Upon the diagonalization of the many-molecule, many-mode Hamiltonian (Eq. 17) in the singly excited subspace S 1 , polariton eigenenergies and eigenstates are obtained, as shown in Fig. for a collection of quasi-continuous parallel cavity modes and Fig. for a single resonant mode. Here, the formation of the upper and lower polariton branches P +/-(Fig. ) or states | + /-⟩ (Fig. ) can be observed. Furthermore, due to the presence of the other molecules (which are all degenerate with the cavity frequency), many degenerate polaritonic states can be found at the same energy as the bare molecular transition. These states are referred to as "dark states" {D i } since they contain no transition dipole moment between the polaritonic ground state nor do they contain any photonic character (in the absence of molecular disorder). The Rabi splitting in this case famously scales as Ω R ∝ N mol . In the presence of molecular fluctuations (e.g., thermal fluctuations), the photonic character becomes distributed amongst the "dark states". Fig. shows the time-dependence of the polariton eigenenergies colored by the (left) photonic character ⟨ψ j | â † â |ψ j ⟩ and (right) non-adiabatic coupling σ j = k ⟨ψ j | d d t |ψ k ⟩ for each polaritonic state j (both on log colorbar scales). The exchange of character mediates population dynamics between such polariton states (i.e., cavity-mediated non-adiabtic couplings). Fig. presents mixed quantum-classical mapping approach to surface hopping (MASH) 148;147 dynamics where a LASER drives population from the collective ground state |G⟩ to the upper polariton |+⟩ (left) without and (right) with cavity loss. The dynamics suggest that the dark states quickly accumulate much of the population at initial times. Wihout cavity loss, the polaritonic states reach equilibrium by 3.0 ps, while with cavity loss, none of the state retain significant population due to the rapid transfer to the ground state and the ongoing polariton-polariton non-adiabatic couplings.