On the other hand, one can also define polariton states as the eigenvector of the adiabatic polariton Hamiltonian Ĥpl = Ĥ -TR , where TR is the nuclear kinetic energy operator (for all phonons). The eigenstates of Ĥpl can be viewed as the adiabatic version of the polariton and dark states because the state character explicitly depends on nuclear configuration { Rα,n }, and it has been used to interpret the photoluminescence spectra or investigate coherences in polariton transport. Our focus is the coherence between |+⟩ and |-⟩ states, which is directly related to the off-diagonal beating in 2DES spectra and has been experimentally explored (see Fig. ). To probe the polariton coherences, we compute the off-diagonal matrix elements of the system-reduced density matrix (RDM), defined as