Analytical ground state for the Jaynes-Cummings model with the   ultrastrong coupling
  We present a generalized variational method to analytically obtain the ground-state properties of the unsolvable Jaynes-Cummings model with the ultrastrong coupling. An explicit expression for the ground-state energy, which agrees well with the numerical simulation in a wide range of the experimental parameters, is given. In particular, the introduced method can successfully solve this Jaynes-Cummings model with the positive detuning (the atomic resonant level is larger than the photon frequency), which can not be treated in the adiabatical approximation and the generalized rotating-wave approximation. Finally, we also demonstrate analytically how to control the mean photon number by means of the current experimental parameters including the photon frequency, the coupling strength, and especially the atomic resonant level. 
Yuanwei Zhang, Gang Chen, Lixian Yu, Qifeng Liang, J.-Q. Liang, and   Suotang Jia
quant-ph cond-mat.other