  We report a first-principles study of the driven dissipative dynamics for
Kerr oscillators in the mesoscopic regime. This regime is characterized by
large Kerr nonlinearity, realized here using the nonlinear kinetic inductance
of a large array of Josephson junctions. The experimentally measured nonlinear
resonance lineshapes of the junction array modes show significant deviations
from steady-state numerical predictions, and necessitate time-dependent
numerical simulations indicative of strong measurement-induced dephasing in the
system arising from the large cross-Kerr effect between array modes. Analytical
and numerical calculations of switching rate corroborate this by showing the
emergence of a slow time scale, which is much longer than the linear decay rate
and is set by fluctuation-induced switching times in the bistable regime.
Furthermore, our analysis shows that the usual quantum-activated escape
treatment is inadequate for prediction of the switching rates at large
frequency shifts caused by strong nonlinearities, necessitating a quantum
treatment that utilizes the full system Liouvillian. Based on our analysis, we
identify a universal crossover parameter that delineates the regimes of
validity of semiclassical and quantum descriptions, respectively. Our work
shows how dynamical switching effects in strongly nonlinear systems provide a
platform to study quantum-to-classical transitions.
