Nanomechanical systems made from phononic crystals can act as highly coherentmicrowave-frequency circuits at cryogenic temperatures. However, generatingsufficient coupling between these devices and microwave superconducting quantumcircuits is challenging due to the vastly different length scales of acousticand electrical excitations. Here we demonstrate a general recipe forcalculating these interactions and show that large piezoelectric coupling ratesbetween microwave superconducting circuits containing Josephson junctions andnanoscale phononic resonances are possible, suggesting a route to phononiccrystal circuits and systems that are nonlinear at the single-phonon level.
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