The currents <strong>j</strong>
M AndersonBrandon
W ClarkCharles
2013
<p><strong>Figure 4.</strong> The currents <strong>j</strong><sub><em>s</em></sub>, <strong>j</strong><sub><em>o</em></sub> and <strong>j</strong> = <strong>j</strong><sub><em>s</em></sub> + <strong>j</strong><sub><em>o</em></sub>, for the ground state of a trapped particle with Weyl coupling, with j=\frac{1}{2}, m=\frac{1}{2} and <em>v</em> = 10. For clarity, we have scaled the currents <strong>j</strong><sub><em>i</em></sub> by {\rm e}^{r^{2}} to account for the Gaussian damping of the wavefunction, and measured distance in units of 1/<em>v</em>, the spin–orbit length. In all figures the plane defined by <em>y</em> = 0 is plotted. The arrows represent flows of the local normalized current vector. The colour density represents the out-of-plane component of the spin textures, with lightest colour representing the maximal out-of-plane current, and the darkest representing maximal in-plane current. All three currents are azimuthally symmetric. (a) The spin currents, <strong>j</strong><sub><em>s</em></sub>, have oscillations on the length scale <em>r</em> ~ 2π/<em>v</em>. On the axis with <em>z</em> = 0, the local spin vector is polarized entirely out-of-plane at solutions to tan <em>rv</em> = −<em>rv</em>/<em>v</em><sup>2</sup>. For large <em>v</em>, the solutions are given approximately by <em>rv</em> = <em>n</em>π, where <em>n</em> is an integer. The odd solutions feature in-plane vortex loops of spin, while the spin forms anti-vortices at the even solutions. (b) The orbital currents, <strong>j</strong><sub><em>o</em></sub>, are dominated by the in-plane component, which is stronger than the out-of-plane components by a factor of <em>v</em>. A small out-of-plane component is largest on the <em>z</em> = 0 axis. In the upper half plane with <em>y</em> > 0, current converges on the point <em>r</em> = 0, while on the lower half plane all the current diverges away from the point <em>r</em> = 0. (c) The total current, <strong>j</strong>, is the sum of orbital and spin currents. The contributions from spin and orbital degrees of freedom are conserved independently.</p> <p><strong>Abstract</strong></p> <p>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.</p>