  Acoustic spontaneous emission into bulk dielectrics can be a strong source of
decoherence in quantum devices, especially when a qubit is in the presence of
piezoelectric materials. We study the dynamics of a qubit coupled to an
acoustic resonator by a piezoelectric film. By varying the surface topography
of the resonator from rough to polished to shaped, we explore the crossover
from fast decay of an excited qubit to quantum-coherent coupling between the
qubit and an isolated phonon mode. Our experimental approach may be used for
precision measurements of crystalline vibrations, the design of quantum
memories, and the study of electro-mechanical contributions to dielectric loss.
