## (a) Position space eigenfunctions for low ground states at large spin–orbit coupling

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**Figure 3.** (a) Position space eigenfunctions for low ground states at large spin–orbit coupling. The red and blue curves correspond to the functions rg_{nj}^{+}(r) and rg_{nj}^{-}(r) respectively. (b) The first three position space eigenfunctions rg_{nj}^{+}(r) for *j* = 1/2 and *n* = 1, 2, 3 (red, blue and black respectively.) (c) The position space eigenfunctions rg_{nj}^{+}(r) in the lowest radial mode for the three lowest values of j=\frac{1}{2},\frac{3}{2},\frac{5}{2}, correspond to the red, blue and black curves respectively.

**Abstract**

We investigate the properties of an atom under the influence of a synthetic three-dimensional spin–orbit coupling (Weyl coupling) in the presence of a harmonic trap. The conservation of total angular momentum provides a numerically efficient scheme for finding the spectrum and eigenfunctions of the system. We show that at large spin–orbit coupling the system undergoes dimensional reduction from three to one dimension at low energies, and the spectrum is approximately Landau level-like. At high energies, the spectrum is approximately given by the three-dimensional isotropic harmonic oscillator. We explore the properties of the ground state in both position and momentum space. We find the ground state has spin textures with oscillations set by the spin–orbit length scale.