Exact Quantum Logic Gates with a Single Trapped Cold Ion

We present an alternative scheme exactly to implement one-qubit and two-qubit quantum gates with a single trapped cold ion driven by a traveling laser 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 vibration of the ion. The conditions to realize these operations, including the duration of each applied laser pulse and Lamb-Dicke parameter, are derived. In our scheme neither the auxiliary atomic level nor the Lamb-Dicke approximation is required. The multiquantum transition between the internal and external degrees of freedom of the ion is considered.