Dissipative Phase Transition in a Parametrically Amplified Quantum Rabi Model with Two-photon decay

We investigate dissipative phase transitions(DPTs) in a parametrically amplified open quantum Rabi model (QRM) with both single-photon and two-photon decay. In the classical oscillator limit, four composite phases emerge, arising from the possible normal or superradiant regimes across the upper and lower spin branches. A mean-field analysis reveals regimes of first- and second-order transitions and a tricritical point. Using adiabatic quantum calculations, we further show that the tricriticality stems from the intrinsic nonlinearity of QRM, unveiled by the interplay of coherent and dissipative two-photon processes. The universality classes of the DPTs are identified, with corresponding critical and finite-size scaling exponents derived through a semiclassical Langevin theory near criticality.