The Spectrum in Qubit-Oscillator Systems in the Ultrastrong Coupling Regime

Recent measurement on an LC resonator magnetically coupled to a superconducting qubit [Phys. Rev. Lett. 105 (2010) 237001] shows that the system operates in the ultra-strong coupling regime and crosses the limit of validity for the rotating-wave approximation of the Jaynes-Cummings model. By using extended bosonic coherent states, we solve the Jaynes-Cummings model exactly without using the rotating-wave approximation. Our numerically exact results for the spectrum of the flux qubit coupled to the LC resonator are fully consistent with the experimental observations. The smallest Bloch-Siegert shift obtained is consistent with that observed in this experiment. In addition, the Bloch-Siegert shifts in arbitrary level transitions and for arbitrary coupling constants are predicted.