  We investigate the dynamics of the bistable regime of the generalized
Jaynes-Cummings Hamiltonian (GJC), realised by a circuit quantum
electrodynamics (cQED) system consisting of a transmon qubit coupled to a
microwave cavity. In this regime we observe critical slowing down in the
approach to the steady state. By measuring the response of the cavity to a step
function drive pulse we characterize this slowing down as a function of driving
frequency and power. We find that the critical slowing down saturates as the
driving power is increased. We compare these results with the predictions of
analytical and numerical calculations both with and without the Duffing
approximation. We find that the Duffing approximation incorrectly predicts that
the critical slowing down timescale increases exponentially with the drive,
whereas the GJC model accurately predicts the saturation seen in our data,
suggesting a different process of quantum activation.
