  We propose a scheme for implementation of logical gates in a trapped ion
inside a high-Q cavity. The ion is simultaneously interacting with a
(classical) laser field as well as with the (quantized) cavity field. We
demonstrate that simply by tuning the ionic internal levels with the
frequencies of the fields, it is possible to construct a controlled-NOT gate in
a three step procedure, having the ion's internal as well as motional levels as
qubits. The cavity field is used as an auxiliary qubit and basically remains in
the vacuum state.
