1. INTRODUCTION

Progress toward practical quantum computational platforms is accelerating, with increasing competition from various industrial efforts [1–4]. Quantum computers promise the ability to efficiently simulate complex quantum systems [5,6], and solve classically infeasible cryptography challenges [7] among other applications. This is in direct contrast to the lengthy and comparatively inefficient computations of our classical methodologies [8]. Quantum photonics is one such platform, and further benefits from its mass-manufacturability (integrated photonics) on chip-scale devices [9,10]. From an applications perspective, it enables the likes of large-scale entangled quantum networks [11,12], next-generation sensing [13], and ultra-precise measurements [14]. The integrated photonics platform grows ever more promising with the developments of many fundamental building blocks needed for a photonic quantum computer [15–20], and is possibly the only platform capable of reaching the number of qubits necessary for true fault-tolerance [21].

Much of the theory surrounding a photonic quantum processor postulates ideal quantum resources [22] for the purposes of their architecture [23,24], as well as for any error-correction schemes [25–27]. To create a photon source in line with these requirements we need high heralding efficiencies and high purities. High-brightness sources are more of a practical requirement, as brightness determines the time frame on which a given computation can be performed [17], with larger circuits needing brighter sources to maintain similar coincidence rates. Together, these metrics maximize the generation probability of sources, which is key for interfering photons from multiple sources, as well as maximizing any subsequent interference between them, making results like [28] achievable. Recent examples targeting improved integrated source metrics showcase versatile [29] and bright sources [30] of indistinguishable photons with limited heralding efficiency, or high-efficiency sources that trade brightness for purity, and are limited to broadband operation [15], leaving them susceptible to additional noise inside the wider filtering band.

Typically, the nonlinear process of spontaneous four-wave mixing (SFWM) is used to probabilistically generate pairs of photons (conventionally referred to as signals and idlers) on the CMOS-compatible silicon-on-insulator (SOI) platform. Devices can be designed to optimize the brightness of this process through long interaction lengths, manifesting as long sections of waveguides [31], or high-field strengths in optical cavities [32], with the under coupled and critically coupled regimes resulting in lower escape efficiencies [33]. These sources use the process of heralding [34] to mitigate the probabilistic nature of SFWM, with valid detection events occurring only with two or more coincident photons. The consequence of heralding is that, due to the energy and momentum-conserving nature of SFWM, any correlation between signal and idler photons projects the heralded photon into a mixed state. The heralded generation of entangled states [35] relies on the interference of multiple indistinguishable photons, and if the heralded single photons are in a mixed state, this will degrade any quantum interference between them [36].

 

Fig. 1. Microscope images of (a) photonic molecule photon-pair source. (b) Bent directional coupler composed of straight coupled waveguides of length ${L_S}$ and bent coupled waveguides of length ${L_C} = r\theta$, where $r$ is the coupler bend-radius, and $\theta$ is the angle swept by the bent region of the coupler. (c) Simulated joint spectral intensity of our designed photonic molecule, where P is corresponding spectral purity, and ${\lambda _S}$ and ${\lambda _I}$ are signal and idler wavelengths, respectively.

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Here we demonstrate a photon-pair source on the SOI platform using the resonantly enhanced SFWM process to maximize brightness and spectral purity. Design optimizations of a photonic molecule architecture (the combination of multiple inter-coupled optical cavities to create a larger device) promise heralding efficiencies much higher than previously demonstrated for ring resonators. Our source was designed to be resilient to fabrication variations such that these devices can be fabricated identically with high yields, as a key target of our approach will be to use large banks of these sources in parallel. This resilience comes in part from the strong coupling regime that we operate in, with target waveguide couplings of between 21% and 33% across the device, but equally from our directional coupler designs. Stronger couplings increase our resilience to fabrication variations because small changes in the coupling will proportionally affect the resonances (linewidth and extinction ratio) less at higher coupling strengths, compared to weaker coupling strengths. Motivated by previous works [37,38], and using the transfer matrix method (TMM), we chose device geometries expected to offer broadband operation, and fabrication resilience, something that has been proposed [37] but never implemented in a full device, to the knowledge of the authors.

2. DESIGN

Simple ring resonators are bound to purities of up to 91.7% due to strict energy conservation conditions between identical linewidth resonances [39,40]. However, this limit can be alleviated if the pump resonance linewidth is broadened relative to the signal/idler fields as in [30,40], or alternatively, if the temporal response of the pump is sharpened [29,41]. Reducing the interaction length or amplitude of the pump field in any way will lead to a decay in the brightness of the SFWM process [40,41].

