  Connecting nanoscale mechanical resonators to microwave quantum circuits
opens new avenues for storing, processing, and transmitting quantum
information. In this work, we couple a phononic crystal cavity to a tunable
superconducting quantum circuit. By fabricating a one-dimensional periodic
pattern in a thin film of lithium niobate and introducing a defect in this
artificial lattice, we localize a 6 gigahertz acoustic resonance to a
wavelength-scale volume of less than one cubic micron. The strong
piezoelectricity of lithium niobate efficiently couples the localized
vibrations to the electric field of a widely tunable high-impedance Josephson
junction array resonator. We measure a direct phonon-photon coupling rate
$g/2\pi \approx 1.6 \, \mathrm{MHz}$ and a mechanical quality factor
$Q_\mathrm{m} \approx 3 \times 10^4$ leading to a cooperativity $C\sim 4$ when
the two modes are tuned into resonance. Our work has direct application to
engineering hybrid quantum systems for microwave-to-optical conversion as well
as emerging architectures for quantum information processing.
