  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.
