which contains the excitonic DOF, the cavity mode, and the light-matter coupling terms. There are a total of N + 1 eigenstate of ĤS in the first excitation subspace (because there are N + 1 basis states, see Eq. 8), among which there are two bright polariton states, commonly referred to as the Upper polariton (UP) state |+⟩ and the Lower polariton (LP) state |-⟩, expressed as