where ĤS is the system Hamiltonian, ÎS is the identity in the system Hilbert space H S , ĤE is the environment Hamiltonian, ÎE is the identity in the environment Hilbert space H E , and ĤI is the interaction Hamiltonian between the system and the environment. For cavity QED systems, ĤS is the PF Hamiltonian ĤPF (Eq. 104) while ĤE describes the far-field photon modes as free bosons 340,341