  We describe a chip-based, solid-state analogue of cavity-QED utilizing
acoustic phonons instead of photons. We show how long-lived and tunable
acceptor impurity states in silicon nanomechanical cavities can play the role
of a matter non-linearity for coherent phonons just as, e.g., the Josephson
qubit plays in circuit-QED. Both strong coupling (number of Rabi oscillations ~
100) and strong dispersive coupling (0.1-2 MHz) regimes can be reached in
cavities in the 1-20 GHz range, enabling the control of single phonons,
phonon-phonon interactions, dispersive phonon readout of the acceptor qubit,
and compatibility with other optomechanical components such as phonon-photon
translators. We predict explicit experimental signatures of the acceptor-cavity
system.
