Chaos and the Quantum Phase Transition in the Dicke Model

TL;DR

This paper explores the quantum chaotic behavior of the Dicke model, revealing how a quantum phase transition influences the transition from integrability to chaos and connecting wavefunction delocalization with macroscopic coherence.

Contribution

It provides an exact effective Hamiltonian description of the quantum phase transition and demonstrates the transition to chaos at finite N through level statistics analysis.

Findings

System transitions from quasi-integrability to chaos near the phase transition

Wavefunction delocalization correlates with macroscopic coherence

Semi-classical model reproduces key features of the quantum phase transition

Abstract

We investigate the quantum chaotic properties of the Dicke Hamiltonian; a quantum-optical model which 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 $N \go \infty$ limit, which we describe exactly in an effective Hamiltonian approach. We then numerically investigate the system at finite $N$ and, by analysing 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 wavefunction indicate that this is connected with a delocalisation of the system and the emergence of macroscopic coherence. We also derive a semi-classical Dicke model, which exhibits analogues of all the important features of the quantum model, such as the phase transition and the concurrent onset of chaos.