## Degenerate optical parametric oscillator for optical computing

Coupled networks of degenerate optical parametric oscillators (OPO’s) have been explored recently to realize a novel form of coherent computing by simulating the classical Ising model, which is a non-deterministic polynomial time (NP)-hard computation problem. The approach is based on the non-equilibrium phase transition that occurs at the oscillation threshold for the OPO, where the oscillating field phase-locks with respect to the pump field to one of two possible states offset by π, analogous to a binary spin system. By utilizing a network of coupled OPO’s, more complex phase-locked states can be achieved which corresponds the ground state of the Ising model, such that the coupling strength of its spins is equivalent to the coupling between the OPO’s. Previous demonstrations based on a χ^{(2)} OPO using bulk optics offers promise for this approach. Alternatively, the χ^{(3)} nonlinearity can be utilized to realize a degenerate OPO with dynamics that mirrors the χ^{(2)} system. The silicon nitride (Si_{3}N_{4}) photonics platform uniquely allows for a robust, compact, integrated system that offers potential scaling to many coupled OPO’s on a single chip. We demonstrate a scheme in which a Si_{3}N_{4} microresonator with normal group-velocity dispersion is dual-pumped to generate a frequency-degenerate signal and idler pair through parametric four-wave mixing (FWM) [Opt. Lett. 40, 5267 (2015)].

Alternatively, we use the generated bi-phase state to realize a quantum random number generator (RNG). RNG’s are a critical component for applications including cryptography, Monte Carlo simulations, statistical sampling, and quantitative finance. While many algorithms exist in computer programming for random number generation, these generate pseudo-random numbers are not truly indeterministic. In our system, since the parametric oscillation is initiated from quantum noise, the system is intrinsically unbiased and only requires the detection of strong, classical signals with no post-processing, greatly reducing the complexity and required computational overhead. We achieve a generation rate of 2 MHz by amplitude modulation of one of the two pump lasers. We verify the generation of the bi-phase state using an asymmetric Mach-Zehnder interferometer to measure the relative phase between adjacent bits in the generated pulse train. In addition, we analyze our sample bits using the National Institute of Standards and Technology (NIST) Statistical Test Suite to verify the randomness of our generated output [Opt. Lett. 41, 4194 (2016)].

## High speed signal processing using space-time duality

The space-time duality of electromagnetic waves originates from the equivalence between the paraxial diffraction of a spatial field and dispersive propagation of a temporal field. The duality implies that spatial optical components such as a lens or prism have temporal counterparts known as a time-lens or time-prism, which can be implemented by applying the appropriate temporal phase shift. We investigate the application of nonlinear optical versions of these temporal elements for ultrafast signal processing. For example, a temporal imaging system consisting of one lens can magnify a waveform or generate the Fourier transform of the waveform allowing for characterization of ultrafast temporal features using a relatively slow photodetector or even a spectrometer [Opt. Lett. 33, 1047 (2008), Nature 456, 81 (2008), Opt. Express 17, 4324 (2009), Opt. Express 18, 14262 (2010)]. Additionally, we have investigated the use of a two-lens temporal telescopic system to compress optical waveforms and generate 2-ps minimum features [Nat. Photonics 3, 581-585 (2009)], to magnify narrow (GHz) spectral features allowing for characterization with a low-resolution spectrometer [Opt. Express 17, 5691 (2009)], and to compensate for dispersion and nonlinearity using spectral phase conjugation [Opt. Express 17, 20605 (2009)].