Fig. presents the potential energy surfaces (PESs) and population dynamics of the photo-isomerization model coupled to an optical cavity, with the system Hamiltonian described in Eq. 82. Here, we explicitly consider the population decay of the photonic DOF through the jump operator L in Eq. 83, which describes the finite lifetime of the cavity mode due to its coupling to the other non-cavity modes. Fig. presents the diabatic PESs for the |g, 0⟩ state (dark blue), the |e, 0⟩ state (cyan), and the |g, 1⟩ state (magenta). These states are diabatic because their electronic character does not depend on the nuclear coordinate R, thus there is no derivative coupling between these states (see Appendix C for details). Conversely, there is a diabatic coupling between the states |e, 0⟩ and |g, 1⟩ which causes coherent population transfer between these states. Fig. presents the adiabatic PESs for the ground |g, 0⟩ state (dark blue), the lower polariton state (middle line, labeled |LP ⟩), and the upper polariton state (upper line, labeled |U P ⟩). These adiabatic states are eigenstates of the polaritonic Hamiltonian ĤS -TR (where ĤS is given in Eq. 82) thus there is no additional diabatic coupling between any of the states. Further, since the electronic character of the upper and lower polariton states changes as a function of the nuclear coordinate R, there exists derivative coupling between the upper and lower polariton states which allows for coherent population transfer between them. Fig. -f presents the population dynamics of diabatic states |g, 0⟩ (dark blue), |e, 0⟩ (cyan), and |g, 1⟩ (magenta). The results of exact Lindblad dynamics (solid lines) are obtained by solving Eq. 12 using the basis {|α, n⟩ ⊗ |χ ν ⟩}, where |α, n⟩ = |α⟩ ⊗ |n⟩, |α⟩ ∈ {|g⟩, |e⟩} are the diabatic electronic states and |n⟩ are the Fock states of the cavity mode. The L-MFE dynamics (dotted) are obtained by treating the electronic and photonic DOFs as the quantum subsystem, with basis |α, n⟩, and the nuclear DOF as a classical DOF. The details of these numerical simulations are provided in Sec. IV.