Semiclassical theory of energy diffusive escape in a Duffing oscillator

Motivated by recent experimental progress to read out quantum bits implemented in superconducting circuits via the phenomenon of dynamical bifurcation, transitions between steady orbits in a driven anharmonic oscillator, the Duffing oscillator, are analyzed. In the regime of weak dissipation a consistent master equation in the semiclassical limit is derived to capture the intimate relation between finite tunneling and reflection and bath induced quantum fluctuations. From the corresponding steady state distributions analytical expressions for the switching probabilities are obtained. It is shown that a reduction of the transition rate due to finite reflection at the phase-space barrier is overcompensated by an increase due to environmental quantum fluctuations that are specific for diffusion processes over dynamical barriers. Moreover, it is revealed that close to the bifurcation threshold the escape dynamics enters an overdamped domain such that the quantum mechanical energy scale associated with friction even exceeds the thermal energy scale.