Characteristic spectra of circuit quantum electrodynamics systems from   the ultrastrong- to the deep-strong-coupling regime
  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. 
Fumiki Yoshihara, Tomoko Fuse, Sahel Ashhab, Kosuke Kakuyanagi, Shiro   Saito, and Kouichi Semba
quant-ph cond-mat.supr-con