Generation of two-mode nonclassical motional states and a Fredkin gate operation in a two-dimensional ion trap

We present an efficient scheme to generate two-mode SU(2) macroscopic quantum superposition (Schr\"odinger cat) states, entangled number states and entangled coherent states for the vibrational motion of an ion trapped in a two-dimensional harmonic potential well. We also show that the same scheme can be used to realize a Fredkin gate operation.