  Demonstrating a device that efficiently connects light, motion, and
microwaves is an outstanding challenge in classical and quantum photonics. We
make significant progress in this direction by demonstrating a photonic crystal
resonator on thin-film lithium niobate (LN) that simultaneously supports
high-$Q$ optical and mechanical modes, and where the mechanical modes are
coupled piezoelectrically to microwaves. For optomechanical coupling, we
leverage the photoelastic effect in LN by optimizing the device parameters to
realize coupling rates $g_0/2\pi\approx 120~\textrm{kHz}$. An optomechanical
cooperativity $C>1$ is achieved leading to phonon lasing. Electrodes on the
nanoresonator piezoelectrically drive mechanical waves on the beam that are
then read out optically allowing direct observation of the phononic bandgap.
Quantum coupling efficiency of $\eta\approx10^{-8}$ from the input microwave
port to the localized mechanical resonance is measured. Improvements of the
microwave circuit and electrode geometry can increase this efficiency and bring
integrated ultra-low-power modulators and quantum microwave-to-optical
converters closer to reality.
