Comparative Study of Indicators of Chaos in the Closed and Open Dicke Model

TL;DR

This study compares static and dynamic chaos indicators in the closed and open Dicke models, revealing the nuanced behavior of chaos signatures and the robustness of the dissipative spectral form factor.

Contribution

It provides a systematic comparison of chaos indicators in both models, highlighting the importance of large spin sizes and the robustness of the dissipative spectral form factor as a diagnostic.

Findings

Spectral form factor deviates from Poissonian predictions in the regular region unless large spin sizes are used.

Dissipative spectral form factor shows quadratic dip-ramp-plateau behavior consistent with GinUE in the superradiant regime.

Liouvillian eigenvalue statistics change from 2-D Poissonian to GinUE at the superradiant quantum phase transition.

Abstract

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 model. 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 (SFF), 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 2-D Poissonian to GinUE behavior.