated in the same SiPIC. By using microring resonators coupled to waveguides that form the arms of an MZI, we measure the on-chip HOM interference to assess the quality of the generated photon pairs. Microring resonators can be thermally tuned in or out of resonance with the pump wavelengths, to turn on and off the microring-based photon sources, respectively. In this way, we can directly compare the on-chip interference of the degenerate photon pairs from the microrings with respect to the one from the waveguides. Our configuration, composed of the degenerate SFWM with two sources put in parallel inside an MZI, allows us to have a configuration not limited by the purity of the sources but by the indistinguishability of the sources.

The paper is organized in this way. In Section 2, we describe the fabricated devices and the experimental setups used. Section 3 reports the measurement results for the photon-pair sources. Section 4 shows the results of the quantum interference experiments for waveguide-based sources. In Section 5, the results of the quantum interference experiments on waveguide- and microring-resonator-based photon-pair sources in the same integrated photonic circuit are presented. Section 6 discusses the characteristic parameters for the different types of photon-pair sources by using both experimental data and simulations. Section 7 makes a final statement about the main results of this paper. Finally, the Appendix A details the theoretical analysis and the numerical calculations.

The generation of photon pairs in a SiPIC can be achieved through SFWM processes in a low-squeezing regime [27–30]. Typically, two pump photons (at wavelengths λP1 and λP2) are converted into two newly generated idler and signal photons (at wavelengths λi and λs) [10]. Non-degenerate SFWM is achieved when identical pump photon wavelengths (λP1=λP2) are used so that the generated photons have different wavelengths [31–34], while in degenerate SFWM the situation is inverted, i.e., different pump photon wavelengths (λP1≠λP2) and equal generated photon wavelengths [λi=λs=2(λP1−1+λP2−1)−1] [35]. In both cases, the state of the output photons is generally in a squeezed state: the two-mode squeezed state for the non-degenerate case and the single-mode squeezed state for the degenerate one [36,37]. Both states have useful properties for quantum applications. For example, the non-degenerate case is used in combination with the heralding procedure to obtain probabilistic single photon sources, while the degenerate case produces a couple of correlated identical single photons. In this work, the latter case has been implemented in a SiPIC, where the generated pairs of degenerate photons are used to obtain HOM interference.

Two SiPICs are used in this work. Figures 1 and 2 show the schematic diagrams of the integrated circuits for the two devices named SiPIC-1 and SiPIC-2, respectively. Pictures of the actual chips together with their packaging modules are shown in Figs. 1(d) and 2(b). The devices are packaged on a metal-based printed-circuit board (MPCB) with 24-port electrical wiring, and they are pig-tailed to a 24-channel fiber array. The MPCB is in contact with a thermo-electric cooler (TEC) to control the chip temperature. SiPIC-1 and SiPIC-2 are based on silicon waveguides with a nominal 450nm×220nm cross section (typically 480nm×210nm after fabrication) and were fabricated with IMEC/Europractice using their passive+ silicon-on-insulator (SOI) platform, similar to our previous report [22]. Dispersion tailoring was not used to optimize SFWM processes since previous experiments [19] show a quite broad generation spectrum for photon pairs in silicon waveguides. The measured waveguide propagation loss is 2 dB/cm, a relevant feature for the sources we want to compare. For microring resonators, this determines the quality of the cavity, whose design has the coupler parameters adjusted to the losses. For waveguide spirals, it gives an effective length of Leff=[1−exp(−αL)]/α, with L being the geometrical length and α being the propagation loss per unit of the length, which enters quadratically in the SFWM generation rate [38]. From the analysis of this quantity, we choose the geometrical length of the long spirals in SiPIC-1 and SiPIC-2, finding 1.5 cm as an optimal value.

Figure 1 (top) shows the schematic experimental setup based on SiPIC-1. Two continuous-wave (CW) tunable laser diodes (CoBrite from IDPhotonics) at λP1=1544.08nm and λP2=1556.18nm are combined by a 3-dB fiber-optic coupler and provide the pump photons. The combined beam is passed through an optical notch filter (NF) [19,22] to eliminate photon noise within a 1.6 nm bandwidth around λi=λs=1550.12nm, which is the wavelength of the on-chip generated photon pairs. Then, the pump photons are inserted into the selected input fiber of the fiber array, which is coupled to the chip by grating couplers and whose coupling loss is measured to be 4.2 dB. In SiPIC-1, different circuits are presented [Figs. 1(a)–1(c)]. These are based on MZIs with two nominally identical photon-pair sources located in their arms. The photon-pair sources are based on 15-mm-long waveguide spirals [Fig. 1(a)] or on 30-μm-radius microring resonators [Figs. 1(b) and 1(c)]. The microrings are coupled to micro-heaters (metal wires on top of the microring), which allow thermal tuning of their resonances.

