Under the condition η < 0. Fig. presents the three lowest polariton eigenenergies of a two-level atom (Eq. 79 without any nuclear DOFs) coupled to a single mode cavity. The figure presents three polariton states |g, 0⟩, |-, 0⟩ and |+, 0⟩. Fig. presents the polaritonic eigenvalues as a function of η = g c /ω c at ∆Eℏω c = 0 (resonance condition). Fig. presents the polaritonic eigenvalues as a function of the detuning ∆Eℏω c with a light-matter coupling strength ℏg c = 1 eV. The eigenenergies are obtained at various levels of theory, including the JC model (yellow) in Eq. 83 that ignores both CRT and DSE, the rotating wave approximation (RWA) Hamiltonian (magenta) that only ignores the CRT term but not the DSE term, the Rabi model (cyan) in Eq. 82 that ignores the DSE, and the full PF treatment (black dashed) in Eq. 70 that includes both the CRT term and DSE. The perturbation theory (PT) (red) which treats CRT perturbatively (see details in Ref. 12) and includes the exact DSE and provides very accurate polariton eigenenergies in the range of the parameter regime investigated here.