Mid-infrared quantum optics in silicon

TL;DR

This paper introduces mid-infrared silicon quantum photonics, demonstrating key components like photon-pair generation and quantum interference beyond 2 μm, enabling scalable and low-loss quantum information technologies.

Contribution

The work pioneers silicon quantum photonics in the mid-infrared, overcoming loss issues at 1.55 μm and demonstrating essential quantum optical components at wavelengths beyond 2 μm.

Findings

Maximum net coincidence rate of 448 ± 12 Hz

Coincidence-to-accidental ratio of 25.7 ± 1.1

Quantum interference visibility of 0.993 ± 0.017

Abstract

Applied quantum optics stands to revolutionise many aspects of information technology, provided performance can be maintained when scaled up. Silicon quantum photonics satisfies the scaling requirements of miniaturisation and manufacturability, but at 1.55 $\mu$m it suffers from unacceptable linear and nonlinear loss. Here we show that, by translating silicon quantum photonics to the mid-infrared, a new quantum optics platform is created which can simultaneously maximise manufacturability and miniaturisation, while minimising loss. We demonstrate the necessary platform components: photon-pair generation, single-photon detection, and high-visibility quantum interference, all at wavelengths beyond 2 $\mu$m. Across various regimes, we observe a maximum net coincidence rate of 448 $\pm$ 12 Hz, a coincidence-to-accidental ratio of 25.7 $\pm$ 1.1, and, a net two photon quantum interference visibility of 0.993 $\pm$ 0.017. Mid-infrared silicon quantum photonics will bring new quantum applications within reach.

Equations

$$\begin{split}\ket{\psi}\mkern 5.0mu=\mkern 5.0mu&\frac{\sin\phi}{\sqrt{2}}\big{(}\ket{0_{s}0_{i}}_{A}\ket{1_{s}1_{i}}_{B}-\ket{1_{s}1_{i}}_{A}\ket{0_{s}0_{i}}_{B}\big{)}\\ +\mkern 5.0mu&\frac{\cos\phi}{\sqrt{2}}\big{(}\ket{1_{s}0_{i}}_{A}\ket{0_{s}1_{i}}_{B}+\ket{0_{s}1_{i}}_{A}\ket{1_{s}0_{i}}_{B}\big{)},\end{split}$$

Full text

The mid-infrared (MIR) is the energy band of vibrations. The molecular ‘fingerprint’ region, 2–20 $\upmu\text{m}$, is characterised by sharp molecular transitions, which lab-on-chip sensors can use to spectrally target molecular species in liquid and gaseous analytes1. Lidar systems can exploit atmospheric transparency and reduced scintillation in the MIR2, as well as the high-power handling and reduced phase error of MIR optical phased arrays for improved reliability3. Much work has been done to bring integrated optics to the MIR, and silicon-on-insulator photonics dominates in the short-wavelength part of the band (the short-wave infrared), up to about 4 $\upmu\text{m}$ 4, 5, 6, 7. Here, silicon benefits from reduced two-photon absorption, facilitating nonlinear optical applications8: optical parametric oscillators and amplifiers9, supercontinuum sources10, 11, and frequency combs12, 13 have all been developed.

Full quantum photonic technology has exquisite performance sensitivity, but stands to revolutionise how we measure, communicate, and ultimately process information18. It requires a huge scaling up for either integration or real-world deployment. As classical optics ventures into the MIR, quantum optics is close behind. Bulk-crystal photon-pair sources have been designed19; experiments with one20, and two21 MIR photons have been shown, with detection provided by avalanche photodiodes and up-conversion.

Silicon photonics, operating mainly around the 1.5-$\upmu\text{m}$ telecommunications band, has exploded in scale and functionality22, and quantum silicon photonics has grown in tandem23. In silicon, quantum-correlated photon pairs are scattered from a bright pump laser via the refractive nonlinearity, by spontaneous four-wave mixing24 (SFWM). Increasingly large interferometers have used these photons to power proof-of-concept quantum protocols25, 26, 27, but to go beyond a handful of photons, ultra-low optical loss is needed.

