Floquet Dissipative Phase Transitions

Dissipative phase transitions (DPTs) are a hallmark of open quantum systems, traditionally characterized through the spectral properties of a time-independent Liouvillian. However, this approach fails for periodically driven systems, where the Liouvillian itself becomes time-dependent. We develop a general framework to characterize DPTs in such systems by analyzing the spectrum of the Floquet propagator—the completely positive, trace-preserving map that evolves the density matrix over one driving period. This allows us to define Floquet dissipative phase transitions through the closure of the Floquet spectral gap, generalizing the concept of the Liouvillian gap to the time-periodic regime.

We first apply this approach to the single-photon driven Kerr resonator. While the standard rotating-wave approximation (RWA) yields a time-independent model with a critical point determined solely by the detuning, our Floquet analysis reveals that beyond-RWA effects introduce a strong dependence on the absolute drive and cavity frequencies, shifting the transition point. We then extend our study to multi-drive configurations, which cannot be reduced to any time-independent form, unveiling previously unexplored nonequilibrium regimes. Finally, we analyze the two-photon driven Kerr resonator beyond the RWA, observing an enhanced sensitivity of the steady-state properties and critical behavior to the drive parameters. This framework paves the way to study dissipative criticality in fully time-periodic quantum systems, with direct applications to the driven Quantum Rabi model and other platforms where counter-rotating terms are non-negligible.