  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.
