  Breather solitons in passive, driven, Kerr-nonlinear optical resonators are
circulating pulses of light that have amplitude, bandwidth, and duration that
oscillate in time over the course of many resonator round trips. These breather
oscillations emerge spontaneously, and their frequency is determined by the
system parameters. Here we propose and theoretically investigate the
possibility that these breather oscillations may become subharmonically
entrained to the periodic perturbation occurring at the resonator round-trip
time $T_R$ that arises from out-coupling and pumping. The result of subharmonic
entrainment is an integer ratio $T_b/T_R=N\gg1$ between the breather period
$T_b$ and the round-trip time. We find that in a regime of intermediate
resonator finesse ($\mathcal{F}\sim30-40$), rigid subharmonic entrainment---or
locking of the breather oscillation to the round-trip time---occurs and is
maintained over a range of system parameters. We observe entrainment with an
integer locking ratio as high as $N=20$, and propose a way to realize the
effect with higher entrainment ratios and at higher finesse. This system's
subharmonic response to governing equations that are invariant under discrete
time translations is similar to recently proposed and reported discrete time
crystals in quantum many-body systems. We discuss a route towards realization
of these breather-soliton time crystals in experiment, and explain how their
incorporation into photonics systems could simplify proposed applications of
microresonator solitons. Importantly, our work explores a new regime of
microresonator Kerr-nonlinear dynamics in which the round-trip-time
perturbation plays a significant role.