Figure 1(e) shows the normalized transmission spectrum of one of the microrings taken by scanning the wavelength of one of the tunable laser diodes. Transmission resonances are observed at 1544.08 nm, 1550.12 nm, and 1556.18 nm, i.e., at the wavelengths of the pump photons and of the generated photon pairs. Their free spectral range (FSR) is about 3 nm, and their loaded quality factor (Q-factor) is about 1.5×104. Micro-heaters are also integrated on one arm of the MZI to act as phase shifters (ϕ) in order to compensate for unwanted phase differences between the two arms. The efficiency of the micro-heater (0.6μm×40μm) of the MZI is measured to be about 25mW/π with an overall resistance of about 50Ω. The efficiency of the heater is relatively poor with respect to our previous report of 12mW/π [39], and this is mainly due to the additional power consumption along the wires from the micro-heater to the bonding pad, which adds to the power dissipated at the contact resistance between the metal wire and the heater layers. We expect that the heating efficiency can be improved simply by increasing the length of the heater.

In the circuit reported in Fig. 1(c) the length of the waveguide in the MZI arms is 240 μm. This implies that photon pairs can also be generated in the waveguide of the MZI of Fig. 1(c). In order to isolate the waveguide contribution, we thermally tuned the microring out of resonance with respect to the pump photons (labeled RingOff). We changed the temperature of the microrings from 11.7°C±0.1°C (RingOn, microring resonant with the pump photons) to 20°C±0.1°C (RingOff) since the microring resonance wavelength temperature dependence is 80 pm/°C [39]. The circuit shown in Fig. 1(c) includes an additional second MZI. The first MZI, similar to the circuits shown in Figs. 1(a) and 1(b), contains the degenerate photon-pair sources and a phase shifter (ϕ1), while the second MZI measures the quantum interference of the generated photon pairs. Note that the two multi-mode interference devices (1×2MMIs) are used after the first MZI to tap and monitor the generated photons. The second MZI also includes a phase-shifter (ϕ2). This is placed 740 μm away from the microrings to avoid any thermal cross talk. At the output of the SiPIC, the photons are out-coupled through gratings using the fiber arrays, filtered off the pump photons with band-pass filters (BPFs, with a 0.8 nm bandwidth centered at 1550.12 nm), and detected by using super-conducting-nanowire single-photon detectors (SNSPDs, EOS from Single Quantum) [19,22]. Then, the SNSPD single-photon events are counted by a time-correlated single photon counter (TCSPC, Logic16 from UQdevices) and analyzed by logical post-selection.

Figure 2(a) shows the schematic experimental setup based on SiPIC-2. With respect to the circuit shown in Fig. 1(c), this is an improvement. It is aimed at a direct comparison between microring and waveguide sources by integrating both an HOM and a reverse HOM experimental set-up with a scheme similar to the one shown in Fig. 1(c). In SiPIC-2, the microrings have 28.5-μm-long radii, the spiral waveguides are 15-mm-long, and the grating couplers have a measured coupling efficiency of 3.3 dB. In addition, SiPIC-2 includes photon pump filters realized by asymmetric MZIs (a-MZIs) and placed after the sources to reduce the accidental coincidence (noisy) counts. The asymmetric length of the a-MZI is designed to be 47.2 μm, which yields a 12.8 nm (1600 GHz) FSR to match four times the nominal FSR of the microrings. Figure 2(c) shows the normalized transmission spectra measured with the photodiodes (PD1–PD4) visible in Fig. 2(a). PD1 and PD2 measure the pump filter characteristics, while PD3 and PD4 measure the transmitted photons. After tuning the resonance of the microring close to the pumping wavelengths (1543.78 nm and 1556.53 nm) and tuning the pump filter MZIs to reject the pump, the following data are observed: a loaded Q-factor of 3×104 (twice larger than the Q-factor of the microrings in SiPIC-1), an FSR of 3.2 nm (400 GHz, similar to the design value), a pump filter FSR of 12 nm (1500 GHz), and a pump rejection ratio larger than 30 dB. This last value is large enough to significantly reduce the accidental counts due to the pump photons in the coincidence measurements.

