1. Introduction

Integrated optical devices would enable practical and powerful implementations of all-optical signal processing and photonic quantum technologies [1,2]. The optical ring resonator is one of the key components which can be used in a broad range of applications such as the classical and quantum light sources’ generation and manipulation with its superior performance in the high nonlinear interaction strength [3,4] as well as multiple linear optical functions of filtering, routing, modulating, switching and biosensing [5–8]. With a small footprint, the ring resonator has become a fundamental building block that significantly enhances the integration density of the photonic chip. Compared with the fixed-coupling-coefficient ring resonator, a reconfigurable ring resonator implies adding tuning mechanisms to the ring, such as control of the resonant frequency, the free spectral range, and the quality factor. The tunable quality factor indicates the feasible control on the working bandwidth or the coupling condition being under-coupled, critically coupled or over-coupled. Therefore a single-AMZI coupled ring resonator can be qualified for a variety of optical processes with different linewidth or optimized for different classical and quantum nonlinear optical processes such as meeting critical coupling condition for continuous wave (CW) pump four-wave mixing(FWM) [9,10] or satisfying pump-pulse dependent coupling conditions for pursuing bright entangled photons which will be derived in this work. Meanwhile, for optimizing a specific nonlinear process, the ring’s quality factor must be tunable since an exact coupling condition cannot be reached by potential tolerances in standard micro-fabrications. Usually, the performance of the nonlinear process wherein multiple resonances interact and simultaneously respond to the fabrication deviation will be significantly affected by the deviation from an ideal coupling condition. For example, the critical coupling condition for the highest efficiency CW FWM requires low values of power coupling between the bus waveguide and the ring in the order of a few percent or less for silicon waveguide devices or other low-loss platforms such as SiN. The standard nm-scale fabrication variations in device dimensions can then significantly affect the power coupling and, hence, the efficiency of resonantly enhanced optical processes.

Therefore a precisely tunable solution to reconfigure the quality factor of microrings in a wide range is highly desired. Tunable rings have been reported, such as devices with MEMs actuators [11] and thermally-controlled transverse temperature gradient coupler [12]. However, the MEMS-actuated ring resonator requires a large footprint and is vulnerable to environmental factors. The temperature gradient coupler can be tuned across the critical coupling point resulting in a high-efficiency FWM, but a large shift of resonance wavelength accompanies with each tuning of the coupling coefficient. The Asymmetric Mach-Zehnder Interferometer (AMZI)-coupled ring resonator was first proposed by Barbarossa et al. in 1995 [13], wherein certain resonant modes can be suppressed through the interference of two possible coupling paths to the resonator [14,15]. Since then, this design has been experimentally demonstrated for realizing the ingenious optical sensor and tunable bandwidth filter [6,16–18]. Recently, several studies have adopted dual-AMZIs coupled resonator to enhance the coincidence efficiency [15,19], the PGR [19], and spectral purity of photons [20–22] as well as the efficiency of classical FWM [23]. The dual-AMZIs structure features in realizing independent coupling condition for the pump and generated photons by configuring the AMZI to one special point that only allows either the pump or the signal(idler) photon entering to the resonator. So such structure’s operating coupling coefficient is not tunable but fixed. In this work, we demonstrate a single-AMZI coupled resonator with reconfigurable quality factors to act as a optimization strategy for classical FWM processes and quantum light sources, enabling high-efficiency wavelength conversion under both CW and pulsed pump situation, as well as high-performance heralded single photons, two-photon source and multi-photon source, as shown in Fig. 1(a).

 

Fig. 1. (a) Schematic description of AMZI-coupled microring resonator and its promising use in optimizing FWM and SFWM process. (b)Theoretical FWM conversion efficiency(normalized) against the ratio between the resonator’s intrinsic and extrinsic quality factor under the CW pump and pulsed pump with different pulse bandwidth with assuming that the pump, signal and idler’s Q factors being identical. (c) Theoretical pulsed pump FWM conversion efficiency assuming that the pump and the signal’s(idler) Q factors are independent. (d) Theoretical two-photon generation rate against the ratio between intrinsic and extrinsic quality factor assuming four beams’ Q factors being identical. (e) Theoretical two-photon generation rate under pulsed pump assuming that the pump and the signal’s(idler) Q factors are independent.

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2. Results

2.1 Theoretical FWM and SFWM efficiency from a ring resonator

The ring resonator’s intrinsic and extrinsic quality factors decided by loss and coupling coefficient, respectively, satisfy

For classical FWM processes, considering that the depletion of the pump and signal are negligible during the generation of the idler beam, the conversion efficiency of CW FWM from a single-bus-waveguide coupled ring resonator has been proven to scale with eight power of enhancement factor [10] as Eq. (11)(see Appendix B)

According to Eq. 11, the conversion efficiency for CW pump FWM will get maximized when the pump, signal and idler are all at the critical coupling point (see Table 1). Assuming both intrinsic($Q_{int,v}$) and extrinsic($Q_{ext,v}$) quality factors for the four interacting beams are identical, the CW pump FWM conversion efficiency defined by the generated idler power over the input signal power varies with the ratio of intrinsic and extrinsic as shown in Fig. 1(b). In the under-coupling regime ($Q_{int,v}<Q_{ext,v},v=p,s,i$), the FWM efficiency drops off quickly. The efficiency also drops off but with a relatively slow speed in the over-coupling regime ($Q_{int,v}>Q_{ext,v}$). It is obvious that variations in both round-trip loss and coupling coefficient due to the fabrication process can decrease the FWM efficiency drastically, thus ring resonators with precisely controlled coupling coefficients are highly recommended.

Table 1. The optimal coupling condition for both FWM conversion efficiency and SFWM PGR depending on the pump laser and whether the quality factor of the pump is dependent with the signal/idler’s or not. The subscript v in the table represents p, s and i.^a

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The conversion efficiency of pulsed pump FWM is different from the CW case due to the impact of pump bandwidth as Eq. 17 (see Appendix C). Assuming both intrinsic($Q_{int,v}$) and extrinsic($Q_{ext,v}$) quality factors for the four interacting beams are identical, the conversion efficiency is shown in Fig. 1(b) with the pulse bandwidth ranging from 0.05 nm to 3 nm. The optimal coupling condition is related to the pulse bandwidth. When the pulse bandwidth is much larger than the resonance’s, the integral of Eq. 17 has an analytic solution (see Appendix C) and the optimized coupling point for pulsed pump FWM is at $Q_{int,v}=4Q_{ext,v}$ (see Tab.1).

Assuming another case that the enhancement factors for the signal and idler are identical but independent with the pump’s, the theoretical conversion efficiency with the pulse bandwidth at 0.4 nm is shown in Fig. 1(c). To obtain the maximum FWM efficiency, the pulse pump should be operated at the very over-coupling regime with the $Q_{int,p}/Q_{ext,p}\approx 8.3$; meanwhile, the converted idler beam should also be operated at the over-coupling regime but with a different condition of $Q_{int,i}/Q_{ext,i}\approx 1.7$ (see Table 1).

