  A non-relativistic system such as an ultracold trapped ion may perform a
quantum simulation of a Dirac equation dynamics under specific conditions. The
resulting Hamiltonian and dynamics are highly controllable, but the coupling
between momentum and internal levels poses some difficulties to manipulate the
internal states accurately in wave packets. We use invariants of motion to
inverse engineer robust population inversion processes with a homogeneous,
time-dependent simulated electric field. This exemplifies the usefulness of
inverse-engineering techniques to improve the performance of quantum simulation
protocols.
