  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.