For spontaneous four-wave mixing processes(SFWM), entangled photon-pairs are generated by a pair of degenerate or nondegenerate pump lasers, the optimal coupling condition for the ring is a different case. For CW pump SFWM, the generated pairs rate scales with the quality factors [19,25]:

There have been several early theoretical works about pulsed pump SFWM. The literature [26,27] discussed the photon pairs generated from the resonator while neglecting the scattering losses in the rings which cannot be ignored in realistic experimental platforms. Vernon [28] included the losses and derived the joint spectral intensity of generated photon pairs without analysing the PGR. Their later work [29] theoretically demonstrated the trade-off between the single photon sources’ heralding rate and efficiency considering a pump pulse with bandwidth of the order of the resonator. Here, starting from the interaction Hamiltonian [30–32], we derived the pulsed pump PGR based on resonator enhanced SFWM and obtained the analytical solution at two special conditions that are the pulse bandwidth equalling to or being much lager than the resonator’s (see Appendix D). Further more, we thoroughly analysed the optimization of the PGR and other parameters including HE and CAR.

Assuming the enhancement factors for four beams being identical, we come to the conclusion that the PGR by pulsed pump SFWM maximizes at the over-coupling regime and the optimal ratio between the intrinsic and extrinsic quality factors gets larger when using a wider band pulsed pump, as shown in Fig. 1(d). When the pulse bandwidth equals to the resonator’s, the optimal coupling point is at $Q_{int,v}=2Q_{ext,v}$ (see Appendix Eq. 33), which is different from the CW case.

Just as Fig. 1(c), assuming the enhancement factor of the signal/idler is independent with the pump’s, we calculate the two-photon generation rate with pulse band width at 0.4 nm, as shown in Fig. 1(e). Obviously, the optimal coupling conditions for the pump and signal/idler are different, with $Q_{int,p}/Q_{ext,p}\approx 10.9$ and $Q_{int,s(i)}/Q_{ext,s(i)}\approx 4.6$ (see Table 1).

2.2 Experimental optimizing CW and pulse pumped four-wave-mixing

Experimental setups for FWM are shown in Fig. 2(a). Two tunable narrow-linewidth CW lasers served as the pump (1454 nm-1641 nm with a linewidth of 50 MHz) and signal (1500 nm-1630 nm with a linewidth of 0.4 MHz) are controlled by two separate polarization controllers (PC) to optimize coupling efficiency between the fiber arrays and gratings on the chip, and then coupled by 3 dB coupler to the input port of the fiber array. The total coupling loss of coupling into and out of the chip is measured to be 7.1 dB. The other output port of the 3 dB coupler is connected to a power meter for monitoring the power of the pump and signal before the chip. The converted idler alongside with residual pump and signal are also coupled out of the chip through vertical coupling between on-chip gratings and off-chip fiber arrays, then a Dense Commercial Wavelength Division Multiplexing(DWDM) separates the pump, signal and idler to three different channels for measurement. The PM2 is used to monitor the pump on resonance, and PM3 is used to measure the converted idler power. When scanning signal wavelengths across the resonant spectrum, the converted idler’s power is recorded. For the pulsed pump FWM experiment, we substitute CW pump by pulsed laser (linewidth 0.6 nm, repetition rates 60MHz) connected by the dotted line in Fig. 2(a). When measuring the efficiency of the pulsed pump FWM and scanning the signal beam across 16 DWDM channels, the converted idlers are measured by an Optical Spectrum Analyzer (OSA) instead of a power meter.

 

Fig. 2. (a) Setup for CW and pulsed pump FWM. PC, Polarization Controller; PM, Power Meter; DWDM, Dense Wavelength Division Multiplexer. OSA, Optical Spectrum Analyzer. (b) Simulations of ring resonator transmission spectrum and effective coupling coefficient. (c) Measured extinction ratio by varying the voltage applied on the AMZI. The inset is the transmission spectrum of the resonator operating at critical coupling point with a −30dB extinction ratio and 0.022nm linewidth. (d-f) CW and pulsed pump FWM experiment(with the signal power fixed at $$ 44 \mu \mathrm{W} $$): (d) Measured conversion efficiency as a function of the ratio $Q_{int,p}/Q_{ext,p}$ with CW pump power of 0.176 mW. The inset shows the measured conversion efficiency as a function of pump power. (e) Conversion efficiency as a function of $Q_{int,p}/Q_{ext,p}$ at an average pulsed pump power of 0.066 mW. (f) Measured conversion efficiency by scanning the signal wavelength across 16 resonant wavelengths of the resonator under pump power of 0.176 mW.

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The AMZI’s path difference $\Delta L$ in this work is designed to be precisely equal to the length of the ring resonator, then, all the ring resonances are modulated by the AMZI simultaneously to operate at the same coupling condition being over-coupled, under-coupled or critically coupled(The detail mechanism of this design is illustrated in the Appendix.). As shown in Fig. 2(b), the resonator’s resonant spectrum and the AMZI’s spectrum with two different phase values $\theta =0$ and $\theta =\frac {2\pi }{3}$ are presented, showing intuitively how the interferometer modulates the resonator’s coupling coefficient.

The calibrated quality factors with the tuning range from $4.3\times 10^{4}$ at the most over-coupling regime ($Q_{int,p}=2.69Q_{ext,p}$) to $9.2\times 10^{4}$ at the under-coupling regime($Q_{int,p}=0.19Q_{ext,p}$)are obtained after measuring 27 resonant spectra by setting different voltages from 0 V to $\sqrt {52}$ V on the AMZI’s thermal optical phase shifter(see Materials and methods and the complete transmission spectra are shown in Fig. 4 in the Appendix.). In the following measurements, we tune the AMZI’s phase to optimize FWM and SFWM process, but we adopt the corresponding $Q_{int,p}/Q_{ext,p}$ as the varying parameter for better understanding of the physics. It indicates that the bandwidth or the coherence time of the converted beams and entangled photons can be controlled which can flexibly match the requirement of practical applications. Furthermore, this design supplies a widely and continuously tunable loaded quality factor that dominates parametric beams’ resonance, hence taking key role in the nonlinear optical process inside the ring.

The extinction ratios are extracted from all of the resonant modes and nearly −30 dB of the extinction ratio can be achieved at a certain voltage as shown in Fig. 2(c) which indicates that critical coupling is approached closely as shown in the inset figure. By fitting the transmission spectrum with Lorentzian function, the total round-trip loss inside the ring is deduced to be around 6.3 dB/cm in contrast to the straight waveguide propagation loss of about 4 dB/cm, which is reasonable since the resonator’s round-trip loss includes propagation loss originated from material loss and waveguide bending loss and coupling loss originated from the coupler between the coupling waveguide and the ring resonator. The measured FSR is about 3.26 nm, matching the 200 GHz spacing of DWDM well.