 

Fig. 2. (a) Directional coupler design space that describes the change in transmittance ($\Delta \kappa$) of specific geometries of couplers for a waveguide separation of 200 nm ${\pm}\; \Delta$Gap (50 nm) and $\lambda = 1550\;{\rm nm}$, as a measure of fabrication tolerance. The solid black contours denote specific transmittances in multiples of 0.05, and the solid white contours denote the maximum wavelength variance (dispersion) of the transmittance over the telecom c-band ($\Delta \lambda = 1530 - 1565\;{\rm nm}$) compared to $\lambda = 1550\;{\rm nm}$. Marked white points indicate the most fabrication-tolerant couplers at each transmittance. (b) Fabrication tolerance and dispersion of a straight evanescent coupler against transmittance, for a waveguide separation of 300 nm and $\lambda = 1550\;{\rm nm}$.

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Figure 1(a) shows our resonator-based photon source design that allows us to engineer the spectral response of the in-resonator pump fields compared to those of the signal/idlers. The coupled resonators (primary–auxiliary) mean that we engineer only the shared resonances between the rings, and restrict photon generation solely to the primary resonator. Promisingly, the idea of a photonic molecule has already proved a versatile and promising design [42–44]. Here, our control over the in-resonator pump spectrum comes from the inter-resonator coupling between our primary resonator, which is simultaneously resonant at all three wavelengths of interest (signal, idler, and pump), and our auxiliary resonator, which is solely resonant at our pump wavelength. Additionally, we implement a loss channel (Fp) for only the pump wavelengths using an asymmetric Mach–Zehnder interferometer (AMZI) to avoid inducing excess loss for the signal/idler photons. As part of the auxiliary resonator, Fp provides control over the auxiliary resonance linewidth and hence, control over the lineshape of the pump in the primary resonator, and for the purposes of source brightness, its field enhancement. In expanding the possible resonant lineshapes of our pump beyond that of a simple Lorentzian [45], we aim to reduce the trivariate trade-off among purity, brightness, and heralding efficiency that is prominently discussed in previous works [33,40,41].

 

Fig. 3. (a) Scanning electron microscope (SEM) images of a cross-section of our (bent) coupler. The cross-section was achieved by using a focused ion beam (FIB) to mill away the surrounding area to give access to the SEM. We overlay a simulation of the modal solution (see Supplement 1, Section I) inside a coupler of this approximate geometry (waveguide cores highlighted) to show the strong interaction with the cladding void. Waveguide separation is 200 nm, height is 220 nm, and width is 500 nm. (b) Transmittance ($\kappa$) measurements of our bent coupler test structures at $\lambda = 1550\;{\rm nm}$, and their maximum variance $\Delta \kappa (\Delta \lambda)$ over the telecom c-band. Data for standard MMI and directional couplers are included, both targeting 3 dB operation. (c) Measured and fitted spectra of our photonic molecule’s signal, pump, and idler resonances.

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The platform chosen for this work uses waveguide geometries of ${500} \times {220}\;{\rm nm}$, and a bend-radius of 10 µm as a compromise between footprint and bend-losses, with estimated propagation losses of $2.40 \pm 1.51\;{\rm dB}/{\rm cm}$ from heralding efficiency measurements (see Supplement 1, Section IV). This allows our source to fit inside a footprint of ${143}\;{\unicode{x00B5}{\rm m}} \times {172}\;{\unicode{x00B5}{\rm m}}$. In assessing the sensitivity of straight-directional couplers (${L_C} = 0$) to both wavelength and fabrication variations (specifically waveguide separations of ${\pm} {50}\;{\rm nm}$), we saw that higher transmittance led to increased sensitivity in both categories [Fig. 2(b)]. This is in stark contrast to the design space of the bent couplers [Fig. 2(a)] where both wavelength and fabrication sensitivity can be minimized by choosing a specific geometry of coupler. For our bent couplers, we chose a waveguide separation of 200 nm to minimize their footprint, as well as their wavelength and fabrication sensitivity.

