No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED

In cavity quantum electrodynamics (QED), the interaction between an atomic transition and the cavity field is measured by the vacuum Rabi frequency Omega(o). The analogous term 'circuit QED' has been introduced for Josephson junctions, because superconducting circuits behave as artificial atoms coupled to the bosonic field of a resonator. In the regime with Omega(o) comparable with the two-level transition frequency, 'superradiant' quantum phase transitions for the cavity vacuum have been predicted, for example, within the Dicke model. In this study, we prove that if the time-independent light-matter Hamiltonian is considered, a superradiant quantum critical point is forbidden for electric dipole atomic transitions because of the oscillator strength sum rule. In circuit QED, the analogous of the electric dipole coupling is the capacitive coupling, and such no-go property can be circumvented by Cooper pair boxes capacitively coupled to a resonator, because of their peculiar Hilbert space topology and a violation of the corresponding sum rule.