  Arrays of coupled phonon cavities each including an impurity qubit in silicon
are considered. We study experimentally feasible architectures that can exhibit
quantum many-body phase transitions of phonons, e.g. Mott insulator and
superfluid states, due to a strong phonon-phonon interaction (which is mediated
by the impurity qubit-cavity phonon coupling). We investigate closed
equilibrium systems as well as driven dissipative non-equilibrium systems at
zero and non-zero temperatures. Our results indicate that quantum many-body
phonon systems are achievable both in on-chip nanomechanical systems in silicon
and distributed Bragg reflector phonon cavity heterostructures in
silicon-germanium. Temperature and driving field are shown to play a critical
role in achieving these phonon superfluid and insulator states, results that
are also applicable to polariton systems. Experimental procedures to detect
these states are also given. Cavity-phoniton systems enable strong
phonon-phonon interactions as well as offering long wavelengths for forming
extended quantum states; they may have some advantage in forming truly quantum
many-body mechanical states as compared to other optomechanical systems.
