  A long-lived coherent state and non-linear interaction have been
experimentally demonstrated for the vibrational mode of a trapped ion. We
propose an implementation of quantum computation using coherent states of the
vibrational modes of trapped ions. Differently from earlier experiments, we
consider a far-off resonance for the interaction between external fields and
the ion in a bidimensional trap. By appropriate choices of the detunings
between the external fields, the adiabatic elimination of the ionic excited
level from the Hamiltonian of the system allows for beam splitting between
orthogonal vibrational modes, production of coherent states and non-linear
interactions of various kinds. In particular, this model enables the generation
of the four coherent Bell states. Furthermore, all the necessary operations for
quantum computation such as preparation of qubits, one-qubit and controlled
two-qubit operations, are possible. The detection of the state of a vibrational
mode in a Bell state is made possible by the combination of resonant and
off-resonant interactions between the ion and some external fields. We show
that our read-out scheme provides highly efficient discrimination between all
the four Bell states. We extend this to a quantum register composed of many
individually trapped ions. In this case, operations on two remote qubits are
possible through a cavity mode. We emphasize that our remote-qubit operation
scheme does not require a high quality factor resonator: the cavity field acts
as a catalyst for the gate operation.
