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- 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 an "inverted" regime where superradiance emerges only at sufficiently low spin-boson coupling. This regime features first- and second-order DPTs separated by a tricritical point, while two-photon dissipation preserves the stability of the superradiant phase. Utilizing an adiabatic approach and the semi-classical Langevin formalism, we further study the steady-state structure beyond the mean-field level. We 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 the corresponding critical and finite-size scaling exponents derived and a scaling ansatz proposed to describe the critical behavior.