  We explore theoretically the physics of dynamic hysteresis for
driven-dissipative nonlinear photonic resonators. In the regime where the
semiclassical mean-field theory predicts bistability, the exact steady-state
density matrix is known to be unique, being a statistical mixture of two
states: in particular, no static hysteresis cycle of the excited population
occurs as a function of the driving intensity. Here, we predict that in the
quantum regime a dynamic hysteresis with a rich phenomenology does appear when
sweeping the driving amplitude in a finite time. The hysteresis area as a
function of the sweep time reveals a double power-law decay, with a behavior
qualitatively different from the mean-field predictions. The dynamic hysteresis
power-law in the slow sweep limit defines a characteristic time, which depends
dramatically on the size of the nonlinearity and on the frequency detuning
between the driving and the resonator. In the strong nonlinearity regime, the
characteristic time oscillates as a function of the intrinsic system parameters
due to multiphotonic resonances. We show that the dynamic hysteresis for the
considered class of driven-dissipative systems is due to a non-adiabatic
response region with connections to the Kibble-Zurek mechanism for quenched
phase transitions. We also consider the case of two coupled driven-dissipative
nonlinear resonators, showing that dynamic hysteresis and power-law behavior
occur also in presence of correlations between resonators. Our theoretical
predictions can be explored in a broad variety of physical systems, e.g.,
circuit QED superconducting resonators and semiconductor optical microcavities.
