Quantum logic operation with a single trapped ion using the nonlinear interaction beyond the Lamb-Dicke limit

Using the nonlinear interaction between internal and external degrees of a trapped ion assisted by a laser beam, we present simplified methods for two-qubit quantum logic operation. The nonlinear interaction arises when the Lamb-Dicke parameter is not small enough. Two special cases, one with the: laser tuned to the first red sideband and the other tuned to the carrier, are discussed in detail.