Propagation and device losses have steadily fallen28, 29, 30, 31, 32, but two-photon absorption (TPA) is intrinsic. TPA allows two photons to excite a crystal electron. It is stimulated absorption, growing with optical intensity. TPA clearly limits pump power, but it also limits the heralding efficiency of single photon sources33, and so is a fundamental limit to the large-scale viability of silicon quantum photonics23. One common approach to avoiding TPA is to replace the guiding material (to e.g. silicon nitride34, 35). Beyond silicon’s two-photon band edge, though, around 2.1 $\upmu\text{m}$, two photons carry insufficient energy to excite a crystal electron, and TPA subsides. A resonant peak in the refractive nonlinearity, here, makes photon-pair sources more efficient, requiring less pump power. Conventional silica cladding remains transparent, and environmental black-body noise is not too large. Linear loss from Rayleigh scattering off etched waveguide side-walls36, 37 is reduced at long wavelengths, and subwavelength features are more readily manufactured32, 38. We plot the TPA coefficient $\beta_{\mathrm{TPA}}$, nonlinear refractive index $n_{2}$, and Rayleigh scattering cross-section, relative to their 1.55 $\upmu\text{m}$ values, in Fig. 1. In this article, we show that silicon photonics in the 2.1-$\upmu\text{m}$ band is one route to ultra-low-loss quantum optics on a chip.

Any new quantum photonic platform needs: a source of quantum light, a way to detect that light, and quantum interference. We report on all three ingredients here. We design and characterise a silicon waveguide able to generate entangled photon pairs, centred on 2.07 $\upmu\text{m}$, and use classical nonlinear optics to verify its design. We deploy a new detector system, optimised for the 2-$\upmu\text{m}$ band, and verify its performance. We then drive SFWM in the designed waveguide and observe quantum-correlated photon pairs. Finally, we embed two such photon-pair sources in a reconfigurable on-chip interferometer and observe quantum interference.

Waveguide design for MIR SFWM

Spontaneous four-wave mixing, where two pump photons are scattered to higher and lower frequencies, conserves energy and momentum. For efficient SFWM, the phase-matching condition must be satisfied. The total wave-vector mismatch, $\Delta k=\Delta k_{\textrm{lin}}-2\gamma P,$ must be zero. Here, $P$ is the peak pump power in the waveguide, and $\gamma=\text{k}_{0}n_{2}/\text{A}_{\text{eff}}$ is the waveguide nonlinear parameter, with $n_{2}$ and $A_{\text{eff}}$ the nonlinear refractive index and effective modal area39, respectively. For frequencies near the pump, $\Delta k_{\textrm{lin}}=-\beta_{2}\,\Delta\omega^{2},$ where $\beta_{2}=\text{d}^{2}k/\text{d}\omega^{2}$ is the waveguide group-velocity dispersion (GVD), and $\Delta\omega$ is the pump-photon angular frequency detuning. For efficient four-wave mixing, $\beta_{2}\leq 0$, i.e. the GVD must be anomalous or zero.

We designed the waveguide shown in Fig. 2a ($510\times 340~{}\text{nm}^{2}$ cross-section, $15^{\circ}$ side-wall angle) based on the variations of $\beta_{2}$ and $A_{\mathrm{eff}}$ with waveguide width shown in Fig. 2c, at $\lambda=2.071~{}\upmu\text{m}$. The highly confined fundamental mode is shown in Fig. 2b. To confirm the phase matching of our source, and estimate its bandwidth, we measure classical stimulated four-wave mixing9, 40. Figure 2f shows a spectral map of the pump and stimulated emission, for various seed laser wavelengths. We observe phase-matched four-wave mixing over at least $60~{}\text{nm}$, limited by grating coupler transmission, implying a very wide SFWM spectrum (Fig. 2e).

