  Here we present a protocol for generating Lissajous curves with a trapped ion
by engineering Rashba- and the Dresselhaus-type spin-orbit 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
enables 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. Finally, we 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.
