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We study the complex nonlinear dynamics of the two-photon Dicke model in thesemiclassical limit by considering cavity and qubit dissipation. In addition tothe normal and super-radiant phases, another phase that contains abundantchaos-related phenomena is found under balanced rotating and counter-rotatingcouplings. In particular, chaos may manifest itself through period-doublingbifurcation, intermittent chaos, or quasi-periodic oscillation, depending onthe value of qubit frequency. Transition mechanisms that exist in these threedistinct routes are investigated through the system's long-time evolution andbifurcation diagram. Additionally, we provide a comprehensive phase diagramdetailing both the existence of stable fixed points and the aforementionedchaos-related dynamics.

Nonlinear dynamics in the balanced two-photon Dicke model with qubit dissipation