Coupled degenerate parametric oscillators have been recently proposed as a physical platform for efficiently solving general Ising spin networks [1], a computationally hard problem. We study theoretically and experimentally the most minimal utilization of such systems, a pair of coupled degenerate oscillators, and show that the observed behavior is much richer than a simple spin-1/2 description.

Parametric oscillators have attracted a significant attention during the last decades due to their potential applications ranging from quantum optics to quantum information and communication. When a parametric oscillator is driven above its threshold by an external pump, it oscillates with a frequency which is exactly half of the frequency of its pump, with phase of either  or  with respect to the pump, thus giving rise to an effective classical spin-1/2 degree of freedom that explicitly breaks time translational symmetry. Because of this feature, networks of many coupled parametric oscillators have been proposed as a potential platform for solving complex combinatorial and minimization problems. Coupled parametric oscillators have been recently employed as simulators of artificial Ising lattices, with the potential to efficiently solve important minimization problems.

We report on a detailed study of the most minimal realization of a coherent Ising machine, i.e., two coupled degenerate parametric oscillators, exploring the entire phase diagram, in terms of pump power, phase and coupling strength, both analytically and experimentally in a radio-frequency (RF) experiment. In addition to a regime where the oscillators act as coupled spin-1/2 degrees of freedom, in the vicinity of the oscillation threshold there is a very wide range of parameters where the spin-1/2 description does not apply and the oscillators never reach steady state, but rather show persistent, full-scale, coherent beats, whose frequency reflects the coupling strength. Our study unveils the rich coherent dynamics of coupled parametric oscillators and shows a routable way to use parametric amplification to conserve coherence indefinitely.

Bibliography

1. T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, Nat. Phot. 10, 415 (2016).