Introduction

The processing speed of modern computers is limited by the fact that program memory and data memory share the same bus. While processors have become faster, the overall speed is limited by the data transfer rate, known as the von Neumann bottleneck1. There has been an increase in demand for solving certain classes of computation problems that scale exponentially in time and energy for the current von Neumann architecture. Many combinatorial optimization problems fall under this category and are classified as non-deterministic polynomial-time (NP) hard and have applications in areas including finance, scheduling, trajectory planning, and artificial intelligence. The ability to solve these problems efficiently has motivated the development of computing accelerators based on digital and physical systems2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29.

Recently, there has been considerable interest in using photonic processors to realize a novel form of coherent computing by simulating the Ising model16,20,21,22. The Ising model was developed for modeling ferromagnetism and is governed by a Hamiltonian that couples discrete variables that represent spin glasses30,31. Solving for the ground state of such a system corresponds to solving an NP-hard computational problem and can provide an architecture for solving other NP-complete problems through polynomial-time mapping to the Ising model32. The Ising Hamiltonian with N spins and no external field is given by \(H=-\mathop{\sum }\nolimits_{ij}^{N}{J}_{ij}{\sigma }_{i}{\sigma }_{j},\) where Jij is the coupling coefficient and σi corresponds to the projection of the ith spin along the z-axis that can have two states  ±1. The physical realization of an Ising machine requires binary degrees of freedom (i.e.,  spins σi) and reconfigurable coupling (i.e., Jij), and initially was studied using a network of injection-locked lasers10,11,12,13,14,15. More recently, investigations have shown that a network of coupled degenerate optical parametric oscillators (DOPOs) based on the χ(2) nonlinearity can be used to realize a hybrid temporally multiplexed coherent Ising machine16,17,18,19,20,21, which includes a recent demonstration of a system of 2000 spins20. The nonlinearity is based on the nonequilibrium phase transition that occurs at the parametric oscillation threshold, resulting in two possible phase states of the DOPO offset by π, and the couplings between the DOPOs are implemented via measurement feedback or optical delay lines. By controlling the coupling between these DOPOs, it is possible to achieve more complex, phase-locked output states that encode the ground state of an Ising model. These demonstrations have utilized a time-multiplexed DOPO system using a 1-km-long fiber ring cavity to simulate the ground state of the Ising model20,21. In addition, extensive experimental and theoretical analysis has been done to characterize the potential performance of such systems23,33,34,35. Furthermore, alternative approaches towards a coherent Ising machine, such as opto-electronic oscillators with self-feedback24,25,26 and spatial light modulation27,28, has been demonstrated and an approach using a dispersive optical bistability has been proposed29.

An alternative approach to realize a DOPO is to use the χ(3) nonlinearity in which a frequency degenerate signal/idler pair is generated via parametric four-wave mixing (FWM)36,37,38,39. Such a scheme has been implemented using silicon nitride (SiN, Si3N4) microresonators and bi-phase state generation has been achieved enabling quantum random-number generation in a chip-scale device (Fig. 1)37,39. The SiN platform is ideally suited for scalability to an all-photonic network of coupled DOPOs since it is CMOS (complementary metal–oxide–semiconductor) process compatible, has low losses in the near infrared, and allows for dispersion engineering, which is crucial for efficient phase-matched nonlinear processes40. Unlike the χ(2) process, the wavelengths of the pump and degenerate signal are spectrally close, allowing for phase matching of the signal through dispersion engineering of the fundamental waveguide mode. In addition, unlike traditional FWM, which requires operation in the anomalous group-velocity dispersion (GVD) regime, the DOPO requires normal GVD for phase matching37, which is more readily accessible across a wider range of photonic platforms. Furthermore, the microresonator-based DOPO system allows for simultaneous oscillation of all DOPOs, enables continuous-wave (cw) operation, and does not rely on long cavity lengths as in the time-multiplexing scheme, which requires phase stabilization to support multiple trains of femtosecond pulses, offering faster computational speeds with lower power consumption in a compact footprint. A recent preliminary study has also reported on numerical simulation of coupled Lugiato–Lefever equations to tackle the MAX-CUT problem, further demonstrating the potential capability of SiN platform for optical computing41.

