  The techniques of cavity optomechanics have enabled significant achievements
in precision sensing, including the detection of gravitational waves and the
cooling of mechanical systems to their quantum ground state. Recently, the
inherent non-linearity in the optomechanical interaction has been harnessed to
explore synchronization effects, including the spontaneous locking of an
oscillator to a reference injection signal delivered via the optical field.
Here, we present the first demonstration of a radiation-pressure driven
optomechanical system locking to an inertial drive, with actuation provided by
an integrated electrical interface. We use the injection signal to suppress
drift in the optomechanical oscillation frequency, strongly reducing phase
noise by over 55 dBc/Hz at 2 Hz offset. We further employ the injection tone to
tune the oscillation frequency by more than 2 million times its narrowed
linewidth. In addition, we uncover previously unreported synchronization
dynamics, enabled by the independence of the inertial drive from the optical
drive field. Finally, we show that our approach may enable control of the
optomechanical gain competition between different mechanical modes of a single
resonator. The electrical interface allows enhanced scalability for future
applications involving arrays of injection-locked precision sensors.
