Trapped ions as vibrational beam splitters: SU(2) states in a two-dimensional ion trap

A scheme for preparing vibrational SU(2) states of motion in a two-dimensional ion trap is described. These anticorrelated two-mode states are formally equivalent to the output photon states of a lossless SU(2) interferometer with number-state inputs. Nontrivial statistics such as the binomial distribution and the discrete ‘‘arcsine’’ distribution can be generated in the vibrational states of trapped ions, and detected by measuring the population inversion of the ion driven by a laser field along a specific direction.