Fig. 1: Conceptual schematic of a network of coupled degenerate optical parametric oscillators.
figure 1

Two-frequency nondegenerate pumps are injected into the network of coupled degenerate optical parametric oscillators (DOPOs) based on silicon nitride (SiN) microresonators. The pumps are split on-chip using multimode interference (MMI) splitters. The DOPO signal is phase matched and generated via four-wave mixing parametric oscillation. Phase-matching conditions result in a bi-phase state for the generated signal. Coupling between DOPOs can be performed using a matrix of reconfigurable Mach–Zehnder interferometers (MZIs).

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In this paper, we demonstrate an integrated photonic circuit that consists of spatial-multiplexed DOPOs on a single SiN chip and present the first demonstration of a coupled DOPO system using cw pumping. We experimentally show reconfigurability of the coupling phase between the two DOPOs through thermal control of the coupling waveguide between the DOPOs and show interference measurements between the DOPOs indicating in-phase and out-of-phase operation. In our system, we solve at a 400-kHz rate with a convergence time of  <310 ns. In addition, we numerically model the coupled DOPO system and confirm the behavior for in-phase and out-of-phase operation and explore the transition region between the two phase states. A distinct transition region between two states is revealed both numerically and experimentally, suggesting a tremendous phase tolerance of such parametric process. We also theoretically investigate our coupled DOPO system using coupled Lugiato–Lefever equations and describe the scalability of such a system to a large number of oscillators on-chip and the challenges in achieving a large-scale photonic coherent Ising machine.

Results

Device characterization and experimental setup

The key components for realizing a network of coupled DOPOs on-chip are, (1) power splitters for routing the pump fields to the microresonators, (2) nonlinear microresonators designed for DOPO generation, and (3) the N × N photonic coupling system between the different microresonators. A microscope image of the SiN device is shown in Fig. 2a and is fabricated using techniques similar to those reported in Luke et al.42. For routing the two-frequency nondegenerate pumps to the microresonators, we employ an on-chip power splitter using multimode interference (MMI)43, where the dimensions of the MMI are designed to allow for 50/50 power splitting ratio. The insertion loss of the MMI splitter is 2 dB. The two SiN microresonators (DOPO1 and DOPO2) have a radius of 45.84 μm, which corresponds to a free spectral range (FSR) of 500 GHz. The loaded quality (Q) factors of the microresonators are 630,000. The condition LD > LNL, where LD = 1/δ2∣β2∣ is the dispersion length and LNL = 1/2γP is the nonlinear length, is critical for achieving maximum gain at the frequency degeneracy point and enabling pure DOPO generation (see Supplementary Note 1)37. Here, β2 is the GVD parameter, γ is the nonlinear parameter, P is the power of each pump, and δ is the pump frequency offset. Based on simulations using a finite-element mode solver, we use a waveguide cross section of 730 × 1050 nm2 such that the two pumps are placed in the normal GVD regime for the fundamental transverse electric (TE) mode to allow for efficient phase matching and maximum gain at the frequency degeneracy point37. The simulated GVD is shown in Fig. 2b, and the corresponding gain for 1 W of pump power is shown in Fig. 2c. The cavity resonance for each microresonator is thermally controlled using integrated platinum resistive microheaters. The coupling between the microresonator and the bus waveguide is designed to have near-critical coupling, where the extinction ratio of the resonances is ~92%. To compensate for the difference in the resonance frequencies of the two resonators due to the fabrication tolerances in microresonator geometry, we use microheaters above each resonator that allow for electrical control of the resonances via the thermo-optic effect44,45. We set the electrical power to the heaters such that the cavity resonances for the two microresonators corresponding to both pump frequencies overlap. We implement reconfigurable unidirectional coupling between the two DOPOs by using a coupling waveguide that directs a fraction of the DOPO1 output field to the input of DOPO2. The coupling strength of the coupling waveguide is adjusted by designing the separation between the bus waveguide and coupling waveguide. In our device, the ratio between the coupled field from DOPO1 and the DOPO2 field is 0.048 (see Supplementary Note 2). In order to minimize reflections, we have implemented a taper design on the end facets. We tune the phase coupling between the two DOPOs from in-phase to out-of-phase by thermally tuning the path length using microheaters.

