Regularity and chaos in cavity QED

TL;DR

This paper investigates the transition between regular and chaotic dynamics in cavity QED models, using semiclassical phase space analysis and quantum participation ratios to quantify chaos in different regimes.

Contribution

It introduces a detailed quantum analysis of chaos in the Dicke model, linking classical Lyapunov exponents with quantum participation ratios across the phase space.

Findings

Chaotic regions show positive Lyapunov exponents and finite participation ratios in the thermodynamic limit.

Regular regions exhibit near-zero participation ratios, indicating localized quantum states.

Quantum chaos can be effectively characterized by the participation ratio in the eigenbasis.

Abstract

The interaction of a quantized electromagnetic field in a cavity with a set of two-level atoms inside can be described with algebraic Hamiltonians of increasing complexity, from the Rabi to the Dicke models. Their algebraic character allows, through the use of coherente states, a semiclassical description in phase space, where the non-integrable Dicke model has regions associated with regular and chaotic motion. The appearance of classical chaos can be quantified calculating the largest Lyapunov exponent in the whole available phase space for a given energy. In the quantum regime, employing efficient diagonalization techniques, we are able to perform a detailed quantitative study of the regular and chaotic regions, where the quantum Participation Ratio (PR) of coherent states on the eigenenergy basis plays a role equivalent to the Lyapunov exponent. It is noted that, in the thermodynamic limit, dividing the Participation Ratio by the number of atoms leads to a positive value in chaotic regions, while it tends to zero in the regular ones.