  Nanomechanical systems made from phononic crystals can act as highly coherent
microwave-frequency circuits at cryogenic temperatures. However, generating
sufficient coupling between these devices and microwave superconducting quantum
circuits is challenging due to the vastly different length scales of acoustic
and electrical excitations. Here we demonstrate a general recipe for
calculating these interactions and show that large piezoelectric coupling rates
between microwave superconducting circuits containing Josephson junctions and
nanoscale phononic resonances are possible, suggesting a route to phononic
crystal circuits and systems that are nonlinear at the single-phonon level.
