Nonlinear nanoelectromechanics of a movable Cooper-pair box
We theoretically study the dynamics of a movable Cooper-pair box coupled to a normal-metal pillar using a semiclassical approach. We analyze the dynamical stability induced by the nonlinear nanoelectromechanical coupling between the mechanical motion and an inelastic Andreev tunneling through linear stability and bifurcation analyses. As a function of $η$, defined as the ratio of electrostatic energy to Josephson coupling energy, the system exhibits reentrant stability. At small $η$, the fixed point loses stability through a supercritical Hopf bifurcation, giving rise to self-sustained vibrations. With a further increase of $η$, a second critical point appears, at which the fixed point regains stability. We show that this second transition corresponds to an inverse subcritical Hopf bifurcation in the adiabatic regime and to an inverse supercritical Hopf bifurcation in the nonadiabatic regime. These results extend previous studies of adiabatic self-vibrations to the nonadiabatic regime and reveal a rich nonlinear dynamical phase diagram arising from the interplay between electronic and mechanical degrees of freedom in superconducting devices.