The overall quality of the integrated sources has been characterized by the measurements of the coincidence-to-accidental ratio (CAR) [19], the heralding rate [19,22], and the g(2)(0) of the heralded photons or gh(2)(0) [10]. To measure the gh(2)(0), we added a 3 dB fiber splitter between the SiPIC-1 outputs and the SNSPDs to realize a Hanbury–Brown–Twiss interferometer [10].

Figure 3(a) shows the measured CAR and heralding rate and Fig. 3(b) the measured g(2)(0) of the heralded photons from the spiral waveguide sources [Fig. 1(a), in short spirals in the following] and from the microring resonator sources [Fig. 1(b), in short rings in the following] in SiPIC-1. The heralding rate for a microring resonator is 5 kHz for 1 mW pump power (−119dB conversion efficiency for heralding), while the heralding rate for a spiral waveguide is 10 kHz for 5 mW pump power (−124dB conversion efficiency for heralding). The coincidence time window of the TCSPC was set equal to 0.2 ns. Measurements show that the spirals have a larger CAR but a smaller heralding rate than the rings at high pump powers. This last indicates that the field enhancement factor in our microring resonators is high enough to compensate for the length of the spirals. In addition, we observe in Fig. 3(b) high-quality heralded single photons at low pump powers [vanishing gh(2)(0)], while multi-photon contributions become relevant at high pump powers (above 2 mW). The multi-photon contribution appears to be more severe for photon pairs generated from the rings than from the spirals.

The circuit in Fig. 1(c) allows for tuning the rings in resonance (RingOn) and out of resonance (RingOff) with λP1 and λP2 and observing the photon-pair generations at the output channels ch 1 and ch 2. Figure 4 shows the CAR, the heralding rate, and the gh(2)(0) for the RingOn and RingOff configurations. Figure 4(a) shows that the heralding rate is larger for the RingOn configuration than for the RingOff configuration. The result for the RingOn configuration is similar to what is shown in Fig. 3(a) for the ring, while the CAR and the heralding rate for the RingOff configuration are lower than those reported for the spirals in Fig. 3(a), which is due to the difference in the waveguide lengths (240 μm versus 15 mm) that enters quadratically in the photon-pair generation rate. To measure gh(2)(0), we set the two rings in the two arms of the first MZI in the RingOn and the RingOff modes, respectively, while we set ϕ2=π of the second MZI. In this way, we can herald by ch 2 detections the single photons at ch 1 or ch 3, i.e., by post-selection we measure the coincidences of ch 1 and ch 3 heralded by ch 2 to get gh(2)(0) [Fig. 4(b)]. The results are similar to what can be observed in Fig. 3(b) when the different waveguide lengths or microrings are considered.

Data for SiPIC-2 are shown in Fig. 5. During the measurements, the phase ϕ2 of the second MZI is fixed at zero [see Fig. 2(a)] and, also in this case, the two modes RingOn and RingOff are possible. When the RingOn mode is selected, the rings are effective as photon-pair sources, while in the RingOff mode the spirals are effective. The CAR is larger than 1000 for the spirals (600 for the rings), which demonstrates the effect of the removal of the pump photons by the on-chip pump filters based on the a-MZIs, as in our previous report [19]. However, the heralding rate for the rings is relatively low due to the additional 3 dB losses caused by the 15-mm-long spirals, which follow the microrings on the arms of the first MZI (a decrease of the coincidence probability by 1/2×1/2=1/4 is estimated) and is due to large two-photon absorption (TPA) losses in the microring because of the high Q-factor.

Regarding TPA, we observe that Figs. 3(a), 4(a), and 5 show a saturation of the quadratic behavior of the heralding rate with respect to the pump power. However, as the following sections show, our comparative experiments are run by using low pump powers to achieve optimal values of CAR and gh(2)(0), where the quadratic behavior of the generation rate is observable. This sets us in a regime where TPA is not limiting the performances. Indeed, using low pump powers, the conversion efficiency of photon-pair generation is low but nonlinear losses given by TPA and free carrier absorption can be neglected with respect to linear scattering loss [40]. Thus, we can affirm that SFWM in SiPIC-1 and SiPIC-2, and especially for rings, should exclusively be performed at low optical pump powers to avoid a steep roll-off in performance at higher powers [41].

