Inversions of qubit energy levels in deep-strongly-coupled qubit-oscillator circuits

The electromagnetic field inside a cavity renormalizes the transition frequency of an atom inside the cavity. The Lamb (n = 0) and n-photon ac-Stark shifts can be written as Δ - Δ_n, where Δ and Δ_n are respectively the transition frequencies of a non-interacting and renormalized atom. In the limit of Δ « ω, where ω is the resonance frequency of the cavity, Δ_n can be written as [1] Δ_n/Δ = exp[-2(g/ω)²]L_n[(2g/ω)²], where g is the coupling constant between the atom and the electromagnetic field in the cavity and L_n is a Laguerre polynomial. This equation indicates that Δ_n changes sign when L_n crosses zero; the atom's ground and excited states are “inverted”. We use circuits of a flux qubit deep-strongly coupled to an LC oscillator [2], where g ≈ ω. The experimentally observed qubit frequencies Δ_n (n = 0, 1, 2) were very close to those given by the equation above. [1] S. Ashhab and F. Nori, PRA 81, 052311 (2010). [2] F. Yoshihara, T. Fuse et al., Nat. Phys. 13, 44 (2017).