# Regularity and chaos in cavity QED. (arXiv:1612.01509v2 [quant-ph] UPDATED)

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.