While the dynamics described by the Lindblad Liouvillian in Eq. A5 are identical to the quantum subsystem dynamics that the Ehrenfest+R method aims to simulate, the particular implementation of the method was developed by additionally considering the role the electromagnetic field should have in spontaneous emission. The authors Chen et al. derive a Hamiltonian interaction term (not present in Eq. A1) that represents the electric dipole coupling between the excited and ground states of the system. This time-dependent interaction term Ĥint is defined as follows