  Radiation pressure within engineered structures has recently been used to
couple the motion of nanomechanical objects with high sensitivity to optical
and microwave electromagnetic fields. Here, we demonstrate a form of
electromechanical crystal for coupling microwave photons and hypersonic phonons
by embedding the vacuum-gap capacitor of a superconducting resonator within a
phononic crystal acoustic cavity. Utilizing a two-photon resonance condition
for efficient microwave pumping and a phononic bandgap shield to eliminate
acoustic radiation, we demonstrate large cooperative coupling ($C \approx 30$)
between a pair of electrical resonances at $10$GHz and an acoustic resonance at
$0.425$GHz. Electrical read-out of the phonon occupancy shows that the
hypersonic acoustic mode has an intrinsic energy decay time of $2.3$ms and
thermalizes close to its quantum ground-state of motion (occupancy $1.5$) at a
fridge temperature of $10$mK. Such an electromechanical transducer is
envisioned as part of a hybrid quantum circuit architecture, capable of
interfacing to both superconducting qubits and optical photons.
