Klein tunneling and Dirac potentials in trapped ions

We propose the quantum simulation of the Dirac equation with potentials, allowing the study of relativistic scattering and Klein tunneling. This quantum relativistic effect permits a positive-energy Dirac particle to propagate through a repulsive potential via the population transfer to negative-energy components. We show how to engineer scalar, pseudoscalar, and other potentials in the $1+1$ Dirac equation by manipulating two trapped ions. The Dirac spinor is represented by the internal states of one ion, while its position and momentum are described by those of a collective motional mode. The second ion is used to build the desired potentials with high spatial resolution.