Trapped-ion Lissajous trajectories by engineering Rashba- and Dresselhaus-type spin-orbit interactions in a Paul trap

Here we present a protocol for generating Lissajous curves with a trapped ion by engineering Rashba- and the Dresselhaus-type spin-orbit (SO) interactions in a Paul trap. The unique anisotropic Rashba $\alpha_{x}$ , $\alpha_{y}$ and Dresselhaus $\beta_{x}$ , $\beta_{y}$ couplings afforded by our setup also enable us to obtain an “unusual” Zitterbewegung, i.e., the semiconductor analog of the relativistic trembling motion of electrons, with cycloidal trajectories in the absence of magnetic fields. We have also introduced bounded SO interactions, confined to an upper-bound vibrational subspace of the Fock states, as an additional mechanism to manipulate the Lissajous motion of the trapped ion. We have also accounted for dissipative effects on the vibrational degrees of freedom of the ion and find that the Lissajous trajectories are still robust and well defined for realistic parameters