The same expression has been used in the development of the stochastic Schrödinger equation (see Eq. 73) that is equivalent to Lindblad dynamics. Thus, when completely ignoring the refiling term Γâρ S â † , one can approximate Lindblad dynamics as the time dependent Schrödinger equation (TDSE) with the complex Hamiltonian Ĥeff , which has been used in several recent works on molecular cavity QED. In the situations where the refilling term is negligible, the dynamics can equivalently be described by using the Schrödinger equation of a wave function evolving with the effective Hamiltonian. However, for the applications considered in current work in Fig. , one cannot ignore the Γâρ S â † term as we do care about the population refilling in the |g, 0⟩ state, as well as the proper decoherence among these states.