For CW pump FWM, the voltage of the resonator is set at 7.5 V so that one resonance is around 1549.4 nm aligned with the pump laser. The pump power in the coupling waveguide estimated by taking the grating coupling loss into consideration is 0.176 mW and the pump resonance is monitored by PM2 to ensure an equal enhancement factor in each measurement. The signal is around the resonance of 1546 nm and the power is fixed at 44 µW. Since grating coupling loss and both on-chip and off-chip propagation losses are approximately identical for signal and idler beams, the FWM conversion efficiency can be defined by the ratio of measured idler output power and signal off-resonance output power. Figure 2(d) shows the measured conversion efficiency against the ratio $Q_{int,p}/Q_{ext,p}$ by tuning the AMZI’s phase, which fits well with the calculated results adopting the measured quality factors based on Eq. 11. The maximum efficiency is obtained when four beams are all nearly critically coupled, consisting with the above theoretical results. When the resonator operated at the under-coupling regime, the conversion efficiency drops drastically, which is the worst case to be avoided. Our sophisticated design with the continuous tuning of the quality factors ensures the precise operation status of the resonator and avoids the decrease of FWM efficiency caused by fabrication variations.

Then, operating the AMZI’s phase shifter at the voltage of $\sqrt {38}$ V to approach the optimal coupling condition (critical coupling), we measure this CW pump FWM conversion efficiency under different pump powers as shown in the inset of Fig. 2(d). The conversion efficiency reaches −30 dB at a CW pump power of 0.66 mW. Such a low-power operation and small-size design will allow for integrating hundreds of this source on a single silicon chip. The conversion efficiency grows quadratically at low pump power but grows slowly at higher power due to the buildup of free carriers absorption (FCA) and two photon absorption (TPA) which can be reduced by integrating a reverse-biased PIN junction across the resonator to sweep out generated free carriers induced by TPA and cut down the nonlinear loss of the device [33,34]. Compared with the −35 dB conversion efficiency using the pump power of 0.8 mW [35]and −33.8 dB conversion efficiency using the pump power of 1 mW [36], our design proves to be a pretty efficient method.

For the pulsed pump FWM, we also measure the conversion efficiency as a function quality factor ratio($Q_{int,p}/Q_{ext,p}$) demonstrated in Fig. 2(e). The repetition rate of the pulse is 60 MHz and the linewidth is 0.6 nm. The operating voltage of the AMZI’s phase shifter for optimizing pulsed pump FWM conversion efficiency is expected to be different from the CW case because the pulse FWM maximizes its conversion efficiency at the over-coupling regime. Since the most over-coupling point of this ring resonator is $Q_{int,p}=2.69Q_{ext,p}$ ( not reaching $Q_{int,p}=4Q_{ext,p}$ yet), the pulsed pump FWM will get maximized at this most over-coupling condition which is proved by the experiment results shown in Fig. 2(e). A strong over-coupling condition can be reached by increasing the single point coupling $k_{1}$ when $k_{1} <0.5$, then theoretically a 4.8 percent enhancement of maximized conversion efficiency is expected when the coupling condition meets $Q_{int,p}=4Q_{ext,p}$ compared to the maximized efficiency here.

At the most over-coupling condition of this ring resonator, the conversion efficiency as a function of average pump power was measured when the signal power keeps 44 µW, which is plotted in Fig. 2(e)’s inset. The conversion efficiency saturates at an average pump power of 0.33 mW, and the conversion efficiency is about −27.8 dB. The bandwidth of our pulsed laser is about 0.6 nm, which is much larger than the resonator bandwidth of about 0.0356 nm. Therefore, most of the pulsed pump directly passed though the coupled waveguide with only partial energy going into the resonator and working in the FWM process. We defined the effective pulse energy by the ratio $\frac {0.33\,mW}{60\,MHz}\frac {0.0356\,nm}{0.6\,nm }$, where 0.33 mW, 60 MHz and 0.6 nm is the average power, repetition rates and spectral width of the pulsed pump respectively, and 0.0356 nm is the spectral width of the resonator. In our case, the effective pulse energy is 0.33 pJ for gaining the −27.8 dB efficiency, which proves this process quite efficient [37]. This way to calculate the effective pulse energy is suggested to value the FWM efficiency under different pulse widths from different literatures.

Due to the periodicity of AMZI’s spectrum, i.e., all the resonances of the AMZI-coupled ring resonator can be simultaneously regulated to work at the same coupling condition. For example, the same critical coupling point for optimizing CW pump FWM or the same over-coupling regime for the optimizing pulsed FWM, we expect such high FWM conversion efficiency to occur for every resonance and multiple channel FWM can be enhanced simultaneously.

To measure pulsed FWM efficiency under different signal wavelengths, the converted idler beam is sent to an optical spectrum analyzer (OSA) instead of a PM as shown in Fig. 2(a). We measure the conversion efficiency by scanning the signal across 16 resonant wavelengths around the pump resonance as shown in Fig. 2(f). Taking the dependence of grating coupler loss on the wavelength into account, the maximized efficiency for each resonant signal beam sketched in Fig. 2(f) is further calculated to have a flatter distribution, indicating that all the FWM processes associated with 16 resonant wavelengths maintain a high conversion efficiency, which can be additionally used for parallel all-optical signal processing.

2.3 Experimental optimizing CW pumped two-photon sources

Figure 3(a) shows the experimental setup for the entangled photon pairs based SFWM utilizing the AMZI coupled resonator. A filter is used for suppressing the pump laser’s sideband noise before coupling the pump into the silicon chip. In SFWM, the third nonlinear interaction annihilates two pump photons while simultaneously generating two daughter photons, viz., the signal photon and the idler photon. Tuning pump laser to the resonant 1549.4 nm, the entangled photon pairs with multiple resonant frequency modes are generated, exported and then pass through the off-chip filter to separate from the pump photons. The PM1 is used to monitor the pump power. The DWDM separates the signal and idler photons. The single-photon detectors’ detection efficiency is about 62%, 64% and 53% for the three channels, respectively.

 

Fig. 3. (a) Setup for CW and pulsed pump SFWM. (b-d) Experimental two-photon results for continuous wave pump SFWM:(b) The measured coincidence as a function of $Q_{int,p}/Q_{ext,p}$ at an average CW pump power of about 0.091 mW. The inset shows the coincidence against the pump power at the optimal coupling point ($Q_{int,p}=1.60\;Q_{ext,p}$); (c) The measured CAR as a function of $Q_{int,p}/Q_{ext,p}$ and the pump power( inset figure ); (d) The HE against $Q_{int,p}/Q_{ext,p}$ corresponding to the coincidence in (b). (e-g) Experimental two-photon results for pulsed pump SFWM: (e) The measured coincidence rates as the function of $Q_{int,p}/Q_{ext,p}$ at the average pulse power of about 0.034 mW. The inset shows the measured coincidence rates with the fitting values as the function of pulse power; (f) The CAR against $Q_{int,p}/Q_{ext,p}$ and the pump power(inset) corresponding to the coincidence in (e); (g) The HE against $Q_{int,p}/Q_{ext,p}$ and the pump power(inset).

