Hopf bifurcation in a flux qubit coupled to a nanomechanical oscillator

We study the nonlinear semiclassical dynamics of a driven flux qubit coupled to a nanomechanical oscillating beam. While both of these systems are dissipative, their mutual coupling can effectively suppress the dissipation. This can lead to a loss of stability and to an emergence of synchronized self-excited oscillations of the system as a whole, at a time scale set by the mechanical beam. In this article we argue this by obtaining a set of semiclassical, nonlinear equations of motion for the two coupled subsystems, and showing that this system undergoes a supercritical Hopf bifurcation as the coupling is increased. We derive analytical expressions for the critical coupling coefficient and the renormalized mechanical dissipation coefficient and frequency, and give a complete characterization of the limit cycle behavior when the qubit and the oscillator are near resonance.