Note that despite the common usage of the Lindblad jump operator LS = â ⊗ Îe ⊗ ÎR (Eq. 81) for describing cavity losses, 14 this jump operator is actually derived by considering the simpler system Hamiltonian ĤS = ℏω c â † â + 1 2 ⊗ Îe ⊗ ÎR without explicit consideration for the matter Hamiltonian or molecule-cavity interactions in ĤQED . Thus, Eq. D3 should be viewed as a phenomenological equation, and the correct Lindblad master equation for cavity QED should be derived starting from the total Hamiltonian ĤT = ĤQED + ĤE + ĤI , where ĤE and ĤI are expressed in Eq. D1 and Eq. D2, respectively. The details of the microscopic derivation of the Lindblad master equation for the Jaynes-Cumming model with cavity losses, as well as for comparison with Eq. D3, can be found in Ref. 70.