where Ĥren and ĥeff B are defined in Eq. 18, Fν and Feff are defined in Eq. 19, and we have subtracted out zero point energy E). The numerical values of the matrix elements for our current model are R LL = ν L | R|ν L = 0.214 a.u., R LL = ν L | R|ν L = -0.933 a.u., and R L L = ν L | R|ν L = -0.702 a.u., and we have found numerically that λ eff λ ν for all light-matter coupling strength η c that we have considered in this work (see Sec. II of the Supplementary Material), and we thus explicitly set λ eff = 0 in our following analysis. In Eq. A2, the second line describes Peierls-type of system-bath coupling (offdiagonal couplings) which includes both the phonon and the photon fluctuations, in which the term