To prove the indistinguishability of the photons generated by the two sources, we performed quantum interference measurements with SiPIC-1 [18]. First, we studied the 15-mm-long spiral-waveguide-based sources in the circuit of Fig. 1(a), and we measured the dependence of the coincidence rates between ch 1 and ch 2 as a function of the phase ϕ of the MZI at a fixed pump power of 1.5 mW (CAR=170). Figure 6 compares the measured counts detected at the two output channels of the MZI in Fig. 6(a), showing the classical light transmission with the coincidence rates between the two channels in Fig. 6(b) and showing the quantum interference between the generated photons. As expected by the theoretical analysis reported in the Appendix A or in Refs. [18,22], the coincidence rates follow sin2(ϕ−π2)=cos2(ϕ), while the classical light transmission follow sin2[(ϕ−π2)/2] or cos2[(ϕ−π2)/2]. Some deviations with respect to the theory are observed due to the use of a first 1×2MMI instead of a 2×2MMI in the MZI. The coincidence measurement result is due to a reverse HOM quantum interference of degenerate photon pairs at the second MMI of the MZI, and its high visibility of 98.8% demonstrates the indistinguishability of the photon pairs generated by the two spirals [10,18]. We use the following formula for the visibility (V): V=Rcmax−RcminRcmax, where Rcmax and Rcmin are the maximum and minimum coincidence rates, respectively. In the Appendix A, such formulas are expressed in terms of coincidence probabilities in Eqs. (A29) and (A40). Then, we used the rings in the circuit of Fig. 1(b). However, because the micro heater of the phase shifter in the MZI is located too close (130 μm) to the microring, a large thermal cross talk caused the microring resonance to shift out of resonance from the pump wavelength when the phase ϕ was varied. This impeded the measurement.

The circuits of SiPIC-1 in Fig. 1(c) and of SiPIC-2 in Fig. 2(a) allow comparing the quantum interference of photon pairs produced by microring resonators or waveguides. Photons are generated in the first MZI, while quantum interference measurements are performed by changing the phase ϕ2 of the second MZI. Actually, this corresponds to a complex HOM measurement sequence [a reverse HOM in the first MZI and a sequence of an HOM and a reverse HOM in the second MZI, see in particular Fig. 2(a)].

Let us first consider SiPIC-1. Given the CAR and gh(2)(0) values measured for the two RingOn and RingOff configurations (Fig. 4), the pump power was set to 1 mW for RingOn (CAR=19, equivalent to 5% of accidental counts) and to 3.2 mW for RingOff (CAR=24, equivalent to 4% of accidental counts). Figure 7(a) shows the measured classical-light transmissions from channel 1 (ch 3 detector) and channel 2 (ch 4 detector) of the second MZI as a function of ϕ2. Figures 7(b) and 7(c) show the measured coincidence rates between channel 1 and channel 2 as a function of ϕ2 in the RingOn and RingOff configurations, which correspond to the quantum interference of photons generated in rings and in spirals, respectively. Note that to compensate for slight variations in the photon-pair generation rate due to the thermal cross talk (heat flow from the micro-heater on the phase shifter in the second MZI to the first MZI), we normalized the coincidence rates between ch 3 and ch 4 to the coincidence rates between ch 1 and ch 2 [see Fig. 1(c)].

As expected and discussed in the Appendix A, the classical light transmissions behave as cos2(ϕ2/2) or sin2(ϕ2/2), while the coincidence rates behave as cos2(ϕ2). For the RingOn configuration [Fig. 7(b)], a visibility of 80% is observed, which is lower than the 98.8% visibility measured for the RingOff configuration [Fig. 7(c)]. The visibility for the RingOff configuration is high considering the presence of 4% of accidental coincidences at a CAR=24. These additional coincidences are due to the classical interference of accidental photons coming from the residual pump, whose interference depends on the phase difference accumulated by the pump photons propagating from the first MZI to the second MZI. Therefore, considering a measured ratio of the ch 3 and ch 4 counts of about 3% for ϕ2=π/2 and 52% for ϕ2=π implies that accidental coincidences are reduced at ϕ2=π/2, yielding a high HOM visibility for the RingOff configuration as observed in Fig. 7(c).

