where ρ is the reduced density operator of the quantum subsystem (electronic-photonic DOFs), V = Ĥpl is the polariton Hamiltonian defined in Eq. 2, and L is a Lindblad jump operator 28 that imparts the impact of the environment (photonic bath) onto the system (cavity mode) with interaction strength Γ (that has a unit of rate or inverse time) and { Â, B} = Â B + B Â represents the anti-commutator. In Eq. 21, we further define two superoperators, L V [•] and L L [•], which are used to describe the evolution of reduced density governed by the quantum subsystem Hamiltonian V and jump operator L, respectively. The Lindblad jump approach ensures the total population conservation as well as the proper decoherence dynamics among states due to the population jumps. For the cavity loss process considered in this work, the jump operator L is defined as 28