3D quaternionic condensation and spin textures with Hopf invariants from synthetic spin-orbit coupling

We study unconventional condensations of two-component bosons in a harmonic trap subject to the 3D $\vec{\sigma}\cdot \vec{p}$-type spin-orbit (SO) coupling. The topology of condensate wavefunctions manifests in the quaternionic representation. The spatial distributions of the $S^3$ quaternionic phase exhibit 3D skyrmion configurations, while those of the $S^2$ spin orientation possess non-zero Hopf invariants. As increasing SO coupling strength, spin textures evolve from concentric distributions to lattice structures at weak interactions. Strong interactions change condensates into spin-polarized plane-wave states, or, superpositions of two plane-waves exhibiting helical spin spirals.