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.