We theoretically explore the dynamic hysteresis behavior of a driven-dissipative photonic resonator with a Kerr-type nonlinearity . In the regime where the semiclassical approach predicts bistability, the exact steady-state density matrix is well known to be unique and a statistical mixture of two states. A direct consequence is that the full quantum treatment predicts no static hysteresis cycle of the excited population as a function of the driving intensity. 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. We show that the dynamic hysteresis can be understood as due to a non-adiabatic response region with connections to the celebrated Kibble-Zurek mechanism for dynamic phase transitions. These theoretical predictions can be explored in a broad variety of physical systems, e.g., circuit QED superconducting resonators and semiconductor optical microcavities.
 W. Casteels, F. Storme, A. Le Boité, and C. Ciuti - Phys. Rev. A 93, 033824 (2016)