Figure 4. Energy splitting and bending behaviors for the coupled atom-field system.

(a) Energy diagram of the bare (left: ground atomic state; right: excited atomic state) and dressed (center) states in the resonant limit (Δ ≪ λ). The dressed states undergo a Rabi splitting of $2\lambda \sqrt{n+1}$ of with n bosons in the excited atomic states, resulting in an energy shift of $\mathrm{\Delta }E=2g\sqrt{n+1}$. When the coupling strength is not very strong, the dressed energy levels could become broadened and even obscure the splitting (marked as fine yellow lines in the interior of the dressed states) given the finite lifetime of the atom-field system owing to the Heisenberg uncertainty relationship between energy and time. (b) In the dispersive limit (Δ ≫ λ), the ground |0,n⟩ and excited |1,n⟩ bare states with n bosons undergo an energy shift of nλ²/Δ and (n + 1)λ²/Δ, resulting in an energy shift of ΔE = (2n + 1)λ²/Δ.