Lower ground state due to counter-rotating wave interaction in a trapped ion system

We consider a single ion confined in a trap under the radiation of two travelling waves of lasers. In the strong-excitation regime and without the restriction of the Lamb-Dicke limit, the Hamiltonian of the system is similar to a driving Jaynes-Cummings model without the rotating wave approximation ( RWA). The approach we developed enables us to present complete eigensolutions, which makes it possible to compare with the solutions under the RWA. We find that the ground state in our non-RWA solution is energetically lower than the counterpart under the RWA. If we have the ion in the ground state, it is equivalent to a spin dependent force on the trapped ion. Discussion is given for the difference between the solutions with and without the RWA, and for the relevant experimental test, as well as for the possible application to quantum information processing.