  We report on spectra of circuit-quantum-electrodynamics (QED) systems in an
intermediate regime that lies between the ultrastrong and deep-strong-coupling
regimes, which have been reported previously in the literature. Our
experimental results, along with numerical simulations, demonstrate that as the
coupling strength increases, the spectrum of a circuit-QED system undergoes
multiple qualitative transformations, such that several coupling regimes are
identified, each with its own unique spectral features. The different spectral
transformations can be related to crossings between energy level differences
and to changes in the symmetries of the energy eigenstates. These results allow
us to use qualitative spectral features to infer certain properties and
parameters of the system.
