  Suspended high-overtone bulk acoustic wave resonators (HBARs) can serve as a
viable optomechanical platform for efficient transduction of signals from the
microwave to the optical frequency domain. In contrast to 1D nanobeam
optomechanical crystals, HBARs benefit from very high RF to phonon injection
efficiency ($\eta_{PIE}\approx$1) and low optical pump induced heating at
cryogenic temperatures. By building small mode volume optical cavities around
these devices, one can in principle achieve optomechanical cooperativities
comparable to 1D nanobeam optomechanical crystals. In this work, we demonstrate
cryogenic operation ($\approx$10 K) of such suspended HBAR devices and show
classical signal modulation upto 3.5 GHz and response times $\approx$ 524 ns
(for the fundamental mode at 340 MHz). While the transduction efficiency is
currently limited by the material and device fabrication processes used in this
work, we show that with reasonable modifications, efficient quantum
transduction is within reach using this approach.
