Integrability Meets Dissipation: Critical Dynamics of Open Driven Systems

Driven quantum systems coupled to an environment are generally non-integrable and typically exhibit relaxational dynamics. We investigate the paradigmatic open Dicke model which describes collective light-matter interactions subject to dissipation. In a certain limit (at large detuning), this model is governed by an effective driven-dissipative Ising model in a transverse field, which is integrable in the absence of dissipation. In the limit of weak dissipation though, integrability is only weakly broken. We show that, in this regime, the system undergoes a dynamical crossover from relaxational dynamics to underdamped critical dynamics, each described by a distinct dynamical exponent. We identify these critical behaviors with the infinite-range classical (stochastic) and quantum (unitary) Ising models at finite temperature, respectively. These results are obtained through a non-equilibrium quantum-to-classical mapping in addition to an efficient numerical analysis that exploits the permutation symmetry.