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Gauge invariance is the cornerstone of modern quantum field theory. Recently, it has been shown that the quantum Rabi model, describing the dipolar coupling between a two-level atom and a quantized electromagnetic field, violates this principle. This widely used model describes a plethora of quantum systems and physical processes under different interaction regimes. In the ultrastrong coupling regime, it provides predictions which drastically depend on the chosen gauge. This failure is attributed to the finite-level truncation of the matter system. We show that a careful application of the gauge principle is able to restore gauge invariance even for extreme light-matter interaction regimes. The resulting quantum Rabi Hamiltonian in the Coulomb gauge differs significantly from the standard model and provides the same physical results obtained by using the dipole gauge. It contains field operators to all orders that cannot be neglected when the coupling strength is high. These results shed light on subtleties of gauge invariance in nonperturbative and extreme interaction regimes, which are now experimentally accessible, and solve all the long-lasting controversies arising from gauge ambiguities in the quantum Rabi and Dicke models., Comment: 8 pages, 3 figures

The ultrastrong-coupling regime, where the atom-cavity coupling rate reaches a considerable fraction of the cavity or atom transition frequencies, has been reported in a flux qubit superconducting quantum circuit coupled to an on-chip coplanar resonator. This regime enables the possibility of manipulating the cavity quantum electrodynamic ground state with controllable physical properties and situations may arise where the resonator field $\hat X= \hat a+\hat a^\dagger$ acquires a nonzero expectation value in the system ground state. We demonstrate that in this case the parity symmetry of an additonal artificial atom with an even potential is broken by the interaction with the resonator. Such mechanism is in close analogy with the Higgs mechanism where the gauge symmetry of the weak force's gauge bosons is broken by the nonzero vacuum expectation value of the Higgs field. The results here presented open the way to controllable experiments on symmetry breaking mechanisms induced by nonzero vacuum expectation values. Moreover the here proposed mechanism can be used as a probe of the ground state macroscopic coherence emerging from quantum phase transitions with vacuum degeneracy., Comment: 7 pages, 4 figures