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Firstly, we show the enhancement of the two-photon rate. Figure 3(b) shows the measured and calculated two-photon coincidence rates as the function of $Q_{int,p}/Q_{ext,p}$ with the signal and idler photons being detected at 1539.6 nm and 1559.2 nm, matching the DWDM channel CH47 and CH23, respectively. As vividly shown, the experimental values agree well with the calculated, which indicates the coincidence rate maximizes at the slight over-coupling regime ($Q_{int,p}=1.60Q_{ext,p}$) and drops at other coupling conditions. At the optimal coupling point, the measured coincidence with the increased pump power is shown with a quadratic fit in Fig. 3(b)’s inset. The raw measured photon pairs coincidence of more than 22 000 Hz has been recorded by using the pump power of about 0.24 mW and the on-chip PGR by subtracting the total off-chip loss is about 1.1 MHz which reaches the top level among the reported results [33,38–44](see Table 2 in the Appendix).

Table 2. Recent results of photon pair generation using silicon microring resonators.

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Secondly, we accomplished CAR characterization [41,45,46]. The coincidence window is about 0.7 ns and the pump power is set as 0.091 mW. The measured CAR as shown in Fig. 3(c) tends to increase at a higher ratio of $Q_{int,p}/Q_{ext,p}$. The inset figure shows the CAR versus input CW pump power, with the highest value exceeding 4800 under the pump power of about 0.018 mW (the measured two-photon coincidence rate is 158 Hz). The measured CAR decreases at higher pump power as expected since the coincidence counts are quadratic in the pump power while the accidental coincidence counts are biquadratic in the pump power. At lower pump power, the CAR also decreases due to the dark counts(with the detector’s two channels being 1.0 kHz and 1.7 kHz respectively). At the pump power of 0.24 mW, the CAR is 327 with the measured coincidence exceeding 22 000 Hz. The reconfigurable ring gives a large tuning range for CAR and PGR, which can meet the requirement for different experiments.

Thirdly, we measured to show that the HE of single-photon sources can also be optimized. In Fig. 3(d), the preparation HE defined by $\frac {N_{cc}}{\eta _{s}N_{i}}$ is shown, where $N_{cc}$ is the measured photon pairs coincidence, $N_{i}$ is the single-photon rates for the idler and $\eta _{s}$ presents the total loss of the signal beam including detection efficiency, off-chip propagation losses and the coupling loss between the chip and the fiber array [29,41]. Generally, the HE increases as the $Q_{int,p}/Q_{ext,p}$ going up, since the theoretical photon extraction rate($\frac {Q_{tot,s}}{Q_{ext,s}}$) from the resonator increases with $Q_{int,s}/Q_{ext,s}$. The discrepancy between theoretical and experimental HE mainly comes from the linear term of the single photon counts in signal/idler channel [45,46], which degrades and saturates the experimental values, since the PGR against the pump power in log/log scale shows the linear fitting coefficient of about 1.91 (very close to the ideal coefficient 2), while the single signal/idler photon rate against the pump power in log/log scale with the linear fitting coefficient of about 1.48. The linear term in the single signal/idler photon rate may mainly come from the pump laser sideband noise, the incomplete filtering of the pump photon in the signal/idler channel and Raman scattering noise from the fiber device, which needs to be further studied. As shown in Fig. 3(d) inset, with the increase of pump power, the HE also increases when the resonator’s coupling condition is fixed at $Q_{int,p}=1.60Q_{ext,p}$. The above results suggest that the HE can also be optimized when being used as heralded single-photon sources.

2.4 Experimental optimizing pulse pumped two-photon sources

We substitute the CW pump with a pulsed laser (bandwidth of 0.4 nm and the repetition rate of 1 GHz ) as shown in the experimental setup of Fig. 3(a) to measure the generated photon pairs coincidence as the function of $Q_{int,p}/Q_{ext,p}$, which follow the theoretical values calculated by the Eq. (31). As the ratio of quality factors goes up, the measured coincidence also increases and maximizes at the most over-coupling point($Q_{int,p}=2.69Q_{ext,p}$) in our system, which can be further increased by tuning the ratio $Q_{int,p}/Q_{ext,p}$ to a larger value, since the optimal coupling condition for pulsed pump SFWM with pulse bandwidth of 0.4 nm is at the over-coupling regime of about $Q_{int,p}=7.3Q_{ext,p}$ as shown in Fig. 1(d). If we design the AMZI coupled ring resonator with a larger single point coupling coefficient $k_{1}$, the effective quality factor’s tuning range should be enlarged to reach the maximum two-photon rate. We also measure the photon pairs coincidence as the function of the average power with the quadratic fit. Under the optimal coupling condition of $Q_{int,p}=2.69Q_{ext,p}$, the coincidence rates exceed 36 000 Hz using the on-chip pump power of about 0.091 mW, as shown in the Fig. 3(e) inset which proves this two-photon source quite efficient.

Figure 3(f) and the inset shows the measured CAR against the $Q_{int,p}/Q_{ext,p}$ and the pulse pump power, corresponding to Fig. 3(e) and the inset, respectively. The CAR tends to increase with increased $Q_{int,p}/Q_{ext,p}$ and will decrease at higher pump powers, as expected. The highest CAR exceeding 4000 was measured under the pump power of 5 µW with the two-photon coincidence 134 Hz.

Figure 3(g) shows both the experimental and theoretical HE as the function of $Q_{int,p}/Q_{ext,p}$. The data discussion is similar with Fig. 3(d).

3. Discussions

This design of AMZI-coupled ring resonator is competent for a variety of third-order nonlinear optical materials, including silicon [47], silicon nitride [48], etc. to optimize the performance of parametric processes inside a ring resonator. For (S)FWM in other third-order nonlinear optical resonators, this AMZI design is easily popularized. An improved power enhancement from (S)FWM in silicon nitride is expected since there is no TPA in this material, and a higher pump power can be applied. Operating crystalline silicon photon-pair sources in the mid-IR band is another promising route to avoiding TPA [49].

The fabrication lab or foundry may offer different round-trip loss and deviations from the designed coupling efficiency by fabrication variations, i.e., the intrinsic and extrinsic quality factors vary from one run to another run and even vary from one unit to another on the same chip. By designing such an AMZI coupled ring resonator, the fabrication variations can be overcome by this electrically tunable solution, thus the parametric process inside the resonator will get optimized with a single ring resonator. In addition, the wide tuning of coupling condition allows a variety of nonlinear processes to be optimized to their best performance for different applications, thus this tunable solution is a optimization strategy. This lower power operation and small footprint design will enable high-density integration of these sources on a single chip, which will pave the practical applications of all-optical signal processing, frequency multiplexed heralded single-photon sources for quantum key distribution [50,51] and wavelength-division multiplexing schemes for entanglement-based quantum communication systems [52].

