where k ∈ {1, • • • , N -1}. The energy of the dark states remain the same as the exciton site energy. Note that the dark states have no overlap with the collective "bright" states, and they do not participate in the interaction with the cavity mode mediated by ĤLM . They are optically dark (has no transition dipole from the ground state). Eq. 14 and Eq. 24 form the polariton basis (or the Tavis-Cummings basis ). Further, we introduce the following discrete Fourier transform for the bath operators bα,k :