Resolution of gauge ambiguities in ultrastrong-coupling cavity quantum electrodynamics

In quantum electrodynamics, the choice of gauge influences the form of light-matter interactions. However, gauge invariance implies that all physical results should be independent of this formal choice. The Rabi model, a widespread description for the dipolar coupling between a two-level atom and a quantized electromagnetic field, seemingly violates this principle in the presence of ultrastrong light-matter coupling, a regime that is now experimentally accessible in many physical systems. This failure is attributed to the finite-level truncation of the matter system, an approximation that enters the derivation of the Rabi model. Here, we identify the source of gauge violation and provide a general method for the derivation of light-matter Hamiltonians in truncated Hilbert spaces that produces gauge-invariant physical results, even for extreme light-matter interaction regimes. This is achieved by compensating the non-localities introduced in the construction of the effective Hamiltonians. The resulting quantum Rabi Hamiltonian in the Coulomb gauge differs significantly in form from the standard one, but provides the same physical results obtained by using the dipole gauge. These results shed light on gauge invariance in the non-perturbative and extreme-interaction regimes, and solve long-lasting controversies arising from gauge ambiguities in the quantum Rabi and Dicke models.