Comparative study of indicators of chaos in the closed and open Dicke model

The Dicke model, renowned for its superradiant quantum phase transition, also exhibits a transition from regular to chaotic dynamics. In this work we provide a systematic, comparative study of static and dynamical indicators of chaos for the closed and open Dicke models. In the closed Dicke model, we find that indicators of chaos sensitive to long-range correlations in the energy spectrum, such as the spectral form factor, can deviate from the Poissonian predictions and show a dip-ramp-plateau feature even in the regular region of the Dicke model unless very large values of the spin size are chosen. Thus, care is needed in using such indicators of chaos in general. In the open Dicke model with cavity damping, we find that the dissipative spectral form factor emerges as a robust diagnostic displaying a quadratic dip-ramp-plateau behavior in agreement with the Ginibre Unitary Ensemble (GinUE) in the superradiant regime. Moreover, by examining the spectral properties of the Liouvillian, we provide indirect evidence for the concurrence of the dissipative superradiant quantum phase transition and the change in Liouvillian eigenvalue statistics from 2D Poissonian to GinUE behavior.