All previous measurements of silicon’s 2.1-$\upmu\text{m}$ optical nonlinearity have been in bulk samples: here we confirm that the high nonlinear figure of merit measured in bulk also exists in nanoscale silicon waveguides. We use self-phase modulation and the Gerchberg-Saxton optical phase-retrieval method to estimate the waveguide nonlinearity16; our results are summarised in Fig. 2d with methods detailed in Supplementary Section S2. We find an effective waveguide nonlinearity of $\gamma=203\pm 26~{}\text{W}^{-1}\text{m}^{-1}$ ($n_{2}=15.3\pm 1.9\times 10^{-18}~{}\text{m}^{2}\,\text{W}^{-1}$ in bulk) and waveguide nonlinear absorption coefficient $\alpha_{\mathrm{TPA}}=24.4\pm 3.3~{}\text{W}^{-1}\text{m}^{-1}$ ($\beta_{\mathrm{TPA}}=0.557\pm 0.07~{}\text{cm}\cdot\text{GW}^{-1}$ in bulk). Here, $\alpha_{\text{TPA}}=\partial\alpha/\partial P$ and $\beta_{\text{TPA}}=\partial\alpha/\partial I$, where $\alpha$ is the waveguide loss coefficient, and the intensity $I=P/A_{\mathrm{eff}}$ for a waveguided power $P$ with calculated $A_{\mathrm{eff}}=0.228~{}\upmu\text{m}^{2}$. Our measurements on waveguides are in agreement with those in bulk15, 14 (see Fig. 1). With the resonant $n_{2}$ enhancement, less pump is required for the same SFWM rate, and so less TPA ultimately results (see Supplementary Fig. S2). This is in addition to the modest reduction in $\beta_{\mathrm{TPA}}$ which we observe.

Observation of correlated photon pairs

For a bright source of quantum-correlated photon pairs, we use SFWM. We start with a picosecond-pulsed pump laser, centred at $2.0715~{}\upmu\text{m}$, which we filter to a width of $1.0~{}\text{nm}$ using a double-pass grating monochromator. Controlling polarisation, we inject this pump into the fundamental TE mode of the waveguide with a vertical grating coupler (VGC; $-7.3$ dB transmission). The waveguide, shown in Fig. 3a, is wrapped into a 7.2-mm square spiral (3.2 dB/cm propagation loss) with 10-$\upmu\text{m}$ minimum radius Euler bends41. Signal and idler photons are emitted in the same spatial mode, coupled off-chip, and separated probabilistically by a 1:1 fibre beam splitter. Both channels are tightly filtered with back-to-back free-space grating monochromators ($\sim$4.5 dB insertion loss per monochromator) to achieve the required $>100~{}\text{dB}$ pump rejection42, 43. The signal and idler filters are $1.0$-nm wide, and separated from the pump by $\pm~{}1.46$ THz ($20.8$ nm). The experimental scheme is shown in Fig. 3b.

To detect single photons from SFWM, sensitive detectors are necessary. Superconducting nanowire single photon detectors (SNSPD) have unrivalled timing jitter, dark count rates (DCR), and system detection efficiency (SDE), and have shown sensitivity up to $5~{}\upmu\text{m}$44. We incorporate superconducting nanowires into a dielectric stack optimised for absorption into the nanowire cavity at $\lambda=2.1~{}\upmu\text{m}$. The SNSPDs were fabricated from a 4-nm thick niobium nitride film, deposited using magnetron sputtering, and patterned using electron-beam lithography into a meander of 100-nm-wide superconducting wire. The nanowires are fibre coupled with anti-reflection coated SM2000 fibre. A second SM2000-fibre span connects these detectors to our optical apparatus ($\sim 30~{}\text{m}$; 1.49 dB and 1.63 dB loss per channel, for detectors $A$ and $B$ respectively). Operating at 780 mK, we find that the SDE of the two detectors plateaus with bias currents around $8~{}\upmu\text{A}$, with peak SDE values of $44\pm 10\%$ and $48\pm 10\%$, for detectors $A$ and $B$, respectively. In the following experiments, we operate with detector bias currents of $7.9~{}\upmu\text{A}$ and $8.1~{}\upmu\text{A}$. We measure a timing jitter of $216~{}\text{ps}$ (full width at half maximum) from photon cross-correlation. The dark count rate and efficiency as a function of bias at $\lambda=2.071~{}\upmu\text{m}$ is shown in Figs. 3c and 3d. We use an attenuated tuneable laser to measure the SDE as a function of wavelength, and plot the results in Fig. 3e.

