One can phenomenologically introduce different sources of dissipation that lead to a spectroscopic broadening of the light-matter eigenspectrum. Let us denote the loss rate for the cavity photon as κ, and the decay rate of the matter excitation as γ (see Fig. for a schematic illustration). For the Markovian dissipation at zero temperature, the cavity-matter density matrix for the JC model is given with the quantum Liouville equation ρ = -i ℏ [H JC , ρ] + κ 2 L â( ρ) + γ 2 L σ (ρ) where L â( ρ) = 2âρâ † -â † âρ -ρâ † â is the dissipative part based on the Lindblad jump operator â with a similar expression with the matter DOFs for L σ (ρ). For the JC model, the approximate evolution of the density matrix under such dissipation can be captured by defining an effective Hamiltonian Ĥ′ JC = ĤJC -iℏ κ 2 â † â -iℏ γ 2 σ † σ, (7) such that ρ ≈ i ℏ ( Ĥ′ JC ρ-ρ Ĥ′ † JC ) when ignoring the 2âρâ † and 2σ ρσ † terms in L â( ρ) and L σ (ρ). Similar to the JC model Hamiltonian, when including dissipation, Ĥ′ JC is block-diagonalized within the {|e, n⟩, |g, n+1⟩} subspace and the matrix elements of Ĥ′ JC within this subspace are written as