Elementary operations for quantum logic with a single trapped two-level   cold ion beyond Lamb-Dicke limit
  A simple alternative scheme for implementing quantum gates with a single trapped cold two-level ion beyond the Lamb-Dicke (LD) limit is proposed. Basing on the quantum dynamics for the laser-ion interaction described by a generalized Jaynes-Cummings model, one can introduce two kinds of elementary quantum operations i.e., the simple rotation on the bare atomic state, generated by applying a resonant pulse, and the joint operation on the internal and external degrees of the ion, performed by using an off-resonant pulse. Several typical quantum gates, including Hadamard gate, controlled-Z and controlled-NOT gates $etc.$, can thus be implemented exactly by using these elementary operations. The experimental parameters including the LD parameter and the durations of the applied laser pulses, for these implementation are derived analytically and numerically. Neither the LD approximation for the laser-ion interaction nor the auxiliary atomic level is needed in the present scheme. 
L. F. Wei, S. Y. Liu and X. L. Lei