To evaluate the population transfer correlation functions, we start by looking at the population transfer rate between two dark states, j ̸ = ±, and k ̸ = ± and ℏω j = ℏω x (dark state energy is identical to the exciton energy). The relevant single-mode bath Hamiltonians Ĥa,j for the a th mode (defined in Eq. 14) is expressed as