  Spontaneous locking of the phase of a coherent phonon source to an external
reference is demonstrated in an optomechanical oscillator based on a
self-triggered free-carrier/temperature limit cycle. Synchronization is
observed when the pump laser driving the mechanical oscillator to a
self-sustained state is modulated by a radiofrequency tone. We employ a
pump-probe phonon detection scheme based on an independent optical cavity to
observe only the mechanical oscillator dynamics. The lock range of the
oscillation frequency, i.e., the Arnold tongue, is experimentally determined
over a range of external reference strengths, evidencing the possibility to
tune the oscillator frequency for a range up to 350 kHz. The stability of the
coherent phonon source is evaluated via its phase noise, with a maximum
achieved suppression of 44 dBc/Hz at 1kHz offset for a 100 MHz mechanical
resonator. Introducing a weak modulation in the excitation laser reveals as a
further knob to trigger, control and stabilise the dynamical solutions of
self-pulsing based optomechanical oscillators, thus enhancing their potential
as acoustic wave sources in a single layer silicon platform.