Fig. 2: Experimental schematic of coupled DOPOs.
figure 2

a Microscope image of the device. The pump waves are split on-chip and sent to DOPO1 (bottom) and DOPO2 (top). A coupling waveguide after DOPO1 is used to send a fraction of the DOPO1 field to DOPO2. b Simulated group-velocity dispersion (GVD) of the SiN microresonator for the fundamental transverse electric (TE) mode. The waveguide cross section is 730 × 1050 nm2. The region of normal GVD is shaded, and the pump wavelengths are indicated with vertical lines. c Calculated parametric gain for 1 W of combined pump power. The pump waves are each located two free spectral ranges (FSRs) from the degeneracy point. d Experimental setup for measurement of coupled DOPO system. Two-frequency nondegenerate pumps are sent into the SiN chip. The output is collected using an aspheric lens and sent to a free-space interferometer to measure the interference signal. EDFA erbium-doped fiber amplifier, BPF bandpass filter, AOM acousto-optic modulator, FPC fiber polarization controller, MMI multimode interference splitter, OSA optical spectrum analyzer.

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  Figure 2d shows the experimental setup for generation and detection of the coupled DOPO signal. The SiN chip is pumped using two-frequency nondegenerate cw pump lasers that are offset by  ±2 FSRs from the degeneracy point at wavelengths of 1543 and 1559.1 nm. In order to ensure that the DOPO builds up from noise for each initiation, we modulate one of the pumps using an acousto-optic modulator (AOM) with 310-ns pulses at a repetition rate of 400 kHz. For simultaneous degenerate oscillation for both DOPOs, we use 72 mW of combined pump power in each bus waveguide and set the electrical power to the heater in DOPO1 to 13.5 mW such that the resonances of the two microresonators are spectrally overlapped. The two DOPO outputs from the chip are collimated using an aspheric lens and sent to our detection setup [dashed red box in Fig. 2a].

Phase characterization of coupled DOPOs

The readout of the coherent phase states of the coupled DOPO system is implemented by directly measuring the interference between the two DOPOs by coupling both outputs into a single fiber collimator and detecting the combined signal on a fast photodiode. The collimated outputs from the DOPOs are combined using a 50/50 beamsplitter and fiber coupled using a collimator. Before combining the beams, a 50/50 beamsplitter is used in each output arm to collect the individual DOPO signals. The signals are each fiber coupled and a 90/10 coupler is used to monitor the time trace and the optical spectrum. The generated DOPO spectra from the microresonators are shown in Fig. 3. In our interference measurement, we manipulate the coupling phase ϕc between the DOPOs below threshold by controlling the electrical power sent to the integrated heater while monitoring the time trace. Figure 4 shows the measured temporal interference signal (green) along with DOPO1 (red) and DOPO2 (blue), for three different heater powers. At 48.3 mW, we observe constructive interference between the two DOPOs, corresponding to in-phase operation (ϕout = 0) [Fig. 4 (top)]. For 54.2 mW of heater power, we observe destructive interference [Fig. 4 (middle)] corresponding to out-of-phase operation (ϕout = π). The transition from in-phase to out-of-phase operation occurs at 50.2 mW of heater power, and we observe both constructive and destructive interference [Fig. 4 (bottom)]. Moreover, we observe in-phase operation (ϕout = 0) for heater powers below 48.3 mW and out-of-phase operation (ϕout = π) for powers above 54.2 mW, indicating that the DOPOs above threshold operate in-phase or out-of-phase for a continuous range of coupling phases ϕc. The convergence time of the DOPO is well within the pump pulse duration of 310 ns. For comparison, we perform interference measurements on a similar device with no coupling waveguide between DOPO1 and DOPO2 and verify that there is no phase correlation between the two signals (see Supplementary Note 3).

