where the |Φ α ⟩ is the polariton state, with the polariton energy E α . Note that here we have not considered any bath degrees of freedom ( ĤR or Ĥex-R ) inside the definition of the polariton state and polariton energies, as opposed to the previous work that explicitly considers them inside the polariton definitions. The lifetime of the cavity mode is finite, due to the coupling between the cavity mode and the far-field photon modes outside the cavity. The detailed discussions for molecular cavity QED with cavity loss are provided in Appendix D of Ref. 47. Here, we use the following Lindblad jump operator to model this process