Considering the photonic part of the light-matter hybrid system, we know these DOFs are explicitly harmonic (i.e., their bare Hamiltonian Ĥph is simply the quantum harmonic oscillator), and their exact eigenstates are known to be the Fock (or number) states of the quantized field. In this way, we already know the solutions to the unperturbed parts of the light-matter Hamiltonian exactly for the photons and approximately for the electrons. If one can directly diagonalize the light-matter Hamiltonian (Eq. 15), then one additionally receives the exact correlation between these DOFs.