  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.
