  A device capable of converting single quanta of the microwave field to the
optical domain is an outstanding endeavour in the context of quantum
interconnects between distant superconducting qubits, but likewise can have
applications in other fields, such as radio astronomy or, in the classical
realm, microwave photonics. A variety of transduction approaches, based on
optomechanical or electro-optical interactions, have been proposed and
realized, yet the required vanishing added noises and an efficiency approaching
unity, have not yet been attained. Here we present a new transduction scheme
that could in theory satisfy the requirements for quantum coherent
bidirectional transduction. Our scheme relies on an intermediary mechanical
mode, a high overtone bulk acoustic resonance (HBAR), to coherently couple
microwave and optical photons through the piezoelectric and strain-optical
effects. Its efficiency results from the combination of integrated Si3N4
photonic circuits with ultra low loss sustaining high intracavity photon
numbers with the highly efficient microwave to mechanical transduction offered
by piezoelectrically coupled HBAR. We develop a quantum theory for this
multipartite system by first introducing a quantization method for the
piezoelectric interaction between the microwave mode and the mechanical mode
from first principles (which to our knowledge has not been presented in this
form), and link the latter to the conventional Butterworth-Van Dyke model. The
HBAR is subsequently coupled to a pair of hybridized optical modes from coupled
optical ring cavities via the strain-optical effect. We analyze the conversion
capabilities of the proposed device using signal flow graphs, and demonstrate
that near quantum coherent transduction is possible, with realistic
experimental parameters.
