where ĤE = j ℏω j ( b † j bj + 1 2 ) accounts for the environmental DOF (far field, non-cavity modes), and the interactions between ĤS and ĤE though the interaction term ĤI = (â † + â) ⊗ j ℏg j ( b † j + bj ) that accounts for the interactions between the cavity mode (see Eq. 36) described by {â † , â} and the far-field modes { b † j , bj }. A detailed discussion of cavity loss (and the expression of ĤE and ĤI ) can be found in Appendix D of Ref. 47. Denoting the total density operator ρT associated with the total Hamiltonian ĤT , we can trace out the environmental DOF and have the system part of the reduced density operator as