is the site index, |G n ⟩ and |X n ⟩ are the ground state and the excited state of site n, respectively, â † and â are the creation and annihilation operators of the cavity photons with frequency ω c , and g c is the single molecule light-matter coupling strength. Experimentally, it was estimated that N = 10 6 -10 12 molecules are collectively coupled to one cavity mode, which was referred to as the collective coupling limit. Each site (molecule) is coupled to its independent phonon environment, which is modeled as the bath Hamiltonian Ĥb + Ĥsb . These baths are assumed to be harmonic and identical for all sites n, expressed as follows