Engineering interactions between superconducting qubits and phononic nanostructures

Nanomechanical systems can support highly coherent microwave-frequency excitations at cryogenic temperatures. However, generating sufficient coupling between these devices and superconducting quantum circuits is challenging due to the vastly different length scales of acoustic and electromagnetic excitations. Here we demonstrate a general method for calculating piezoelectric interactions between quantum circuits and arbitrary phononic nanostructures. We illustrate our technique by studying the coupling between a transmon qubit and bulk acoustic-wave, Lamb-wave, and phononic crystal resonators, and show that very large coupling rates are possible in all three cases. Our results suggest a route to phononic circuits and systems that are nonlinear at the single-phonon level.