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

We performed measurements of the photon correlation $\left[g^{\left(2\right)}\left(\tau \right)\right]$ in driven nonlinear high-$Q$ silicon photonic crystal microcavities. The measured $g^{\left(2\right)}\left(\tau \right)$ exhibits damped oscillatory behavior when the input pump power exceeds a critical value. From a comparison between experiments and simulations, we attribute the measured oscillation of $g^{\left(2\right)}\left(\tau \right)$ to self-pulsing (a limit cycle) emerging from an interplay between the photon, carrier, and thermal dynamics. Namely, the oscillation frequency of $g^{\left(2\right)}\left(\tau \right)$ 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 $g^{\left(2\right)}\left(\tau \right)$ 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 $g^{\left(2\right)}\left(\tau \right)$ up to the order of microseconds, which could be a demonstration of the stabilization of a stochastic limit cycle through pumping.

Figure 1

(a) Illustration of a limit cycle emerging from a driven dissipative nonlinear system (left) and a laser scanning microscope image of the high-$Q$ Si PhC microcavity (right). (b) Laser transmission spectrum of the cavity showing the resonance of a fundamental mode. (c) Output intensity $I_{\mathrm{out}}$ as a function of input power $P_{\mathrm{in}}$, where $P_c$ is the critical laser input power for self-pulsing. BS and SP represent bistable and self-pulsing regions, respectively. Here, $P_{\mathrm{in}}$ is the fiber output power of the tunable semiconductor laser.

Figure 2

(a) Examples of measured photon correlations with the start-stop HBT interferometer $P_2\left(\tau \right)$ with a fitting curve by Eq. (1) and reconstructed normalized second-order photon-correlation functions $g^{\left(2\right)}\left(\tau \right)$. (b) $g^{\left(2\right)}\left(0\right)$ (top), the oscillation frequency ${\omega }_r$ (middle), and the coherence time ${\tau }_r$ (bottom) of measured $g^{\left(2\right)}\left(\tau \right)$. The critical input power of self-pulsing is $P_c\simeq 0.6$ mW. (c) Real-time trajectories of the light output measured with an avalanche photodiode (APD) for two pump powers. For measurements, the detuning was fixed as $\delta \simeq -2$.

Figure 3

(a) Simulated self-pulsing (SP) and bistable (BS) regions as a map of detuning $\delta$ and laser input power $P_{\mathrm{in}}$, where the solid red circle represents the onset of SP for $\delta =-2$. (b) Simulated $g^{\left(2\right)}\left(\tau \right)$ and real-time evolution of carrier $n\left(t\right)$, thermal effect $\theta \left(t\right)$, and the light output $I\left(t\right)={\left|\alpha \left(t\right)\right|}^2$ for $P_{\mathrm{in}}=1.3$. (c) $g^{\left(2\right)}\left(0\right)$ (top), oscillation frequencies ${\omega }_r$ (middle), and the coherence time ${\tau }_r$ (bottom) of the simulated $g^{\left(2\right)}\left(\tau \right)$. For simulations, the detuning $\delta =-2$ was used. The critical input power of self-pulsing is $P_c=0.6$. While (a) is the result with the deterministic coupled-mode equations, (b) and (c) are the results with stochastic coupled-mode equations with noise terms.