Appendix A. Device design and fabrication

Figure 1(a) is the sketch of the AMZI-coupled ring resonator which contains two coupling points and its equivalence of straight waveguide side-coupled resonator with tunable coupling coefficient. To understand how this design operates, we first think of the individual components. Being a cavity, the ring will only support specific wavelengths of light meeting the resonant condition $2\pi Rn_{eff}=m\lambda (m=0,\pm 1,\pm 2\cdots )$, whose resonant frequency is separated by the free spectral range (FSR) $\omega _{FSR}=\frac {V_{g}}{R}$, where $R$ is radius of the ring resonator and $n_{eff}$ is the wavelength dependent effective refractive index. The spectrum of an AMZI is sinusoidal with the difference in optical path length between the two paths whose FSR is decided by $\Delta L n_{eff}=n\lambda (n=0,\pm 1,..)$, where $\Delta L$ is the path difference between two paths of AMZI. By considering the cavity and the AMZI together and using Transfer-Matrix method [53], the input (before the first coupling point of AMZI to the ring) and output amplitude (after the second coupling point of AMZI to the ring) of this device are related by the matrix:

(5)$$e^{-\alpha l_{1}/2-i \beta l_{1}}\left(\begin{array}{cc}{-k_{1}+ t_{1}e^{{-}i\phi}} & {i\sqrt{k_{1}t_{1}}+i\sqrt{k_{1}t_{1}} e^{{-}i\phi}}\\ {i\sqrt{k_{1}t_{1}}+\sqrt{k_{1}t_{1}}e^{{-}i\phi}} & {-k_{1} e^{{-}i\phi}+ t_{1}}\end{array}\right)$$

with $\phi =\beta \Delta L+\theta -i\Delta L\alpha /2$, where $l_{1}$ is the length of the short arm, $\beta$ is the propagation constant of the waveguide and $\theta$ is the phase difference between AMZI’s two paths including initial phase difference and electrically tunable phase. $k_{1}$ and $t_{1}$ represents power coupling and transmission coefficient of single coupling point with $k_{1}+t_{1}=1$, so effective power coupling coefficient can be expressed as [14,15]

(6)$$k(\theta)=e^{-\alpha l_{1}}k_{1}(1-k_{1})(1+e^{-\Delta L\alpha}+2e^{-\Delta L\alpha/2}cos(\beta\Delta L+\theta)).$$

Tuning $\theta$ determines the constructive and destructive interference between the ring resonator’s and the AMZI’s transmission spectrum, thus tuning the coupling coefficient effectively. So an AMZI coupled ring resonator is equivalent to a single waveguide side-coupled ring resonator but with a tunable coupling coefficient, varying from the maximum value of $4k_{1}(1-k_{1})$ to the minimum value of 0 ($\Delta L\alpha \ll 1$). It is easy to deduce that when the AMZI’s path difference is not the integers of $2\pi R$, the adjacent two resonant modes may be affected by the AMZI differently, showing different Q-factor tuning characters which will be useful in some nonlinear optical processes [19,22,23].

The AMZI coupled resonators are fabricated on a silicon-on-insulator (SOI) chip. The radius of the ring is 28 µm with the waveguide’s cross-section height and width being 220 nm and 500 nm, respectively. The AMZI long path just used the common bends with a radius of 17 µm connected with straight waveguide of 69 µm and the narrowest gap of the coupling region is 180 nm. The path difference between AMZI’s two paths is 179 µm which is identical with the resonator’s circumference length, ensuring that the effective coupling coefficients for the adjacent resonance wavelengths are identical. Thus, the quality factors for the pump, signal and idler beams are approximately the same and can be modulated simultaneously with the thermo-optic phase shifter integrated on the waveguide of AMZI’s long arm. A phase shifter also tunes the ring’s resonance to align the resonant wavelength with the channel of DWDM.

In the main text, we give the tuning range of the resonator’s quality factors by operating the AMZI phase shifter’ voltage. Here, we give the detail transmission spectra and its’ full width at half maximum(FWHM) at each voltage. As shown in Fig. 4, the FWHM firstly gets lager and then decreases, which indicates the resonator experiences from overcoupling to undercoupling.

 

Fig. 4. The transmission spectra. $V^{\land }$2 represents the square of the voltage operating at the AMZI’s phase shifter. $\Delta \lambda$ is the transmission spectraum’s FWHM.

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Appendix B. The key parameters of the quantum light source compared with other works

In the main text, recent results of photon pair generation using silicon microring resonators were not given. Here, we compared the on-chip PGR and CAR optimized in this work with others, as shown in Table 2.

Appendix C. Theoretical derivations of FWM in a ring resonator

Appendix C.1 CW pump FWM

Although the theory for CW pump FWM is clear and has been experimentally verified in other works [10], we give simple derivation as the base for the following discussion and the pulsed pump FWM process. The coupled-mode equation in the CW pump FWM process is given as

(7)$$\begin{aligned} \frac{d}{d z} a_{\mathrm{p}}(\omega_{p},z)&=(i \beta_{\mathrm{p}}-\alpha_{p}/2 )a_{\mathrm{p}}(\omega_{p},z),\\ \frac{d}{d z} a_{\mathrm{s}}(\omega_{s},z)&=(i \beta_{\mathrm{s}}-\alpha_{s}/2) a_{\mathrm{s}}(\omega_{s},z),\\ \frac{d}{d z} a_{\mathrm{i}}(\omega_{i},z)&=(i \beta_{\mathrm{i}}-\alpha_{i}/2) a_{\mathrm{i}}(\omega_{i},z)+i \gamma a_{\mathrm{p}}(\omega_{p},z)a_{\mathrm{p}}(\omega_{p},z) a_{\mathrm{s}}^{*}(\omega_{s},z), \end{aligned}$$

where $a_{p},a_{s},a_{i}$ represent the intra-cavity amplitudes of the pump, signal, and idler at a single frequency. $\gamma$ is the nonlinear parameter defined by $\frac {2\pi }{\lambda }\frac {n_{2}}{A_{eff}}$, where $n_{2}$ is the Kerr nonlinear coefficient, $\lambda$ is the wavelength and $A_{eff}$ is the effective mode area. $\beta _{v}$ is propagation constants of the pump, signal and idler beams with the phase mismatch given by $\bigtriangleup \beta =2\beta _{p}-\beta _{s}-\beta _{i}-2\gamma P$, where $-2\gamma P$ originates from self-phase modulation and cross-phase modulation. Considering the enhancement of ring resonator, the input bus-waveguide power amplitude($S_{in}$) and intra-cavity power amplitude are related by

(8)$$a_{v}(\omega_{v},0)=f_{v}(\omega_{v})S_{in,v}(\omega_{v}) \quad(v=p, s),$$

In the following, we assume the total round-trip loss and coupling coefficient are identical for the four interacting waves($\alpha =\alpha _{p,s,i}$). Then Eq. 7 can be solved as

