In the JC model Hamiltonian (yellow), the ground state does not shift with increasing η, while the |+, 0⟩ and |-, 0⟩ states linearly split as a function of η. This behavior can be easily understood by examining the JC eigenspectrum in Eqn. 5. The Rabi model (cyan), which only accounts for the CRT, overestimates the negative energy corrections and incorrectly decreases energies for all states. Thus, the Rabi model predicts that the ground state energy becomes unstable. The RWA Hamiltonian (magenta), which ignores the CRT but includes the DSE, overestimates the energy correction in the positive direction and shifts all states upward. The perturbative treatment (red) that includes CRT as a perturbation as well as the DSE performs well and is nearly identical to the exact PF curve within the range of the η or ℏ∆ω c presented here. Note that in Fig. , for ℏ∆ω c < -0.5 eV, the polariton eigenenergy for |-, 1⟩ becomes lower than |+, 0⟩. As a result, a trivial crossing is formed between the 3 rd and 4 th polaritonic eigenenergies as a function of ℏ∆ω c at ℏ∆ω c ≈ -0.5 eV.