Photon correlation measurements of stochastic limit cycles emerging from high-$Q$ nonlinear silicon photonic crystal microcavities

We performed measurements of photon correlation [$g^{(2)}(\tau)$] in driven nonlinear high-$Q$ silicon (Si) photonic crystal (PhC) microcavities. The measured $g^{(2)}(\tau)$ exhibits a damped oscillatory behavior when input pump power exceeds a critical value. From comparison between experiments and simulations, we attribute the measured oscillation of $g^{(2)}(\tau)$ to self-pulsing (a limit cycle) emerging from an interplay between photon, carrier, and thermal dynamics. Namely, the oscillation frequency of $g^{(2)}(\tau)$ corresponds to the oscillation period of the limit cycle, while its finite coherence (damping) time originates from the stochastic nature of the limit cycle. From the standpoint of phase reduction theory, we interpret the measured coherence time of $g^{(2)}(\tau)$ as the coherence (diffusion) time of a generalized phase of the limit cycle. Furthermore, we show that an increase in laser input power enhances the coherence time of $g^{(2)}(\tau)$ up to the order of microseconds, which could be a demonstration of the stabilization of a stochastic limit cycle through pumping.