where the coupling constant is A 0 √ hωc = 2 hω 3 c χ as we used in Eq. 2. Note that we have used the fact that Ûφ ĤM Û † φ = ĤM , i.e., Ûφ is identity for the matter DOFs. Hence, the role of Ûφ is to switch pc and qc , since for a photon field, they are interchangeable due to the pure harmonic nature of the quantized field. The PF Hamiltonian has the advantage as a pure real Hamiltonian and the photonic DOF can be viewed and computationally treated as "nuclear coordinates".