This NAC in the polariton representation (Eq. 42) is only used for computing the direction of re-scaling the velocity. Further, since the energy of the entire hybrid system is dissipated to the photonic environment during the cavity loss process, there is no energy conservation for the cavity loss process, and as such, we do not perform the velocity scaling for nuclear DOFs if the hopping occurs during the cavity loss process governed by Eq. 27b. This is consistent with the previous work of TSH dynamics in simulating polariton dynamics. In the TSH simulation used in this work, we set one of the polariton states as the initial active state, which means the initial coefficients c I (0) for the state |E I ⟩ is set to be one, and the rest of the coefficients are set to be zero. These coefficients {c J (0)} in the polariton state basis can be unitary-transformed into the coefficients {c i (0)} in the adiabatic-Fock state basis to perform the Lindblad dynamics for each nuclear initial condition.