Photon pairs are detected in coincidence, with detector output pulses time correlated using standard coincidence counting logic. We observe a characteristic peak at zero relative delay from photons produced in the same SFWM event. A representative histogram is shown in Fig. 3f. We varied the launched optical power, fit each time correlation histogram with a Gaussian function, and integrated over 3 standard deviations (389 ps). These data are plotted in Fig. 3g. At low pump powers, we estimate an on-chip per-pulse generation efficiency of $11~{}\mathrm{MHz}/\mathrm{W}^{2}$, for watts of peak power, which at the repetition rate of our laser gives a generation probability of $0.28/\mathrm{W}^{2}$ (peak). These estimated on-chip rates naturally exclude loss (see Methods). We measure a maximum net coincidence rate of $112\pm 3~{}\text{Hz}$, with peak coincidence to accidental ratio (CAR) of $25.7\pm 1.1$, at a coincidence rate of $1.1~{}\text{Hz}$. We define $\mathrm{CAR}=(X_{\mathrm{raw}}-X_{\mathrm{acc}})/X_{\mathrm{acc}}$, where $X_{\mathrm{raw}}$ is the integrated coincidence count in the histogram central peak, and $X_{\mathrm{acc}}$ is the average integrated coincidence count in the side (accidental) peaks (see Supplementary Section S4 and Fig. S4 for singles rates). Dark counts limit the maximum CAR $\propto\eta/\mathrm{DCR}$, so higher system transmission ($\eta$) and lower DCR will offer further improvements to the CAR. On a longer 17.5-mm waveguide, we measured a peak net coincidence rate of $224~{}\text{Hz}$ (Supplementary Fig. S2c). Due to the simple separation of the signal and idler with a beam splitter, the actual pair-production rates are $4\times$ the measured net rates. Thus, we observe true rates of $448~{}\text{Hz}$ and $896~{}\text{Hz}$ for the two waveguides, respectively (see Supplementary Section S4).

On-chip quantum interference

The indistinguishability of generated photons, confirmed by high-visibility quantum interference between them, is an essential platform resource. The Hong-Ou-Mandel (HOM) effect causes two indistinguishable single photons to strictly bunch at the outputs of a balanced beam splitter45. Here, we demonstrate on-chip quantum interference using a time-reversed HOM experiment46, with experimental setup and waveguide circuit shown in Figs. 4a and 4b.

After filtering the pump as before, we couple it onto the chip through a VGC. We set the on-chip peak pump power to $0.32~{}\text{W}$, corresponding to a CAR of $19.3$ and $\sim 0.03$ pairs per pulse. The pump field is equally split between the two photon-pair sources, with a balanced $1\times 2$ multimode interference (MMI) coupler. Both 7.39-mm sources are coherently pumped and the relative phase $\phi$ between the two arms is varied with a thermo-optic phase modulator (see Supplementary Section S1 for details of integrated optics). The biphoton state then interferes on a balanced directional coupler ($R=0.49\pm 0.02$). Coherent pumping of both sources produces SFWM photon pairs in superposition, and the quantum state at the directional coupler output is

[TABLE]

with signal and idler frequency ($s,i$) and spatial modes ($A,B$) in subscript (see Supplementary Section S5 for more details). SFWM photons are then frequency demultiplexed on-chip with asymmetric Mach-Zehnder interferometers. Off chip, we reject the pump laser, and use coincidence detection to estimate $|\bra{1_{s}0_{i}}_{A}\bra{0_{s}1_{i}}_{B}\ket{\psi}|^{2}$.