(9)$$\begin{aligned} a_{\mathrm{p}}(\omega_{p},z) &=a_{\mathrm{p}}(\omega_{p},0) e^{\left(i \beta_{\mathrm{p}}-\alpha/2\right) z},\\ a_{\mathrm{s}}(\omega_{s},z) &=a_{\mathrm{s}}(\omega_{s},0) e^{\left(i \beta_{\mathrm{s}}-\alpha/2\right) z},\\ a_{\mathrm{i}}(\omega_{i},z)&=e^{\left(i \beta_{\mathrm{i}}-\alpha/2\right) z}[a_{\mathrm{i}}(\omega_{i},0)+i \gamma a_{\mathrm{p}}^{2}(\omega_{p},0) a_{\mathrm{s}}^{*}(\omega_{s},0) \int_{0}^{z} dze^{-\alpha z} e^{i \Delta \beta z}], \end{aligned}$$

The converted idler power output from the resonator is related with the transmission matrix,

(10)$$\begin{pmatrix} S_{v,out}(\omega_{v})\\a_{k}(\omega_{v},0) \end{pmatrix}=\begin{pmatrix}t & ik\\ik & t\end{pmatrix}\begin{pmatrix}S_{v,in}(\omega_{v})\\a_{v}(\omega_{v},L)\end{pmatrix}(v=p,s,i)$$

and can be further deduced as

(11)$$\eta_{FWM}= \frac{\left|S_{\mathrm{i,out}}(\omega_{i})\right|^{2}}{\left|S_{\mathrm{s,in}}(\omega_{s})\right|^{2}}=(\gamma PL_{eff})^{2}\left|\mathrm{f}_{\mathrm{p}}(\omega_{p})\right|^{4}\left|\mathrm{f}_{\mathrm{s}}(\omega_{s})\right|^{2}\left|\mathrm{f}_{\mathrm{i}}(\omega_{i})\right|^{2},$$

where $L_{eff}=e^{-\alpha L}\left |\frac {e^{-\alpha L}e^{i\bigtriangleup \beta L}-1}{-\alpha L+i\bigtriangleup \beta L}\right |^{2}$ and $P= S_{p,in}(\omega _{p})$.

Obviously, the optimal coupling condition for CW pump FWM is critical coupling and the theoretical converted efficiencies are identical for the two situations decided by whether the pump’s enhancement factor is independent with the signal/idler’s or not.

Appendix C.2 Pulsed pump FWM

Here we derive pulsed pump FWM conversion efficiency for microring resonators in the frequency domain based on the coupled-mode equation

(12)$$\begin{aligned} \frac{d}{d z} a_{\mathrm{p}}(\omega_{p},z)&=(i \beta_{\mathrm{p}}-\alpha_{p}/2 )a_{\mathrm{p}}(\omega_{p},z),\\ \frac{d}{d z} a_{\mathrm{s}}(\omega_{s},z)&=(i \beta_{\mathrm{s}}-\alpha_{s}/2) a_{\mathrm{s}}(\omega_{s},z),\\ \frac{d}{d z} a_{\mathrm{i}}(\omega_{i},z)&=(i \beta_{\mathrm{i}}-\alpha_{i}/2) a_{\mathrm{i}}(\omega_{i},z)+i \gamma\int d\omega_{p} a_{\mathrm{p}}(\omega_{p},z)a_{\mathrm{p}}(\omega_{s}+\omega_{i}-\omega_{p},z) a_{\mathrm{s}}^{*}(\omega_{s},z), \end{aligned}$$

So just similar to the derivation for CW pump FWM, the output idler power amplitude of a pulsed pump FWM is given by

(13)$$\begin{aligned} &S_{i,out}(\omega_{i})=i\gamma e^{(i\beta_{i}-\alpha/2)L}\frac{e^{-\alpha L}e^{i\bigtriangleup\beta L}-1}{-\alpha +i\bigtriangleup \beta }f_{s}^{*}(\omega_{s})S_{s,in}^{*}(\omega_{s})f_{i}(\omega_{i})\\ &\times\int d\omega_{p1} f_{p}(\omega_{p1})f_{p}(\omega_{p2})\times S_{in,p}(\omega_{p})S_{in,p}(\omega_{s}+\omega_{i}-\omega_{p1}), \end{aligned}$$

The frequency distribution of the pulsed pump takes the form of Gaussian as an approximation to the shape of the mode-locked laser pulse

(14)$$S_{p,in}(\omega_{p})=\sqrt{\frac{P_{0}}{\sqrt{\pi}\sigma}}e^{-\frac{(\omega_{p}-\omega_{p0})^{2}}{2\sigma^{2}}},$$

where $P_{0}$ is the average power of the pulse, $\omega _{p0}$ is the central frequency and $\sigma$ relates to the frequency bandwidth($\Delta \omega =2\sqrt {ln2}\sigma$). The final output idler power is given by

(15)$$P_{idler}=\int S_{i,out}^{2}(\omega_{i})d\omega_{i},$$

with the conversion efficiency defined by

(16)$$\eta=\frac{p_{idler}}{P_{signal}}.$$

Substituting Eq. 13, Eq. 14, Eq. 15 to Eq. 16, we finally get the conversion efficiency of pulsed pump FWM

(17)$$\begin{aligned} \eta&=(\gamma L_{eff}P_{0})^{2}\left|f_{s}(w_{s})\right|^{2} \int d \omega_{i}|f_{i}(w_{i})\int dw_{p}f_{p}(w_{p}) f_{p}(w_{s}+w_{i}-w_{p})\\ &\times G_{p,in}(w_{p}) G_{p,in}(w_{s}+w_{i}-w_{p})|^{2}, \end{aligned}$$

where $G_{p,in}(\omega _{p})=S_{p,in}(\omega _{p})/\sqrt {P_{0}}$.

In the main text, we analyzed the optimal coupling condition for pulsed pump FWM in two situations decided by whether the pump’s enhancement factor is independent of the signal/idler’s or not. The maximized conversion efficiencies achieved in the two situations differ. Actually, the theoretical maximum efficiency in the independent situation is twice larger than the other situation.

If the pulse bandwidth is much lager than the resonance’s, the resonator’s group dispersion is negligible and the enhancement factors for the four interacting beams are identical ($r_{ext}=r_{ext,v},\;r_{tot}=r_{tot,v}\;(v=p,s,i$)), then Eq. 17 can be analytically solved

(18)$$\eta=(\gamma L_{eff}P_{0})^{2}(\frac{2V_{g}}{L})^{4}\frac{\sqrt{2^{3}\pi^{3}}}{3\sigma^{2}\omega_{p0}}\frac{ Q_{tot}^{5}}{Q_{ext}^{4}},$$

Thus, the optimal coupling condition in this situation is $Q_{int}=4Q_{ext}$.

