  A quantum mechanical superposition of a long-lived, localized phonon and a
matter excitation is described. We identify a realization in strained silicon:
a low-lying donor transition (P or Li) driven solely by acoustic phonons at
wavelengths where high-Q phonon cavities can be built. This phonon-matter
resonance is shown to enter the strongly coupled regime where the "vacuum" Rabi
frequency exceeds the spontaneous phonon emission into non-cavity modes, phonon
leakage from the cavity, and phonon anharmonicity and scattering. We introduce
a micropillar distributed Bragg reflector Si/Ge cavity, where Q=10^5-10^6 and
mode volumes V<=25*lambda^3 are reachable. These results indicate that single
or many-body devices based on these systems are experimentally realizable.
