Exact quantum logic gates with a single trapped cold ion

We present an alternative scheme to exactly implement one-qubit and two-qub it quantum gates with a single trapped cold ion driven by a travelling lase r field. The internal degree of freedom of the ion acts as the target qubit and the control qubit is encoded by two Fock states of the external vibrat ion of the ion. The conditions to realize these operations, including the d uration of each applied laser pulse and Lamb-Dicke parameter, are derived. In our scheme neither the auxiliary atomic level nor the Lamb-Dicke approxi mation is required. The multiquantum transition between the internal and ex ternal degrees of freedom of the ion is considered.