All simulations presented in the main text are restricted in the Fock state basis |0 and |1 . Under the strong light-matter interactions, this can become problematic and more Fock states are required to reach convergence. In Fig. we provide the convergence test of the Fock states with the intermediate coupling case (g c = 0.005). Fig. presents the population dynamics of the |e0 state with the exact quantum dynamics. The results obtained using n = 6 and n = 8 Fock states are nearly identical and only display a small difference after 30 fs, and the n = 10 Fock state results are identical to the n = 8 result (which is on top of the n = 10 result). Fig. presents the same calculations using the Ehrenfest dynamics, with the general nuclear gradient expression in Eq. 28 and Eq. 30. As we already presented in the main text, the Ehrenfest simulation results (or FSSH results) are very close to the exact simulation results for the SM1-cavity coupling model investigated here, the Ehrenfest dynamics thus presents almost identical trend of the convergence for the population dynamics.