Figure 8 shows the measured HOM interferences for the SiPIC-2 when the phase ϕ2 of the second MZI is varied [Fig. 2(a)]. For the RingOn configuration [Fig. 8(a)], an HOM interference visibility of 94% is measured when a pump power of 0.3 mW (corresponding to a CAR=74 in Fig. 5) is used. The RingOn visibility decreases to 89% for a pump power of 0.6 mW (CAR=15). Accidental counts impact at 1.3% for a CAR=74 and 6.3% for a CAR=15. So the 94% HOM interference visibility is equivalent to a 95.3% visibility when the accidental noise is removed. For the RingOff configuration [Fig. 8(b)], HOM interference visibilities of 99% for 0.6 mW pump power (CAR=557) and of 91% for a pump power of 5 mW (CAR=12) are observed. The accidental counts impact at 0.2% for a CAR=557 and 7.7% for a CAR=12. So the 99% HOM interference visibility is equivalent to 99.2% visibility when the accidental counts are excluded. Measurements show again that the HOM interference visibility depends on the CAR of the measured photon pairs. In addition, accidental photons due to residual pump classical interference yield a measured ch 1 to ch 2 ratio of 97% for ϕ2=0 and of 90% for ϕ2=π/2, showing the effective role of the pump filters and less relevance of this noise for SiPIC-2 than for SiPIC-1. Finally, the better HOM interference visibilities for SiPIC-2 rather than the HOM interference visibilities for SiPIC-1 are related to the enhanced CAR due to the integrated pump-filters and to an improved fabrication uniformity of the microrings in SiPIC-2 as compared to SiPIC-1.

The two types of photon-pair sources (waveguides and microring resonators) were simulated taking into account the input pump photon spectra as well as the band-pass filters (BPFs) at the output channels. The joint spectral intensity (JSI) of the generated pairs was calculated, as detailed in Appendix A.1. Figure 9 shows the JSI for the different sources in SiPIC-1 and SiPIC-2 (15-mm-long and 240-μm-long spiral waveguides, and microring resonators with Q-factors 1.5×104 and 3×104). In particular, it is worth noting that even in a degenerate SFWM process the generated photon pair wavelengths have a dispersion around λs≃λi due to the spectral width of the pump laser lines and the generation band of the FWM process.

The different shapes of the JSI for the photon pairs generated by the spiral waveguides [Figs. 9(a) and 9(b)] are mainly due to their different lengths. The longer waveguide acts as an additional narrow band-pass filter that modifies the JSI. This modification can be understood from the phase matching function reported in Eq. (A10) and contained in the joint spectral amplitude (JSA), Eq. (A9). It is also evident that the JSI shapes of the waveguides are wider than the shape of the microring resonators. This implies a higher correlation in the generated photons from the waveguides than from the microring resonators. These wide JSIs can be explained by the filtering of the broad waveguide generation band (between the two pump photon wavelengths) by the BPF. Filtering also lowers the photon-pair source brightness. In the case of the microring resonators, the effect of the BPF is negligible because of the narrow resonance spectral widths of the microrings. Moreover, their high Q-factor values make their JSIs narrow and increase their photon-pair generation probabilities, which depend on a cubic law of the Q-factor value [38]. However, high Q-factor values make the microring resonator more sensitive to thermal cross talk and TPA.

Table 1 summarizes the measured and simulated properties of the two types of sources considered in this work. In the simulations for the JSI and purity, the two waveguides/microring resonators are assumed both identical to their nominal design values. This implies that the simulated values are not affected by the small variations of the nominal parameters present in the fabricated structures. The purity has been calculated by using Eq. (A4) and the JSA overlaps have been calculated using the experimental results of the visibility, following the procedure discussed in Appendices A.2 and A.3, and more specifically Eq. (A33).Table 1.

Summary of the Relevant Properties of the Photon-Pair Sources in SiPIC-1 and SiPIC-2

In Appendix A.4, we discuss how unwanted residual pump noise can be included into the description of the visibility in terms of the JSA overlaps. The qualitative effect consists in lower values of the visibility because of lower values of the signal-to-noise ratio. This behavior is found experimentally in Fig. 8, where we observe different values of visibility for different CAR values using an asymmetric MZI (a-MZI) to filter the residual pump. In Table 1, we report the best values of observed visibility, and the computation of the JSA overlaps does not take into account the noise contribution discussed in Appendix A.4. Thus, the reported value of the computed JSA overlaps is a lower bound, since the assumption of negligible noise contribution might not be correct for our experiments.