3. RESULTS AND DISCUSSION

A. Robustness

Any practical photon-pair source must retain its mass-fabricability, and as such, it needs to be resilient to fabrication inaccuracies. The intrinsic design of our source plays a major role in its robustness as targeting higher heralding efficiency using a ring-resonator necessitates operating in the strongly over-coupled regime [33], which, if targeting an all-around high-performance photon source, is one of many constraints we need to operate within. This regime comes with inherent tolerance to fabrication as small deviations will proportionally impact the design less. In particular, without Fp, we would have to work with very small, and therefore sensitive couplings between the primary and auxiliary resonators. This is analogous to the limited purity gains (97%) from a small amount of pump backscattering in ring resonators [45]. In our design, Fp allows us to work with much higher inter-ring coupling strengths and also leads to higher purities (see Supplement 1, Section II).

As for the couplers themselves, SEM images of the devices [Fig. 3(a)] highlighted the limitations of the fabrication technology and confirmed the anticipated presence of voids (absence of cladding). Voids are a common issue when depositing thick films using chemical vapor deposition (CVD) on structures with aspect ratios larger than 1:1, and the likelihood of voids is increased inside devices with small waveguide separations [46]. Accordingly, this can be alleviated by reducing the aspect ratio of the coupling region (ratio of waveguide height and separation [47]), which is a means of adding some fabrication tolerance to straight directional couplers at the cost of device footprint. Regardless, we can infer their robustness by characterizing the coupler test structures across the chip [Fig. 3(b)]. These devices are located on average 2.9 mm apart to minimize the chances that the devices are locally correlated due to their proximity inside the write-field during device lithography and fabrication. Our devices for comparison were a straight directional coupler and a standard multimode interferometer (MMI) both for 3 dB splitting at 1550 nm. We measured each device’s transmission across the telecom c-band to test their performance and saw almost negligible dispersion compared to a straight coupler [Fig. 3(b)]. Each coupler performed well with a small spread in transmittance across the chip, which fell within expectations [Fig. 2(a)], and the dispersion of each coupler continued to compete with that of an MMI [Fig. 3(b)]. This is further evident in the spectrum of the photonic molecule [Fig. 3(c) — full spectra in Supplement 1, Section II], where over ${\sim}10\;{\rm nm} $, we see very little change in the linewidth of the resonances. Interestingly, the straight coupler’s transmittance is lower than expected for the device, which could still be explained by a cladding void that forms later in the PECVD process.

B. Brightness

We experimentally characterized our photon-pair source by pumping it with a 340 pm (9 ps) bandwidth pulsed laser at a 51 MHz repetition rate, to excite the pump resonance of the ring, which has a bandwidth of 200 pm [Fig. 3(c)]. To estimate the source brightness, we varied the power of the pulsed laser using a variable optical attenuator (VOA) and measured the dependence of coincidences and singles with on-chip power. On-chip average power was estimated using a 90:10 fiber coupler, and the insertion loss of our grating couplers (3.8 dB — see Supplement 1, Section IV). We kept the pump power low to avoid excess nonlinear loss from two-photon absorption (TPA [48]), which is annotated in Fig. 4(e), where the expected coincidence-to-accidental ratio (CAR) deviates from the ideal fit at about 0.15 mW. We can solve for heralding efficiencies at the detectors and the effective nonlinearity (${\gamma _{{\rm eff}}}$) by quadratically fitting the singles and coincidence rates [Fig. 4(a)—Supplement 1, Section III). The ${\gamma _{{\rm eff}}}$ that we extract is $4.4 \pm 0.1\;{\rm MHz}\,{{\rm mW}^{- 2}}$. The effective nonlinearity essentially characterizes the on-chip generation rate, and therefore the brightness of the photon-pair source. By performing a similar measurement after taking the auxiliary ring off-resonance, we see that ${\gamma _{{\rm eff}}}$ improves to $15.5 \pm 0.4 \; {\rm MHz}\,{{\rm mW}^{- 2}}$ (see Supplement 1, Section III), a ${3.5} \times$ increase in brightness. Comparatively, the discussion below Eq. (5) from [40] predicts that, for our target purity, they expect a ${46} \times$ decrease in brightness using their design [40] compared to the generation rate of a single ring. This originates from a Q-factor decrease of ${3.33} \times$, rather than the ${10} \times$ originally used in [40]. However, our design achieves the target purity with a significantly lower (${3.5} \times$) decrease in brightness, demonstrating the power of our design with over an order of magnitude improvement in the expected brightness.

