  Quantum acoustodynamics (QAD) is a rapidly developing field of research,
offering possibilities to realize and study macroscopic quantum-mechanical
systems in a new range of frequencies, and implement transducers and new types
of memories for hybrid quantum devices. Here we propose a novel design for a
versatile diamond QAD cavity operating at GHz frequencies, exhibiting effective
mode volumes of about $10^{-4}\lambda^3$. Our phononic crystal waveguide cavity
implements a non-resonant analogue of the optical lightning-rod effect to
localize the energy of an acoustic mode into a deeply-subwavelength volume. We
demonstrate that this confinement can readily enhance the orbit-strain
interaction with embedded nitrogen-vacancy (NV) centres towards the
high-cooperativity regime, and enable efficient resonant cooling of the
acoustic vibrations towards the ground state using a single NV. This
architecture can be readily translated towards setup with multiple cavities in
one- or two-dimensional phononic crystals, and the underlying non-resonant
localization mechanism will pave the way to further enhance optoacoustic
coupling in phoxonic crystal cavities.