Fig. 3: Optical spectra of two DOPO signals.
figure 3

The measured spectra of the primary DOPO (top, DOPO1) and the secondary DOPO (bottom, DOPO2).

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Fig. 4: Phase characterization of coupled DOPOs.
figure 4

Temporal ensemble of interference measurements of the combined DOPOs (green) along with measurements of DOPO1 (red) and DOPO2 (blue). Inset shows the phase chart of the coupling phase ϕc. We indicate the coupled DOPO operation regime for in-phase (pink), out-of-phase (blue), and transition regime (gray). a Constructive interference (in-phase, ϕout = 0) is observed for a heater power of 48.3 mW. b Destructive interference (out-of-phase ϕout = π) is observed for a heater power of 54.2 mW. c Transition from in-phase to out-of-phase operation for heater power of 50.2 mW. For clarity, DOPO2 and the interference are offset from DOPO1 in power by 40 and 80, respectively.

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Theoretical analysis of coupled DOPO system

We theoretically investigate this coupled DOPO system using coupled Lugiato–Lefever equations with two pump waves41,46,

$${t}_{{\rm{R}}}\frac{\partial {E}_{1}}{\partial t}\,= \, \left[-\alpha -i{\delta }_{0}-{\delta }_{1}\frac{\partial }{\partial \tau }+iL{\mathop {\sum}\limits_{k\ge 2}}\frac{{\beta }_{k}}{k!}{\left(i\frac{\partial }{\partial \tau }\right)}^{k}+i\gamma L| {E}_{1}(t,\tau ){| }^{2}\right]{E}_{1}(t,\tau )\\ +\sqrt{\theta }{A}_{{\rm{in}}}\left({e}^{-i{\Omega }_{0}\tau }+{e}^{i{\Omega }_{0}\tau }\right),$$
(1)
$${t}_{{\rm{R}}}\frac{\partial {E}_{2}}{\partial t}\,= \, \left[-\alpha -i{\delta }_{0}-{\delta }_{1}\frac{\partial }{\partial \tau }+iL{\mathop {\sum}\limits_{k\ge 2}}\frac{{\beta }_{k}}{k!}{\left(i\frac{\partial }{\partial \tau }\right)}^{k}+i\gamma L| {E}_{2}(t,\tau ){| }^{2}\right]{E}_{2}(t,\tau )\\ +\sqrt{\theta }{A}_{{\rm{in}}}\left({e}^{-i{\Omega }_{0}\tau }+{e}^{i{\Omega }_{0}\tau }\right) \ +\kappa {E}_{1}(t,\tau ),$$
(2)

where tR is the roundtrip time in the resonator, α is the total roundtrip loss, δ0 is the effective phase detuning, δ1 is the mode-dependent detuning, Ω0 corresponds to the pump offset frequency from the degeneracy point, θ is the transmission coefficient between the resonator and the bus waveguide, L is the cavity length, γ is the nonlinear parameter, and βk corresponds to the kth-order dispersion coefficients of the Taylor expansion of the propagation constant. Here, τ represents the temporal coordinate within the time scale of a single round trip and t represents the long-time-scale evolution over many round trips. The term with Ain describes the bichromatic pump waves, and κ represents the complex coupling coefficient from the primary DOPO (DOPO1) to the secondary DOPO (DOPO2) (see Supplementary Note 4). We add a noise of one photon per spectral mode with random phase onto the pump47. The simulation parameters are similar to that of our experiment, including the coupling between the DOPOs being unidirectional. Our simulations (Fig. 5) show oscillation behavior for three different values of the coupling phase ϕc. For ϕc = 0°, constructive interference between the DOPOs is favored [Fig. 5 (top)], indicating that the DOPOs oscillate in-phase. We observe similar in-phase behavior for  −70° < ϕc < 70°. In contrast, for ϕc = 180°, we observe destructive interference [Fig. 5 (middle)] from the DOPOs oscillating π out-of-phase. Likewise, similar out-of-phase behavior is observed for 110° < ϕc < 250°. These predictions are consistent with our experimental results. In addition, we have numerically explored the transition region between in-phase and out-of-phase operation. Figure 5 (bottom) shows the simulated interference for ϕc = 90°. Here, we observe that the secondary DOPO no longer oscillates out-of-phase with respect to the primary DOPO and becomes frustrated with the oscillation becoming uncorrelated, which is also consistent with our experimental observations for the transition region (heater power of 50.2 mW). Work is ongoing to determine the degree of phase tolerance as the number of DOPOs is increased.

