  We propose a hybrid system with quantum mechanical three-body interactions
between photons, phonons, and qubit excitations. These interactions take place
in a circuit quantum electrodynamical architecture with a superconducting
microwave resonator coupled to a transmon qubit whose shunt capacitance is free
to mechanically oscillate. We show that this system design features a
three-mode polariton--mechanical mode and a nonlinear transmon--mechanical mode
interaction in the strong coupling regime. Together with the strong
resonator--transmon interaction, these properties provide intriguing
opportunities for manipulations of this hybrid quantum system. We show, in
particular, the feasibility of cooling the mechanical motion down to its ground
state and preparing various nonclassical states including mechanical Fock and
cat states and hybrid tripartite entangled states.
