  Bulk acoustic wave (BAW) resonators are attractive as intermediaries in a
microwave-to-optical transducer, due to their long coherence times and
controllable coupling to optical photons and superconducting qubits. However,
for an optomechanical transducer to operate without detrimental added noise,
the mechanical modes must be in the quantum ground state. This has proven
challenging in recent demonstrations of transduction based on other types of
mechanical resonators, where absorption of laser light caused heating of the
phonon modes. In this work, we demonstrate ground state operation of a
Brillouin optomechanical system composed of a quartz BAW resonator inside an
optical cavity. The system is operated at $\sim$200 mK temperatures inside a
dilution refrigerator, which is made possible by designing the system so that
it self-aligns during cooldown and is relatively insensitive to mechanical
vibrations. We show optomechanical coupling to several phonon modes and perform
sideband asymmetry thermometry to demonstrate a thermal occupation below 0.5
phonons at base temperature. This constitutes the heaviest ($\sim$494 $\mu$g)
mechanical object measured in the quantum ground state to date. Further
measurements confirm a negligible effect of laser heating on this phonon
occupation. Our results pave the way toward low-noise, high-efficiency
microwave-to-optical transduction based on BAW resonators.