We observe characteristic half-period interference fringes in the coincidences46, 47, as the on-chip phase $\phi$ is varied, consistent with Eq. 1 (see Fig. 4c). This quantum interference has a net visibility of $V=0.993\pm 0.017$ (with $0.862\pm 0.014$ raw). We calculate the visibility as $V=(X_{\mathrm{max}}-X_{\mathrm{min}})/(X_{\mathrm{max}}+X_{\mathrm{min}})$, where $X_{\mathrm{max}}$ and $X_{\mathrm{min}}$ are the maximum and minimum coincidence count rates of the sinusoidal fit. This compares favourably to performance at 1.55 $\upmu\text{m}$ on chip46, 48, 49, 50, and at 2.1 $\upmu\text{m}$ in bulk 51. We observe coincidence rates of up to $5.5\pm 0.2~{}\text{Hz}$ at the interference peak. Simultaneously measuring all four chip outputs would double the observed rate. For perfect interference, the raw visibility is limited to $V\leq\mathrm{CAR}/(2+\mathrm{CAR})=0.91$.

Discussion and conclusions

Despite operating in the 2-$\upmu\text{m}$ band, we see from Fig. 2d that two-photon absorption remains, albeit reduced from its strength at $1.55~{}\upmu\text{m}$. This is to be expected at room temperature and at $\lambda=2.07~{}\upmu\text{m}$, as silicon’s indirect band gap gives a TPA cut-off around $2.21~{}\upmu\text{m}$ (Fig. 1). Our waveguided estimates of both nonlinear absorption and refraction are in broad agreement with literature values for bulk silicon15, 14. At low temperatures, a blue-shift in the band edge causes a blue-shift in the TPA cut-off52, to 2.15 $\upmu\text{m}$. Future room-temperature experiments will benefit from 2.2-$\upmu\text{m}$ laser development (e.g. semiconductor disk lasers53), while experiments integrating SNSPDs on-chip or in-package will benefit from this low-temperature shift of the TPA cut off.

Black-body radiation from our room-temperature apparatus was a source of noise in our measurements. As seen in Figs. 3c and 3d, the ‘dark’ count rate, collected with the lights and laser turned off, plateaus, rather than growing exponentially with bias, because environmental black-body photons nonetheless illuminate the detectors (see Supplementary Figs. S3a,b). Black-body noise would be suppressed by a cold filter. Both CAR and raw fringe visibility would benefit from such a filter.

In demonstrating a bright source of photon pairs, efficient single-photon detectors, and high-visibility quantum interference, we have provided all the necessary ingredients for a dense, manufacturable, and high-performance platform for applied quantum optics. The mid-infrared presents a new approach to scalable optical quantum information processing in silicon.

Acknowledgements

We thank D. Bonneau, A. McMillan, B.D.J. Sayers, B. Kuyken, M. Nedeljkovic, K. Erotokritou, G. Taylor, and R.H. Hadfield for their valuable inputs throughout the long gestation of this work, and are grateful to L. Kling, and G.D. Marshall for their advice and support. The chip reported here was fabricated as part of the CORNERSTONE project, and we are especially grateful to C. Littlejohns for his efforts therein.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data and computer code that support the findings of this study are available at the University of Bristol’s data repository, data.bris . Other information is available from the authors upon reasonable request.

