  We systematically study the chaotic signatures in a quantum many-body system
consisting of an ensemble of interacting two-level atoms coupled to a
single-mode bosonic field, the so-called extended Dicke model. The presence of
the atom-atom interaction also leads us to explore how the atomic interaction
affects the chaotic characters of the model. By analyzing the energy spectral
statistics and the structure of eigenstates, we reveal the quantum signatures
of chaos in the model and discuss the effect of the atomic interaction. We also
investigate the dependence of the boundary of chaos extracted from both
eigenvalue-based and eigenstate-based indicators on the atomic interaction. We
show that the impact of the atomic interaction on the spectral statistics is
stronger than on the structure of eigenstates. Qualitatively, the
integrablity-to-chaos transition found in the Dicke model is amplified when the
interatomic interaction in the extended Dicke model is switched on.