Appendix D. Theoretical derivations of SFWM in a ring resonator

Appendix D.1 CW pump SFWM

We will deduce the entangled photon pairs rate from a ring resonator under the CW pump starting from the interacting Hamiltonian

(19)$$H(t)=\frac{3 \varepsilon_{o}}{4} \int d r \chi^{(3)} E_{s}^{(-)}(r, t) E_{i}^{(-)}(r, t) E_{p}^{(+)}(r, t) E_{p}^{(+)}(r, t)+h . c .$$

where $\varepsilon _{o}$ is vacuum permittivity and $\chi ^{(3)}$ is the third nonlinear coefficient. The signal, idler and pump’s fields are as follows:

(20)$$\begin{aligned} E_{j}^{(+)}(r, t)&=i \sqrt{\frac{\hbar \omega_{j0}}{2 \varepsilon_{o} n_{j} S c}} \int \frac{d \omega_{j}}{\sqrt{2 \pi}} \hat{a}(\omega_{j}) f_{j}(\omega_{j}) e^{i \beta_{j} z-i \omega_{j} t}(j=s, i)\\ E_{p}^{(+)}(r, t)&=\sqrt{\frac{2 P_{0}}{\varepsilon_{o} n_{p} S c}} f_{p}(\omega_{p}) e^{i \beta_{p} z-i \omega_{p} t} \end{aligned}$$

where $\hbar$ is the Planck constant, S is the effective waveguide area, c is the vacuum speed of light, $n_{j}$ is refractive index, $P_{0}$ is the average pump power and the $\hat {a}_{j}$ is the annihilation operator for the generated pairs. Substituting the Eq. 20 to Eq. 19 will get:

(21)$$\begin{aligned} H(t)&=\gamma P_{0}\frac{\hbar}{2\pi}\iiint dzd\omega_{s}d\omega_{i}f_{p}(\omega_{p})f_{p}(\omega_{p})f_{s}^{*}(\omega_{s})f_{i}^{*}(\omega_{i}) e^{i\Delta\beta L-i(2\omega_{p}-\omega_{s}-\omega_{i})t}\hat{a}^{+}(\omega_{s})\hat{a}^{+}(\omega_{i})+h . c .\\ &= \gamma P_{0}LSinc(\Delta\beta L/2)\frac{\hbar}{2\pi}\iint d\omega_{s}d\omega_{i}f_{p}(\omega_{p})f_{p}(\omega_{p})f_{s}^{*}(\omega_{s}) f_{i}^{*}\\ &\times(\omega_{i}) e^{{-}i(2\omega_{p}-\omega_{s}-\omega_{i})t}\hat{a}^{+}(\omega_{s})\hat{a}^{+}(\omega_{i})+h . c ., \end{aligned}$$

where $\gamma =\frac {3}{4}\frac {1}{\varepsilon _{o} n_{p} S c^{2}}\sqrt {\frac {\omega _{s0}\omega _{i0}}{n_{s}n_{i}}}$, which is consistent with the nonlinear coefficient of the classical FWM. The quantum state can be derived from the interaction picture

(22)$$\left|\psi\right\rangle={-}\frac{i}{\hbar} \int_{-\infty}^{+\infty} d t H(t)|0\rangle$$

The integral over time can be calculated as

(23)$$\int_{-\infty}^{+\infty} d te^{{-}i(2\omega_{p}-\omega_{s}-\omega_{i})t}=2\pi\delta(2\omega_{p}-\omega_{s}-\omega_{i}),$$

which gives the energy-conservation relation. In the next discussion, we will neglect the term of $Sinc(\Delta \beta /2)$, since the phase mismatch is quite small considering the signal/idler is near the pump’s resonant mode.

Finally, the output photon pairs rate from the resonator using the CW pump SFWM simplified as:

(24)$$\begin{aligned} N&=\left\langle\psi \mid\psi\right\rangle =(\gamma P_{0}L)^{2}\int d\omega_{s}\lvert f_{s}(\omega_{s})\rvert^{2}\lvert f_{i}(2\omega_{p}-\omega_{s})\rvert^{2}\lvert f_{p}(\omega_{p})\rvert^{4}\\ &=(\gamma P_{0}L)^{2}16\pi(\frac{V_{g}}{L})^{4}(\frac{r_{ext,p}}{r_{tot,p}^{2}})^{2}\frac{r_{ext,s}}{r_{tot,s}}\frac{r_{ext,i}}{r_{tot,i}}\frac{r_{tot,s}+r_{tot,i}}{\Delta\omega^{2}+(r_{tot,s}+r_{tot,i})^{2}}, \end{aligned}$$

where $\Delta \omega =2\omega _{p0}-\omega _{s0}-\omega _{i0}$ is the resonant frequency detuning of the pump, signal and idler, which will also be neglected in the next analyse. Thus, we get the simplified form:

(25)$$N=\frac{16 V_{\mathrm{g}}^{4} \gamma^{2} P_{\mathrm{0}}^{2} Q_{\mathrm{tot,p}}^{4}Q_{\mathrm{tot,s(i)}}^{3}}{\omega_{\mathrm{p0}}^{3} \pi R^{2} Q_{\mathrm{ext}, \mathrm{p}}^{2}Q_{\mathrm{ext}, \mathrm{s(i)}}^{2}},$$

When the pump’s extrinsic Q factor is independent with the signal/idler’s (i.e. the enhancement factors for the pump and signal/idler are independent), the optimal coupling condition is the pump at the critical coupling point($Q_{int,p}=Q_{ext,p}$) and the signal/idler at the over-coupling point($Q_{int,s(i)}=2Q_{ext,s(i)}$) [19,24]. When the enhancement factors for the pump and signal/idler are identical, the optimal coupling condition is the pump/signal/idler at over-coupling point ($Q_{int}=4/3Q_{ext}$). Particularly, the maximum two-photon rate achieved in the former situation is 10% lager than the latter.

Here we take the derivation of optimal coupling condition for PGR from Eq. (4) as an example. Taking the terms independent with the Q factors as a constant A, the PGR function will be simplified as $N_{cc}=A (Q_{tot}^{7})/(Q_{ext}^{4} )$. Substituting the relation $Q_{tot}=(Q_{int} Q_{ext})/(Q_{int}+Q_{ext} )$ into the simplified PGR function, then, we get $N_{c c}=A \frac {Q_{\text {int }}^{7} Q_{\text {ext }}^{3}}{\left (Q_{\text {int }}+Q_{\text {ext }}\right )^{7}}=A \frac {Q_{\text {int }}^{3}\left (Q_{\text {int }} / Q_{\text {ext }}\right )^{4}}{\left (Q_{\text {int }} / Q_{\text {ext }}+1\right )^{7}}$. Assuming the intrinsic Q factor $Q_{int}$ is fixed and the extrinsic Q factor $Q_{ext}$ is controllable, we take $x=Q_{int}/Q_{ext}$ as the independent variable to obtain the PGR function in the form : $N_{c c}=A Q_{\text {int }}^{3}\frac { x^{4}}{(x+1)^{7}}$ . Taking the derivative with respect to the variable x, the maximum PGR is achieved with variable x being at 4/3, that is $Q_{int}=4/3 Q_{ext}$ given in our main manuscript.