In the case of the waveguide source, the purity goes from a simulated value of 20% in the unfiltered case to the value reported in Table 1 when a BPF is used. The simplicity in the fabrication of a waveguide source has its bottleneck in the necessity to use a high-quality BPF to produce photons with high spectral purity. Indeed, the results reported in the Table 1 show similar values for the waveguide devices of SiPIC-1 and SiPIC-2 despite that the fabrication of the latter resulted in devices with better performances. Indeed, the features of the microring resonators greatly improved from SiPIC-1 to SiPIC-2 due to a better fabrication uniformity achieved with the second process.

One may wonder why the visibility in the case of the waveguide is larger than the visibility of the microring resonator case, even if the purity shows an opposite trend. This point constitutes one of the main interesting aspects of the experiments described in Sections 4 and 5. In many published experiments the sources are independent [20,21,24], while in our case the two sources are not independent, and therefore the visibility is not linked to the purity of the individual sources. For example, in the case of independent sources, one typically uses non-degenerate SFWM to generate two pairs of idler and signal photons. The wave function is the product of the two pair wave functions, and the heralding procedure is the partial trace with respect to the heralding twins. The visibility is then given by the purity of each heralded photon. For standard microring resonators this means that the visibility cannot exceed 92% [42,43]. Therefore, the property of being independent brings the overall wave function of the state to be separable, while in our case the state is not separable. To verify this, we can consider the state generated by the pair of sources, Eq. (A23) in the Appendix A for small squeezing (the meaning of the different symbols is given in the appendix), |Ψ⟩II∼∫dωidωs[F1(ωi,ωs)a^1†(ωi)a^1†(ωs)+F2(ωi,ωs)a^2†(ωi)a^2†(ωs)]|vac⟩,(1)where F1 and F2 are the JSAs of the two sources. Crucially, |Ψ⟩II is a superposition between pairs of photons generated in the two sources. What is generated in our case consists of a photon pair in a superposition of two paths, while in the case of independent sources and non-degenerate FWM, two pairs are generated, and after heralding, a state with one heralded photon in each path is obtained. The Appendix A shows how the visibility is connected to the indistinguishability of the sources, represented by the JSA overlap ∫dωidωsF1(ωi,ωs)⋅F¯2(ωi,ωs). In particular, given our definition of observed visibility, the visibility can be written as V=2∫dωidωsF1(ωi,ωs)F¯2(ωi,ωs)1+∫dωidωsF1(ωi,ωs)F¯2(ωi,ωs),(2)which is presented in Eq. (A33) and derived in Appendices A.2 and A.3. Therefore, even if the purity is low, what matters is how much the two sources are indistinguishable. This is quantified by the integral overlap between the JSA of the two sources, and its relation to the visibility is given by Eq. (A33). We note that for F1=F2 the state generated by the couple of sources is exactly the input state used to perform a reverse HOM experiment [see Eq. (1)], and Eq. (2) gives V=1. The result of this visibility proves that there is good matching between the spectral shapes of the photon pair generated by the waveguides in SiPIC-1 and SiPIC-2 and by the microring resonators in SiPIC-2. The microring resonators in SiPIC-1 show a significantly lower JSA overlap and a lower measured visibility. This can be because some geometrical parameters can be slightly different between the two microring resonators. Therefore, the variability in the performances of the microring resonators between the two SiPIC models gives an idea about the higher requirements on the fabrication process needed to reach a high level of uniformity in the components to assure proper indistinguishability of the different integrated sources when microrings are used. We note that the source purity is not affected by the Q-factor of the microring and reaches a theoretical maximum value of 92% [42,43].

