  The quantum Rabi model is in the scientific spotlight due to the recent
theoretical and experimental progress. Nevertheless, a full-fledged
classification of its coupling regimes remains as a relevant open question. We
propose a spectral classification dividing the coupling regimes into three
regions based on the validity of perturbative criteria on the quantum Rabi
model, which allows us the use of exactly solvable effective Hamiltonians.
These coupling regimes are i) the perturbative ultrastrong coupling regime
which comprises the Jaynes-Cummings model, ii) a region where non-perturbative
ultrastrong and non-perturbative deep strong coupling regimes coexist, and iii)
the perturbative deep strong coupling regime. We show that this spectral
classification depends not only on the ratio between the coupling strength and
the natural frequencies of the unperturbed parts, but also on the energy to
which the system can access. These regimes additionally discriminate the
completely different behaviors of several static physical properties, namely
the total number of excitations, the photon statistics of the field, and the
cavity-qubit entanglement. Finally, we explain the dynamical properties which
are traditionally associated to the deep strong coupling regime, such as the
collapses and revivals of the state population, in the frame of the proposed
spectral classification.
