Generation of C-NOT gate, swap gate and phase gate based on a two-dimensional ion trap

We study the realization of quantum logic gates in the interacting system of a two-dimensional trapped ion and two laser beams. We simplify the Hamiltonian of the system by Choosing the frequency detuning of laser fields and the transition frequency between the internal states of the ion, and furthermore, we describe the realization of a C-NOT gate, a swap gate and a phase gate. During these processes, the system needs to meet the Lamb-Dicke limit, and the Rabi oscillation frequency should be far smaller than the phonon frequency.