Fig. 5: Numerical modeling of coupled DOPO system.
figure 5

Simulated temporal interference between two DOPOs with unidirectional coupling. The plot shows the combined DOPO (green) and the individual DOPOs, DOPO1 (red) and DOPO2 (blue) for coupling phase ϕc of 0° (in-phase, ϕout = 0), 180° (out-of-phase, ϕout = π), and 90° (transition from in-phase to out-of-phase) from top to bottom.

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Scalability of coupled DOPO system

Lastly, we discuss the potential of a large-scale DOPO photonic system in light of power consumption and computing speed. Figure 6a shows the combined pump power required for a single microresonator-based DOPO to oscillate as a function of the intrinsic Q-factor of the SiN microresonator for critical coupling. The measured values for the current device and a similar single DOPO device are denoted with a diamond (see Supplementary Note 5). For a Q-factor of 10 million, oscillation can occur with 1  mW of combined pump power, implying that 1000 DOPOs can be pumped simultaneously with an on-chip optical power of 1 W, offering promise for scaling to a large number of DOPOs (see Supplementary Note 6). Here, since the power scales as 1/Q2 and the lifetime scales as Q, the energy required for the entire computation scales as 1/Q. With reduction of surface roughness, Q-factors of 37 million have been achieved in high-confinement SiN microresonators48, which could further reduce the pump power to 80 μW per DOPO (labeled with a triangle in Fig. 6a), at the expense of computing rate. In addition, Fig. 6b, c shows the simulated DOPO oscillation time for two different Q-factors at near-threshold pump powers. For a single DOPO in a microresonator with an intrinsic Q of 1.26 million (our experiment), we observe a convergence time of 174 ns with a cavity lifetime of 0.52 ns based on our numerical model. For a microresonator with an intrinsic Q of 10 million, we observe convergence times of 540 ns. Experimentally in our 2-DOPO system, we achieve a convergence time <310 ns with a computing rate of 400 kHz. Furthermore, we have performed numerical modeling of a N = 100 DOPO system and observe an annealing time Tann = 0.2 μs with a 35% ground-state success probability (see Supplementary Note 7). We calculate a time to solution, \({T}_{{\rm{sol}}}=\frac{\mathrm{log}\,(0.01)}{\mathrm{log}\,(1-P)}=2.1\) μs6,23. As a comparison, the fiber-based DOPO system has a Tsol = 3.3 ms for solving cubic MAX-CUT problems and Tsol = 2.3 ms for dense MAX-CUT with N = 10023, suggesting that the spatial-multiplexing approach offers promise for accelerating the computing time of an Ising model. As the number of DOPOs in the network increases, the anneal time that corresponds to the pump turn-on time must be controlled to slow down the DOPO dynamics to prevent freeze-out effects that prevent the system from reaching the ground-state solution23,35. More specifically for our system of microresonator-based DOPOs, it has been reported elsewhere that the rate at which the DOPOs are tuned into resonance is a decisive parameter that determines the success probability of finding the ground-state solution (see Supplementary Note 8)41. As the pump-to-resonance detuning dictates the power build-up within the microresonator, this observation is consistent with previous studies where the pump power was ramped up at a controlled rate to improve the performance of their systems20. In other words, the detuning is the more natural control parameter for our microresonator-based DOPOs, which must be carefully tuned. This fact will be subject to a more comprehensive investigation and reported elsewhere. We also observe that the oscillation threshold is reduced for a 2-DOPO system for both in-phase and out-of-phase couplings41, which leads to apparent faster convergence times as compared to a single DOPO. Investigations are ongoing for the optimal turn-on time as the problem size further increases for our spatial-multiplexed DOPO system. Finally, we have developed a preliminary 4-DOPO system and have observed DOPO generation in a single microresonator as shown in Supplementary Note 9. Future work will implement reconfigurable phase coupling between the DOPOs via integrated heaters.