Methods

Dispersion calculation

Finite difference eigenmode simulations were performed to determine the optimal waveguide geometry 54. The waveguide height was fixed to $340~{}\text{nm}$, with a $15^{\circ}$ sidewall angle. The dispersion was calculated with a fractional frequency offset of 0.0001 to obtain the group velocity dispersion for the fundamental transverse electric mode at a centre wavelength of $\lambda=2.071~{}\upmu\text{m}$. A waveguide with a cross-sectional area of $510\times 340~{}\text{nm}^{2}$ simultaneously achieves anomalous group-velocity dispersion ($\beta_{2}\leq 0$) required for phase matching, while minimising the effective modal area, see Fig. 2c.

Device fabrication

The device in this report was a $340~{}\text{nm}$ crystalline silicon(100)-oriented silicon-on-insulator platform, with a $2~{}\upmu\text{m}$ buried oxide and LPCVD $\text{SiO}_{2}$ $1~{}\upmu\text{m}$ top cladding. The silicon structures were formed with a 248 nm deep-UV lithography and dry etching. Resistive filaments of titanium nitride with a $3~{}\upmu\text{m}$ width were deposited on the top cladding with a metal lift off process, used for the on-chip thermal phase shifters. The fabrication of the chip was courtesy of the Cornerstone foundry based at the University of Southampton.

Phase retrieval

In the waveguide nonlinearity measurements, the transmitted power and spectrum of our passively mode-locked Ho-doped fibre pulsed laser (AdValue photonics) were taken 16. Firstly, the pump input pulses are filtered with a single pass monochromator centred at $\lambda=2.0715~{}\upmu\text{m}$. The pulses in the time domain are characterised by taking an intensity autocorrelation (Femtochrome FR-103PD) and fitted with a hyperbolic secant-squared ansatz giving an input pulse duration of $\tau=4.82~{}\text{ps}$ (see Supplementary Section S2 for more detail). The laser is then connected to the $17.5-\text{mm}$ spiral with an 9:1 optical tap to monitor the input power. The optical power launched into the chip is controlled with a knife edge variable optical attenuator, allowing the intensity-dependent transmission and spectrum of the output pulse to be measured. For each power launched, a frequency spectrum is recorded on an OSA (Yokogawa AQ6375) at the chip output.

To retrieve nonlinear phase from the output spectrum, sequential Fourier and inverse Fourier transforms are performed to transform between the time and frequency domains (Gerchberg-Saxton algorithm 55). By retaining the phase information, but replacing the amplitude with our measured secant-pulse envelope (time domain) or power spectrum (frequency domain), the algorithm is found to converge on a steady state. Using this algorithm the nonlinear phase corresponding to the measured spectral broadening is retrieved for each pulse. Finally, the phase profiles are fitted with a secant-shaped phase model and combining this with the free carrier dispersion and absorption effects found from fitting the transmission data, the nonlinear phase and hence the waveguide nonlinearity are found. To determine the value of $n_{2}$ in Fig. 1, we calculated the effective modal area using the method from 39. In addition to determining the refractive nonlinearity, the transmitted power is also used to determine the TPA parameter by using a fit to the inverse transmission against the input power, as described in Ref. 56.

Stimulated four-wave mixing

We pump the waveguide source with a filtered picosecond pulsed laser centred at $\lambda=2.0715~{}\upmu\text{m}$ and a tuneable continuous wave (CW) laser (Sacher Lion) as the stimulating field. The two lasers are combined on a 1:1 fibre beam splitter before the chip with power monitoring optical taps to normalise the stimulated FWM at the chip output. Tuning the frequency of the CW laser, the stimulated FWM from a 7.2-mm spiral is measured on an OSA.

Detector characterisation

Electrical output traces of the RF pulses from the SNSPDs are recorded for several bias settings. The peak voltage of the pulse changes linearly as a function of detector bias and a linear model is fit to this data: 7.46 and 10.84 $\text{mV}\upmu\text{A}^{-1}$ for detectors $A$ and $B$, respectively (Supplementary Fig. S3). This is combined with a constant offset of 15 and 20 mV, determining the discrimination voltage as a function of bias for detectors $A$ and $B$, respectively. Detectors $A$ and $B$ are connected internally to cryostat ports D4 and D3, respectively.

