Generation of Schr?dinger cat state of a single trapped cold ion

The fidelity of the generated Schr?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.