  Bulk acoustic resonators support robust, long-lived mechanical modes, capable
of coupling to various quantum systems. In separate works, such devices have
achieved strong coupling to both superconducting qubits, via piezoelectricity,
and optical cavities, via Brillouin interactions. In this work, we present a
novel hybrid microwave/optical platform capable of coupling to bulk acoustic
waves through cavity-enhanced piezoelectric and photoelastic interactions. The
modular, tunable system achieves fully resonant and well-mode-matched
interactions between a 3D microwave cavity, a high-frequency bulk acoustic
resonator, and a Fabry Perot cavity. We realize this piezo-Brillouin
interaction in x-cut quartz, demonstrating the potential for strong
optomechanical interactions and high cooperativity using optical cavity
enhancement. We further show how this device functions as a bidirectional
electro-opto-mechanical transducer, with quantum efficiency exceeding
$10^{-8}$, and a feasible path towards unity conversion efficiency. The high
optical sensitivity and ability to apply large resonant microwave field in this
system also offers a new tool for probing anomalous electromechanical
couplings, which we demonstrate by investigating (nominally-centrosymmetric)
CaF$_2$ and revealing a parasitic piezoelectricity of 83 am/V. Such studies are
an important topic for emerging quantum technologies, and highlight the
versatility of this new hybrid platform.