The transmission of a tuneable CW laser (Sacher Lion) centred at $\lambda=2.07~{}\upmu\text{m}$ is measured with a InGaAs photodiode (Thorlabs S148C) at a fibre just before the cryostat to calibrate out any fibre losses between the source and the detectors. A motorised knife edge attenuator is then used to calibrate the knife edge position as a function of transmitted power at the photodiode. Individual transmission of four neutral density filters are then recorded before inserting them in the beam path and reducing the net photon flux down to the single photon regime.

Finally, the photodiode input fibre is connected to the superconducting detector, and singles counts from a Picoquant PicoHarp timetagger are integrated for $0.5~{}\text{s}$ for each optical power launched. The estimated input and output photon flux is fit with a linear slope model and is used to determine the system detection efficiency (see Supplementary Section 3 for error analysis).

Correlated photon pair measurement

We pump the waveguide source shown in Fig. 3a with a passively mode-locked Ho-doped fibre pulsed laser (AdValue Photonics) centred on $2.0715~{}\upmu\text{m}$, producing $2.9$-ps pulses, at a $39.4$-MHz repetition rate, with $487~{}\text{W}$ of peak power coupled into fibre from free space. The laser is filtered with a monochromator in a double-pass configuration (Fig. 3b), producing $5.78$-ps pulses, with $24.4$-W peak power. Filtered pump is then coupled into the waveguide with a V-groove array and vertical grating couplers. After propagating through the waveguide spiral, the signal and idler photons are passively demultiplexed on a fibre beam splitter. Back-to-back single-pass monochromators suppress the pump. Finally, the photon pairs are detected by our SNSPDs. A time-interval analyser (PicoQuant PicoHarp) with a 4-ps resolution correlates detector output pulses. For each power launched, a histogram was integrated for 540 and 180 seconds, in the low and high power regimes, respectively. Each integration was subdivided into 20-second intervals.

At low pump powers, nonlinear absorption and dispersion play a reduced role, greatly simplifying our model, and allowing us to unambiguously estimate a source efficiency, exclusive of channel losses57. In the low-power regime, SFWM simply scales with $P^{2}$, where $P$ is the peak pump power. Here, we consider $P<0.5~{}\mathrm{W}$ to be ‘low power’. We estimated the on-chip pair-generation efficiency and probability by analysing the pair-generation versus pump-power data of Fig. 3g (net coincidences, $X$) and Supplementary Fig. S4 (singles of each channel, $C_{0}$, $C_{1}$). All three datasets were fit with a polynomial of the form $aP^{2}+bP+c$, though with $b=c=0$ for the net coincidences data (as accidentals are already subtracted). The on-chip rate per unit peak-power-squared is then estimated as $a_{C_{0}}a_{C_{1}}/a_{X}$, where $a_{*}$ is the best fit parameter $a$ for the indicated dataset. This efficiency is converted to probability by dividing by the laser repetition rate, and converted to average power by dividing by the square of the effective duty cycle (about $2.6\times 10^{-4}$ for our filtered laser).

Thermo-optic phase modulator calibration

We used a 16-bit DC voltage source (Qontrol Systems) to drive the on-chip thermo-optic phase modulators. Injecting CW laser light centred on $\lambda=2.049~{}\upmu\text{m}$ (Eblana photonics quantum cascade laser) into the input vertical grating coupler, the optical output of the chip was monitored with a photodiode while the phase-voltage was varied, see Supplementary Fig. S1. The interferometers voltage-squared versus transmission was fit with a sinusoidal envelope. In the time-reversed HOM experiment, 30 evenly spaced squared-voltages were set on the source interferometer phase shifter.