Fig. 6: Scaling to larger number of coupled DOPOs.
figure 6

a Combined pump power required for single DOPO as a function of the intrinsic Q-factor of the microresonator. The two different pump powers (and the corresponding Q) used in our experiments is denoted with diamonds. Triangle (green) denotes pump power based on the Q of state-of-the-art SiN microresonators48. Simulated DOPO convergence for b Qint = 1.26 × 106, which corresponds to a cavity lifetime τp = 0.52 ns, and c Qint = 10 × 106, which corresponds to τp = 4.1 ns, for critical coupling.

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Discussion

In conclusion, we demonstrate phase reconfigurable all-optical coupling between microresonator-based DOPOs in an integrated silicon nitride platform. We experimentally observe that the system is highly tolerant to the coupling phase between the DOPOs, offering flexibility in setting up the system when scaling to larger number of DOPOs. Since the system does not rely on time multiplexing, and the computation time is comparable to the build-up time of a single DOPO, it is possible to rapidly test the fidelity of the final state. To enable reconfigurable coupling between arbitrary DOPOs, a fraction of the DOPO power will be sent to a matrix of 2N × 2N Mach–Zehnder interferometers (MZIs) to perform N × N arbitrary linear transformations49,50,51. To achieve this on a single device layer, intersections between waveguides are required. Our transmission measurements (see Supplementary Note 10) indicate 0.45 dB loss per intersection, which can be further optimized. In addition, we are investigating a multilayer design52 where the coupling matrix based on MZIs is located in a different material plane, which uses silicon for the waveguides. Due to the larger thermo-optic effect in silicon, the MZIs can be further miniaturized such that each MZI element resides on a footprint of 100 μm × 100 μm (0.01 mm2)53, which is significantly smaller than that required for a SiN MZI (0.7 mm2)54. Such a design would enable all-to-all coupling among the DOPOs with arbitrary weightings and the resulting system should in principle allow the implementation of an arbitrary Hamiltonian. Our results provide the initial building blocks for the realization of a large-scale DOPO network for studying nontrivial coupled oscillator dynamics, offering potential towards the realization of a chip-based photonic Ising machine.

Methods

Experimental setup

The setup is shown in Fig. 2a. Two tunable cw lasers (New Focus Velocity) are amplified using an erbium-doped fiber amplifier (EDFA) to use as pumps. The high-wavelength pump is modulated using an AOM, where the modulation depth and frequency are carefully chosen such that the DOPO signal reaches the noise level each time the AOM turned off. The pulse duration, repetition rate, and the extinction ratio are largely dependent on the quality factor of the microresonator39. For our system, with our pulse parameters we use an extinction ratio of 5:1 for our AOM. This extinction ratio is sufficient to reduce the pump power to a level below the oscillation threshold and, this together with our choice of pulse duration and repetition rate, allows for the DOPO signal to decay to a level below the noise before the initiation of the next period. In order to suppress the amplified spontaneous emission from the EDFA at the DOPO wavelength, we use 9-nm-wide bandpass filters after each EDFA. Fiber polarization controllers are used in each arm to set the input polarization to TE. For the chip output, the individual DOPO arms and the interferometer output are collected using a fiber collimator and the DOPO wavelength is filtered using a tunable bandpass filter with a 0.8-nm bandwidth and sent to an InGaAs amplifier photodetector (150-MHz bandwidth) and a 1-GHz real-time oscilloscope.