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We revisit the derivation of Rabi- and Dicke-type models, which are commonlyused for the study of quantum light-matter interactions in cavity and circuitQED. We demonstrate that the validity of the two-level approximation, which isan essential step in this derivation, depends explicitly on the choice of gaugeonce the system enters the ultrastrong coupling regime. In particular, while inthe electric dipole gauge the two-level approximation can be performed as longas the Rabi frequency remains much smaller than the energies of allhigher-lying levels, it can dramatically fail in the Coulomb gauge, even forsystems with an extremely anharmonic spectrum. We extensively investigate thisphenomenon both in the single-dipole (Rabi) and multi-dipole (Dicke) case, andconsidering the specific examples of dipoles confined by double-well and bysquare-well potentials, and of circuit QED systems with flux qubits coupled toan LC resonator.

Breakdown of gauge invariance in ultrastrong-coupling cavity QED