Fig. 3: Knotting states of three-fold symmetry.
From: Topological atom optics and beyond with knotted quantum wavefunctions
a The spin-orbit invariant wavefunction \({\zeta }^{{{{{{{{\rm{C}}}}}}}}}={({\zeta }_{2}{e}^{i\phi }\,\,\,0\,\,\,0\,\,\,{\zeta }_{-1}\,\,\,0)}^{T}\) shows a rotation of the spherical harmonics by − 2π/3 on a full azimuthal traversal while the overall phase changes by 2π/3. In the inset box mF denotes the magnetic sublevels of the hyperfine states. We depict the populations and phases of the wavefunction components \(\left\vert F,{m}_{F}\right\rangle\), showing the azimuthal phase of the component in \(\left\vert 2,2\right\rangle\) and the uniform phase of the component in \(\left\vert 2,-1\right\rangle\). b We show a head-on view of the three-fold symmetric spin alignment in terms of spherical harmonics using lines depicting the alignment measure for the cyclic phase wavefunction. c K−1,3 is visualized in 3D following a mapping of the coupled rotation of spin and orbital angular momentum onto the 3D torus. d The experimentally realized atomic wavefunction shows the rotation of the spin state on an azimuthal traversal. We show a 3D reconstruction of the K−1,3 torus knot from the experimental data. The color bar depicts the phase in all panels.
