which relates to the cavity-mediated non-adiabatic couplings. Due to the light-matter couplings, the |g, 0⟩ and |g, 1⟩ are coupled through the gradient on the permanent dipole moment µ gg as well as through the non-adiabatic coupling multiplied by the transition dipole moment µ ge , in total, proportional to µ gg (R ) +2d ge (R )µ ge (R ). This can be seen directly in the X g0,g1 curve in Fig. given that the permanent dipole moment µ gg has a constant gradient near R = 0.0 and thus provides a plateau for X g0,g1 in the same region. Most importantly, the redistribution of d ge among all the basis states allows for a complex and interesting non-adiabatic events that would be absent without the light-matter coupling. Finally, Fig. shows the quantum dynamics simulation using the exact nuclear gradients (solid curves, Eq. 62) in addition to showing the approximate population dynamics using the approximate gradients of the Jaynes-Cummings Hamiltonian in the |e, 0⟩, |g, 1⟩ subspace (dashed curves, see Ref. 116 for more details). Clearly, additional states beyond the JC subspace will be explored by the quantum dynamics of the hybrid system due to the NACs among these states. The population of the |e1⟩ state is mainly contributed from the population transfer from the |e0⟩ state due to the lightmatter coupling originating from the permanent dipole µ ee . In addition, the population transition between |e0⟩ and |g1⟩ is mediated by the cavity-induced coupling near R = 0.0, where the PES exhibits a strong mixing between these two states as shown in Fig. .