  We study the interaction of a two-level atom with a single-mode nonlinear
electromagnetic resonator, considering coupling strengths ranging from zero to
the so-called deep strong coupling regime. When the qubit-resonator coupling is
very strong, the standard Kerr model for the resonator becomes questionable.
Moreover, recently, it has been shown that extra care is needed when
constructing gauge-independent theories in the presence of approximations as
the truncation of the Hilbert space of the matter system. Such a truncation can
ruin gauge invariance leading to non-physical results, especially when the
light-matter interactions strength is very high. Here we face and solve this
issues to provide a consistent nonlinear-resonator quantum Rabi model
satisfying the gauge principle.
