The successes of superconducting quantum circuits at local manipulation of quantum information and photonics technology at long-distance transmission of the same have spurred interest in the development of quantum transducers for efficient, low-noise, and bidirectional frequency conversion of photons between the microwave and optical domains. We propose to realize such functionality through the coupling of electrical, piezoelectric, and optomechanical resonators. The coupling of the mechanical subsystems enables formation of a resonant mechanical supermode that provides a mechanically-mediated, efficient single interface to both the microwave and optical domains. The conversion process is analyzed by applying an equivalent circuit model that relates device-level parameters to overall figures of merit for conversion efficiency $eta$ and added noise $N$. These can be further enhanced by proper impedance matching of the transducer to an input microwave transmission line. The performance of potential transducers is assessed through finite-element simulations, with a focus on geometries in GaAs, followed by considerations of the AlN, LiNbO$_3$, and AlN-on-Si platforms. We present strategies for maximizing $eta$ and minimizing $N$, and find that simultaneously achieving $eta>50~%$ and $N < 0.5$ should be possible with current technology. We find that the use of a mechanical supermode for mediating transduction is a key enabler for high-efficiency operation, particularly when paired with an appropriate microwave impedance matching network. Our comprehensive analysis of the full transduction chain enables us to outline a development path for the realization of high-performance quantum transducers that will constitute a valuable resource for quantum information science.
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