We conclude this section by reporting some numerical results obtained by simulating small deviations of one of the two sources with respect to the nominal design values. In the case of the spiral-waveguide-based source, differences with respect to losses, length, and width of the waveguides can be responsible for a lower JSA overlap. These characteristics determine the generation band of the process since they enter into the phase matching function [Eq. (A10)]. Let us note that the BPFs select 100 GHz (50 GHz at 3 dB bandwidth) around 1550.12 nm as it is described by the filters’ spectral amplitude in Eq. (A33). The filtering acts as a quantum eraser [44], deleting the information about possible differences in the generation bands and making the structures less sensitive to deviations from nominal values. Thus, we expect that a narrower BPF will result in higher visibility. Given the spectra of our filters, from our simulations we do not observe any significant change in the JSA overlaps by varying the relative width of the waveguides. Indeed, by considering a width variation of 5% with respect to the nominal value, the JSA overlap only decreases to 99.98%. Instead, varying by 10% the effective length Leff of one of the two spiral waveguides with respect to the other, a JSA overlap of 98% is found. Comparing these values with Table 1, we infer that differences in propagation loss and/or length can be responsible for lower values of JSA overlaps. The case of the microring-resonator-based sources is not affected by the action of the filters, and even if their footprint is much smaller, deviations in the coupling coefficients and ring losses can have a big impact on the JSA overlap. In Ref. [45], it has been shown that devices uniformity within 1% is achievable by using advanced processes in the same PIC. In a microring resonator, the coupling gap uniformity plays a very important role [46–48], and it can determine a different coupling regime of the device. As an example, from our numerical simulations based on Eq. (A11), we observe that the JSA overlaps decrease to 82% for a 25% Q-factor deviation and to 67% for a 50% deviation. Figure 12 at the end of the Appendix A shows the simulated JSA overlaps as a function of the variations of Leff for the spiral waveguides and the variations of the Q-factors for the microring resonators. Finally, the outcomes of our numerical analysis support our experimental results, where variations of the nominal design feature have a stronger effect on the microring resonators by decreasing the JSA overlap. Clearly, for both kinds of sources, large variations result in an increase in their distinguishability, allowing an answer to the question “Where did the pair generate?” and suppressing the visibility of the HOM interference. Generally, we expect and observe that in the resonant case the impact of the non-uniformity of the devices is more severe.

We performed Hong–Ou–Mandel fringe interference experiments of degenerate photon pairs generated on-chip from nominally identical and non-independent probabilistic sources. Non-independency of the sources is achieved by putting them on the two arms of an MZI, while the HOM interference is realized by a second MZI. Through such a configuration, it is possible to simultaneously excite both sources and create a superposition of twin photons from the two sources, which is actually a path-entangled state. The photon-pair sources we studied and compared are microring resonators and waveguides. At best, we measured the on-chip HOM visibility of 94% for microring resonators and 99% for waveguides. The visibility of the HOM interference gives information about the indistinguishability of the twin photons generated, which is in turn the indistinguishability of the two nominally identical sources. Crucially, the non-independency of the degenerate photon-pair sources results in the visibility being limited only by the indistinguishability and not by the purity. As a matter of fact, this places us in an ideal situation to address the question that titles this work, whose answer is the following: in quantum silicon photonics, spiral waveguide results have better performances in indistinguishability than microring resonators. This result is quantified by the computed JSA overlaps of 98% with waveguides and 89% with microring resonators; see Table 1.

The purity of the generated photon pairs plays a significant role when the generated photons have to interfere with photons produced by other independent photon sources [20,21]. However, in a generic quantum application it is desirable to have large values for both the purity and the indistinguishability because of the requirement of high visibility for dependent as well as independent sources. In our devices, we have seen two opposite cases: on one hand, a high visibility and JSA overlap but a low purity and low brightness (waveguide-based photon-pair sources), while on the other hand a low visibility and JSA overlap but a high purity (microring based photon-pair sources). This prevents qualifying one source as better than the other a priori. However, in the NISQ era of quantum computing one can simply choose what is needed to solve a task. Indeed, NISQ involves small numbers of qubits and algorithms that are focused on solving specific problems. One example is the variational quantum eigensolver [49,50], where what matters is the degree of indistinguishability and not the purity parameter. More generally, in applications where we create a photon pair in a superposition between different sources’ paths, similarly to what happens in our SiPICs, the quantum interference is influenced by how much more indistinguishable the sources are. On the other hand, in applications like Boson sampling [51,52], where a large number of single (heralded) photons are required and the probability to inject a single photon state is proportional to the purity, one needs high values of purity. This brings us to the idea of application-tailored sources, i.e., sources designed for a specific algorithm and chosen to balance the required properties and the implementation efforts. Following this idea, the ideal would be to have a library of photon-pair sources classified according to their main characteristics and to choose the photon-pair source depending on the target class of problems and on the available fabrication resources.

From this point of view, our method can be understood as a quantifier for the indistinguishability of two sources, and it can be generalized for the indistinguishability test of more than two sources since the pair-wise test does not guarantee the overall indistinguishability of the sources [53,54]. The generalization can be achieved by parallelizing more sources and bringing the superposition of pairs into a suitable network of MZIs. Related to that, other applications of the experimental scheme and the theoretical model can be found in the many-particle quantum interference investigations [55–57].