  We consider the steady states of a harmonic oscillator coupled so strongly to
a two-level system (a qubit) that the rotating wave approximation cannot be
made. The Hamiltonian version of this model is known as the $E\otimes\beta$
Jahn-Teller model. The semiclassical version of this system exhibits a fixed
point bifurcation, which in the quantum model leads to a ground state with
substantial entanglement between the oscillator and the qubit. We show that the
dynamical bifurcation survives in a dissipative quantum description of the
system, amidst an even richer bifurcation structure. We propose two
experimental implementations of this model based on superconducting cavities: a
parametrically driven nonlinear nanomechanical resonator coupled capacitively
to a coplanar microwave cavity and a superconducting junction in the central
conductor of a coplanar waveguide.
