Fig. 4: Gauge invariance of the Renormalized QRM.
From: Renormalization and low-energy effective models in cavity and circuit quantum electrodynamics
Mean square error of the eigenvalues of the first 5 excited states with respect to the full model, as a function of the gauge parameter η, for g/ωc = 0.8, m = 1, γ = 60, and ωc = 3ω10. The QRM (red dashed) breaks gauge invariance, showing that the dipole gauge (η = 1) is the most accurate. On the other hand, the RQRM \({\hat{{{\mathcal{H}}}}}^{{\prime} (\eta )}\) (green dotted) in Eq. (19) is not only gauge invariant but also provides more accurate results. For completeness, we also compare these models with the gauge-preserving QRM \({\hat{{{\mathcal{H}}}}}^{(\eta )}\) (blue dash-dotted) in Eq. (15), which, however, does not take into account the renormalization of the higher energy levels.
