We describe how two vibrational degrees of freedom of a single trapped ion can be coupled through the action of suitably-chosen laser excitation. We concentrate on a two-dimensional ion trap with dissimilar vibrational frequencies in the x- and y-directions of motion, and derive from first principles a variety of quantized two-mode couplings, concentrating on a linear coupling which takes excitations from one mode to another. We demonstrate how this can result in a state rotation, in which it is possible to transfer the motional state of the ion from say the x-direction to the y-direction without prior knowledge of that motional state.
@article{arxiv.quant-ph/9708008,
title = {Engineering two-mode interactions in ion traps},
author = {J. Steinbach and J. Twamley and P. L. Knight},
journal= {arXiv preprint arXiv:quant-ph/9708008},
year = {2009}
}