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For certain {\it specific} (or {\it"magic"}) Lamb-Dicke (LD) parameters,Monroe {\it et al} showed [Phys. Rev. {\bf A 55}, R2489 (1997)] that atwo-qubit quantum operation, between the external and internal degrees offreedom of a single trapped ion, could be implemented by applying a singlecarrier laser pulse. Here, we further show that, such a two-qubit operation(which is equivalent to the standard CNOT gate, only apart from certain phasefactors) could also be significantly-well realized for {\it arbitrarily}selected LD parameters. Instead of the so-called "$\pi$-pulses" used in theprevious demonstrations, the durations of the pulses applied in the presentproposal are required to be accurately set within the decoherence times of theion. % We also propose a simple approach by using only one off-resonant (e.g.,blue-sideband) laser pulse to eliminate the unwanted phase factors existed inthe above two-qubit operations for generating the standard CNOT gates.

Simplified approach to generate controlled-NOT gates with single trapped ions for arbitrary Lamb-Dicke parameters