Routes to chaos in the balanced two-photon Dicke model with qubit dissipation

We study the semiclassical limit of the two-photon Dicke model with both cavity decay and qubit dissipation, and extend a recent analysis of its stationary points [Garbe et al., Sci. Rep. 10, 13408 (2020)], where a large unstable region was found. By considering the explicit dynamical evolution, we show that the unstable region actually hosts a rich nonlinear behavior. At variance with other types of Dicke models (without qubit dissipation or with one-photon interactions), the occurrence of chaos does not rely on a large counterrotating interaction. Furthermore, new routes to chaos appear in addition to period-doubling bifurcations, i.e., intermittent chaos and quasi-periodic oscillations. The transition mechanisms under these three distinct routes are investigated in detail through the system's long-time evolution, the optical field power spectrum, Lyapunov exponents, and bifurcation diagrams. Additionally, we provide a comprehensive phase diagram detailing the existence of stable fixed points, limit cycles, and the aforementioned chaos-related dynamics.