  Recent measurement on an LC resonator magnetically coupled to a
superconducting qubit[arXiv:1005.1559] 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 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.
