Engineering two-mode interactions in ion traps

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 that 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.