Generation of Kerr non-Gaussian motional states of trapped ions

Non-Gaussian states represent a powerful resource for quantum information protocols in the continuous variables regime. Cat states, in particular, have been produced in the motional degree of freedom of trapped ions by controlled displacements dependent on the ionic internal state. An alternative method harnesses the Kerr nonlinearity naturally existent in this kind of system. We present detailed calculations confirming its feasibility for typical experimental conditions. Additionally, this method permits the generation of complex non-Gaussian states with negative Wigner functions. Especially, superpositions of many coherent states are achieved at a fraction of the time necessary to produce the cat state.