Chaos and the quantum phase transition in the Dicke model.

TLDR: A semiclassical Dicke model is derived that exhibits analogues of all the important features of the quantum model, such as the phase transition and the concurrent onset of chaos, and it is demonstrated that the system undergoes a transition from quasi-integrability to quantum chaotic.

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 $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ 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.