Abstract
We consider a theoretical model for a nonlinear nanomechanical resonator
coupled to a superconducting microwave resonator. The nanomechanical resonator
is driven parametrically at twice its resonance frequency, while the
superconducting microwave resonator is driven with two tones that differ in
frequency by an amount equal to the parametric driving frequency. We show that
the semi-classical approximation of this system has an interesting fixed point
bifurcation structure. In the semi-classical dynamics a transition from stable
fixed points to limit cycles is observed as one moves from positive to negative
detuning. We show that signatures of this bifurcation structure are also
present in the full dissipative quantum system and further show that it leads
to mixed state entanglement between the nanomechanical resonator and the
microwave cavity in the dissipative quantum system that is a maximum close to
the semi-classical bifurcation. Quantum signatures of the semi-classical
limit-cycles are presented.
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