Chaos and the quantum phase transition in the Dicke model

TLDR

We investigate the quantum‑chaotic properties of the Dicke Hamiltonian, a quantum‑optical model of a single‑mode bosonic field coupled to an ensemble of N two‑level atoms. We derive an effective Hamiltonian for the Dicke model’s zero‑temperature quantum phase transition, numerically study finite‑N level statistics to show a transition from quasi‑integrability to quantum chaos driven by precursors of the phase transition, and construct a semiclassical model reproducing the phase transition and concurrent onset of chaos. The Dicke model exhibits a zero‑temperature quantum phase transition in the thermodynamic limit, and finite‑N numerical level statistics reveal a transition from quasi‑integrability to quantum chaos driven by precursors of that transition, with wave‑function analysis indicating delocalization and emergence of macroscopic coherence.

Abstract

We investigate the quantum-chaotic properties of the Dicke Hamiltonian; a quantum-optical model that describes a single-mode bosonic field interacting with an ensemble of N two-level atoms. This model exhibits a zero-temperature quantum phase transition in the N --> infinity limit, which we describe exactly in an effective Hamiltonian approach. We then numerically investigate the system at finite N, and by analyzing the level statistics, we demonstrate that the system undergoes a transition from quasi-integrability to quantum chaotic, and that this transition is caused by the precursors of the quantum phase transition. Our considerations of the wave function indicate that this is connected with a delocalization of the system and the emergence of macroscopic coherence. We also derive a semiclassical Dicke model that exhibits analogues of all the important features of the quantum model, such as the phase transition and the concurrent onset of chaos.