 

Fig. 4. (a) Measured brightness of our photonic molecule. Fits of the raw data are used to obtain the brightness (${\gamma _{{\rm eff}}}$) and the heralding efficiencies (${\eta _{s,i}}$) for the signal and idlers, respectively. On-chip rates have been inferred from our estimates of ${\eta _{s,i}}$. (b) Measured joint spectral intensity of the source; ${\Delta _{{\rm Idler},{\rm Signal}}}$ are the respective linewidths of the idler and signal resonances. (c) Purity obtained from unheralded second-order correlation measurements for a range of filter bandwidths ${\Delta _{{\rm Filter}}}$. (d) Histogram for the unheralded second-order correlation function measurement for the starred data point in (c). (e) Measurements of the coincidence-to-accidental ratio (CAR) and heralded second-order correlation function; the point at which two-photon absorption (TPA) becomes non-negligible is annotated.

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C. Purity

We characterized the source spectral purity using stimulated emission tomography (SET [49]) to extract the joint spectral intensity (JSI) of the photon-pair source, followed by a Schmidt decomposition on the corresponding joint spectral amplitude (JSA) to obtain an estimate of spectral purity. Additionally, we took measurements of the unheralded second-order correlation function ($g_{\textit{uh}}^{(2)}$ [50]) to corroborate our estimate of purity, where ${\rm purity},\; {\rm P} = g_{\textit{uh}}^{(2)}(0) - 1$. This is based on the assumption that we are operating in the low-gain regime (i.e., low probability of pair generation), and that we use broadband gate detection [50]. We satisfy these assumptions by using a coincidence window of 500 ps, capturing the entire coincidence histogram of each pulse as our detector timing resolution is 80 ps, as well as keeping our pump power low. In this way, the unheralded $g_{\textit{uh}}^{(2)}$ provides an estimate of the effective number of modes (or purity) by using the photon statistics of one beam of a twin-beam squeezer. The heralded $g_h^{(2)}$ is an additional metric that quantifies the photon-number purity and is due to the probabilistic nature of SFWM combined with our non-photon-number-resolving detection. It is the conditional probability of generating two photons given a successful heralding event, and is a measurement of the single-photon nature of light from the source.

For our SET measurements, we used a continuous-wave (CW — T100S-HP) laser with a linewidth of 3.2 am to perform a wavelength scan (1 pm resolution) over the signal resonance, measuring the output power at the idler resonance using a high-resolution (0.16 pm) optical spectrum analyzer (OSA — Waveanalyzer 1500s). The result of this measurement is presented in Fig. 4(b), and the corresponding purity is ${99.1}\; {\pm}\; {0.1}\%$, in line with the predictions of [40] and unequivocally demonstrating the unentangled nature of our photons. While the JSI discards phase information of the JSA, previous works [29] have reported that for high-purity JSIs, the purity estimate of the JSI converges with that of the true JSA, for which our simulations suggest a difference of only 0.01% (see Supplement 1, Section V). To obtain the error on our JSI measurement, we used a supersampling technique to take advantage of the huge precision afforded to us by our equipment, allowing us to reduce the effect of noise from both our OSA and our tunable CW laser. This reduced the uncertainty in our measurement, and at a final resolution of 4 pm, plateaued in both error and purity (see Supplement 1, Section VI), allowing us to say with confidence that our measurement contains the true value of purity.

Table 1. Comparing Source Metrics of Both Pure and Impure Sources^a

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While $g_{\textit{uh}}^{(2)}$ measurements can be subject to more noise due to its unheralded nature [51], it can serve to verify a source’s spectral purity in a scenario more closely resembling a realistic use-case. We pass our pump through a 400 pm bandwidth filter with 60 dB extinction to ensure no excess noise contaminates our measurement. Taking our signal photons, we filter them to varying degrees using the tunable filter, send them into a 50:50 fiber beamsplitter, and measure the CAR between the two arms of the beamsplitter [52]. This ratio should be two if the photon is in a pure state, i.e., exhibiting perfect thermal statistics. Our results are shown in Fig. 4(c). The best measured $g_{\textit{uh}}^{(2)}$ is ${1.98}\; {\pm}\; {0.02}$ for a filter bandwidth twice as wide as our source, which ensures the highest removal of any broadband nonlinear noise surrounding the source due to the 100 µm of waveguide either side of our source. We chose this length of input waveguide to provide a practical estimate of how this source could perform as part of a larger circuit according to our other works [53]. However, finding the best compromise between noise removal and filter bandwidth (${\Delta _{{\rm Filter}}}/{\Delta _{{\rm Signal}}} = 3$), we observe a $g_{\textit{uh}}^{(2)}$ of ${1.97} \;{\pm}\; {0.02}$ [Fig. 4(d)] where we measure our highest CAR between signal and idler photons of ${1644} \;{\pm}\; {263}$ [Fig. 4(e)]. Only filtering to this extent also avoids any serious degradation of the heralding efficiency, which becomes an issue for filters approaching the bandwidth of the source [54]. This measurement agrees with that of our measured JSI and further certifies the purity of our photons. Finally, we measure a minimum $g_h^{(2)}$ value of ${0.0029} \;{\pm}\; {0.0021}$ [Fig. 4(e)] at an average power of ${16} \;{\pm}\; {0.8}\;{\unicode{x00B5} \rm W}$, well inside the single-photon regime.

