Generation of Schrodinger cat state of a single trapped cold ion

The fidelity of the generated Schr\"{o}dinger Cat state (SCS) of a single trapped ion in the Lamb--Dicke approximation is discussed. The results show that the fidelity significantly decreases with the values of Lamb--Dicke parameter $\eta $ and coherent state amplitude $\alpha $ increasing. For $\eta = 0.20$ and $\alpha = 3$, the typical values of experimental parameters, the fidelity is rather low (30{\%}). A scheme for generating the SCS is proposed without making the Lamb--Dike approximation in laser--ion interaction, and the fidelity of the generated SCS is about 99{\%} for the typical values of experimental Lamb--Dicke parameters.