  Microwave to optical transduction has received a great deal of interest from
the cavity optomechanics community as a landmark application for
electro-optomechanical systems. In this Letter, we demonstrate a novel
transducer that combines high-frequency mechanical motion and a microwave
cavity for the first time. The system consists of a 3D microwave cavity and a
gallium arsenide optomechanical crystal, which has been placed in the microwave
electric field maximum. This allows the microwave cavity to actuate the
gigahertz-frequency mechanical breathing mode in the optomechanical crystal
through the piezoelectric effect, which is then read out using a telecom
optical mode. The gallium arsenide optomechanical crystal is a good candidate
for low-noise microwave-to-telecom transduction, as it has been previously
cooled to the mechanical ground state in a dilution refrigerator. Moreover, the
3D microwave cavity architecture can naturally be extended to couple to
superconducting qubits and to create hybrid quantum systems.