Appendix D.2 Pulsed pump SFWM

There are lacking both theoretical and experimental demonstration on the optimal coupling conditions for the pulsed pump SFWM. Thus, we will also deduce the PGR of the pulsed case from the interacting Hamiltonian equation. In the pulsed pump SFWM process, the signal and idler fields are the same as Eq.14 but the pump’s fields should be modified as follows:

(26)$$\begin{aligned} E_{j}^{(+)}(r, t)&=i \sqrt{\frac{\hbar \omega_{j0}}{2 \varepsilon_{o} n_{j} S c}} \int \frac{d \omega_{j}}{\sqrt{2 \pi}} \hat{a}(\omega_{j}) f_{j}(\omega_{j}) e^{i \beta_{j} z-i \omega_{j} t}(j=s, i)\\ E_{p}^{(+)}(r, t)&=\sqrt{\frac{2 W}{\varepsilon_{o} n_{p} S c}} \int \frac{d \omega_{p}}{\sqrt{2\pi}} f_{p}(\omega_{p}) l_{p}\left(\omega_{p}\right) e^{i \beta_{p} z-i \omega_{p} t}. \end{aligned}$$

where $W$ is the single pulse energy and $l_{p}$ is the normalized frequency spectrum that is $\int d\omega _{p} \lvert l_{p}(\omega _{p})\rvert ^{2}=1.$ Substituting the Eq. 26 to Eq. 19 just as the CW case, we will get:

(27)$$\begin{aligned} H(t)&=\frac{\hbar\gamma WL}{(2\pi)^{2}}\iiiint d\omega_{s}d\omega_{i}d\omega_{p1}d\omega_{p2}f_{s}^{*}(\omega_{s})f_{i}^{*}(\omega_{i}) f_{p}(\omega_{p1}) f_{p}(\omega_{p2})l_{p}(\omega_{p1})l_{p}(\omega_{p1})\\ &\times e^{{-}i(\omega_{p1}+\omega_{p2}-\omega_{s}-\omega_{i})t}\hat{a}^{+}(\omega_{s})\hat{a}^{+}(\omega_{i})+h . c . \end{aligned}$$

Finally, by substituting the Eq. 27 to Eq. 22, the two-photon wave function from the resonator using the pulsed pump SFWM is:

(28)$$\vert\psi\rangle=\gamma WL\frac{1}{2\pi}\iint d\omega_{s}d\omega_{i}\phi(\omega_{s},\omega_{i})\hat{a}^{+}(\omega_{s})\hat{a}^{+}(\omega_{i})\left|00\right\rangle,$$

where $\phi (\omega _{s},\omega _{i})$ is the two-photon spectral correlation function

(29)$$\phi(\omega_{s},\omega_{i})=I(\omega_{s},\omega_{i})f_{s}^{*}(\omega_{s})f_{i}^{*}(\omega_{i}),$$

and the integral of the pump frequency gives

(30)$$I(\omega_{s},\omega_{i})=\int d\omega_{p}f_{p}(\omega_{p})f_{p}(\omega_{s}+\omega_{i}-\omega_{p})l_{p}(\omega_{p})l_{p}(\omega_{s}+\omega_{i}-\omega_{p})$$

So the PGR per pulse is given as

(31)$$N=(\gamma WL\frac{1}{2\pi})^{2}\iint d\omega_{s}d\omega_{i}\lvert \phi(\omega_{s},\omega_{i})\rvert^{2}$$

Next, we will give the analytical solution of Eq. 31 at two special frequency bandwidth of the pulse.

Case I:The bandwidth of the pulse is the same as the resonator’s, assuming the pump pulse in the form of Lorentzian function $l_{p}(\omega _{p})=\frac {1}{\sqrt {\pi }}\frac {\sqrt {r_{tot,p}}}{i(\omega _{p}-\omega _{p0})+r_{tot,p}}$. The integral result of Eq. 30 is as follows:

(32)$$I(\omega_{s},\omega_{i}) = \frac{2V_{g}\pi}{L}\frac{r_{ext,p}}{r_{tot,p}}\frac{1}{\pi}\frac{2r_{tot,p}}{(2r_{tot,p})^{2}+(\omega_{s}+\omega_{i}-2\omega_{p0})^{2}.}$$

Assuming the quality factors of the pump, signal and idler are identical and substituting Eq. 32 and Eq. 29 into Eq. 31, we will final pairs rate per pulse:

(33)$$N=(\gamma WL)^{2}\frac{3}{8}(\frac{V_{g}}{L})^{4}\frac{r_{ext}^{4}}{r_{tot}^{6}} =\frac{3\gamma^{2}W^{2}V_{g}^{4}}{8\pi^{2}R^{2}\omega_{p0}^{2}}\frac{Q_{tot}^{6}}{Q_{ext}^{4}},$$

where $r_{ext}=r_{ext,v},\;r_{tot}=r_{tot,v},(v=p,s,i),\;\Delta \omega =0$. The optimal coupling condition in this case can be easily deduced, that is $Q_{int}=2Q_{ext}$.

Case II:The bandwidth of the pulse is much lager than the resonator’s, assuming the pump pulse in the form of Gaussian function $l_{p}(\omega _{p})=\sqrt {\frac {1}{\sqrt {\pi }\sigma }}e^{-\frac {(\omega _{p}-\Omega _{p})^{2}}{2\sigma ^{2}}}\;(2\sqrt {ln2}\sigma \gg 2r_{tot,p})$. The integral result of Eq. 30 is as follows:

(34)$$I(\omega_{s},\omega_{i}) = \frac{2V_{g}\pi}{L}\frac{r_{ext,p}}{r_{tot,p}}\frac{2r_{tot,p}}{2r_{tot,p}-i(\omega_{s}+\omega_{i}-2\omega_{p0})} l(\omega_{p0})l(\omega_{s}+\omega_{i}-\omega_{p0})$$

Just as the dealing process of Case I, the PGR is given as follows:

(35)$$N=(\gamma WL)^{2}\frac{2\pi}{\sigma^{2}}(\frac{V_{g}}{L})^{4}\frac{r_{ext}^{4}}{r_{tot}^{4}} =\frac{\gamma^{2} W^{2} V_{g}^{4}}{2\pi R^{2}\sigma^{2}}\frac{Q_{tot}^{4}}{Q_{ext}^{4}}$$

So the optimal coupling condition will gets lager as the pulse bandwidth increasing, as shown in Fig. 1(b).

Similar to the pulsed pump FWM in the main text, we also analyzed the optimal coupling condition for pulsed pump SFWM in two situations decided by whether the pump’s enhancement factor is independent with the signal/idler’s or not. The maximum two-photon generation rates achieved in the two situations differ a little. Numerical calculations show that the theoretical maximum rate pumped by 0.4 nm bandwidth pulse in the independent situation is about 14% larger than the other situation.

We emphasize that our calculation is general for third nonlinear materials including silica, silicon, silicon nitride, etc. The PGR function (Eq. (24), Eq. (33), Eq. (35)) will supply quite accurate estimation value for quantum light source experiment.

Funding

National Natural Science Foundation of China (No. 11627810, No. 11690031, No.61632021); National Key Research and Development Program of China (No. 2017YFA0303700, No.2019YFA0308700).

Acknowledgments

The authors acknowledge the Open Funds from the State Key Laboratory of High Performance Computing of China(NUDT).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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