Abstract
We propose a scheme for implementation of logical gates in a trapped ion inside a high finesse cavity. The ion is 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 levels as well as vibrational (motional) levels as qubits. The cavity field is used as an auxiliary qubit and basically remains in the vacuum state.
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