Topological spinor vortex matter on spherical surface induced by non-Abelian spin-orbital-angular-momentum coupling

We provide an explicit way to implement non-Abelian spin-orbital-angular-momentum (SOAM) coupling in spinor Bose-Einstein condensates using magnetic gradient coupling. For a spherical surface trap addressable using high-order Hermite-Gaussian beams, we show that this system supports various degenerate ground states carrying different total angular momenta $\mathbf{J}$, and the degeneracy can be tuned by changing the strength of SOAM coupling. For weakly interacting spinor condensates with $f=1$, the system supports various meta-ferromagnetic phases and meta-polar states described by quantized total mean angular momentum $|\langle \mathbf{J} \rangle|$. Polar states with $Z_2$ symmetry and Thomson lattices formed by defects of spin vortices are also discussed. The system can be used to prepare various stable spin vortex states with nontrivial topology, and serve as a platform to investigate strong-correlated physics of neutral atoms with tunable ground-state degeneracy.