D. Efficiency

To get the best idea of the source’s full efficiency, we use a filtering bandwidth of 500 pm. The heralding efficiency that we measure off-chip is ${\eta _s} = 7.2 \;{\pm}\; 0.2\%$, and ${\eta _i} = 5.6 \;{\pm}\; 0.2\%$, where ${\eta _{s,i}}$ indicate the efficiencies of the signal and idler photons, respectively. However, the intrinsic heralding efficiency of the source is the limiting factor for any on-chip implementation of single-photon detectors, which would remove coupling-related insertion losses. We calculate the losses of our setup by measuring the insertion loss from the output of our laser power reference to our detectors both with and without the chip to isolate the insertion loss of the signal and idler channels. Most of our losses come from our filtering (see Supplement 1, Section IV), which adds insertion losses of approximately 6 dB in total and consists of a fiber DWDM (1 nm bandwidth) for coarse filtering, and more noise-isolating filtering using a tunable filter (XTA-50). Additionally, we have characterization data from the superconducting nanowire detectors we are using to be able to account for non-unity detection efficiencies equivalent to insertion losses of ${-}{1.060} \;{\pm}\; {0.044}\;{\rm dB}$ and ${-}{0.814}\;{\rm dB} \;{\pm}\; {0.026}\;{\rm dB}$ (see Supplement 1, Section IV). After this analysis, the post-source heralding efficiencies that we estimate are ${\eta _s} = 92.1 \;{\pm}\; 3.2\%$ and ${\eta _i} = 94.0 \;{\pm}\; 2.9\%$, which are comparable even with waveguide implementations of intrinsically pure sources ($\eta = 91 \;{\pm}\; 9\%$— [15]).

4. CONCLUSIONS

The photonic molecule architecture of our resonant photon-pair source brings gains in all of the key metrics that are required to build a fault-tolerant photonic quantum computer (summarized in Table 1). Our measured purity of ${99.1} \;{\pm}\; {0.1}\%$, certified using $g_{\textit{uh}}^{(2)}$ and SET measurements, is fundamental in the creation of quantum resources through entangling operations [2]. We improve on the previous ring resonator literature purity of 95.0 ± $1.5 \%$ [30], and remain on par with the large intrinsically pure waveguide sources (${99.04} \;{\pm}\; {0.06}\%$ [15]). Additionally, with a competitive maximum estimated heralding efficiency of ${94.0} \;{\pm}\; {2.9}\%$, especially when compared to previously reported pure resonators ($\eta = 52.4\%$ [30]), our source successfully operates within on-chip loss thresholds of a fusion-based architecture [23]. Finally, the simultaneous improvement in purity and heralding efficiency has significant implications for the future of quantum teleportation, which can be boosted well beyond existing implementations ([32] ${\rm purity} = {91} \;{\pm}\; {4}\%$, $\eta = 50\%$). This technology can be used to create quantum repeaters that are fundamental to the operation of any future large-scale quantum network [57]. Our source surpasses expectations set by previous works [29,30,40,41], and reduces the trade-off among brightness, heralding efficiency, and purity, with a measured ${\gamma _{{\rm eff}}} = 4.4 \;{\pm}\; 0.1\;{\rm MHz}\,{{\rm mW}^{- 2}}$, beating expected brightness degradation by over an order of magnitude [40]. Finally, our source is reasonably resilient to fabrication defects [Fig. 3(a)] and variances, both through the implementation of fabrication-tolerant directional couplers with dramatically reduced dispersion, and the intrinsic over-coupled design of our source. Therefore, our results present a bright and efficient ring resonator source of pure photons that has scalability at the core of its design. State-of-the-art propagation losses [16] could improve source heralding efficiencies to values in excess of 99%. Our design can therefore be fully realized through the maturity of the fabrication process, as all of our source metrics can only improve with reduced propagation losses due to higher Q-factors [33], leading to a scalable and truly optimizable source.