Fig. 5: Tying Solomon’s knot and a discrete four-fold symmetry.
From: Topological atom optics and beyond with knotted quantum wavefunctions
a We show the spherical harmonic representation of the spin-orbit invariant wavefunction \({\zeta }^{{{{{{{{\rm{BN}}}}}}}}}={({\zeta }_{2}\,\,\,0\,\,\,0\,\,\,0\,\,\,{\zeta }_{-2}{e}^{i2\phi })}^{T}\) that shows a rotation by π on an azimuthal traversal. 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 uniform phase of the component in \(\left\vert 2,2\right\rangle\) and the \({\ell }^{{\prime} }=2\) azimuthal phase of the component in \(\left\vert 2,-2\right\rangle\). b Visualizing the orientation of the atomic wavefunction that combines a discrete four-fold symmetry with a condensate phase in terms of two disjoint Möbius-type topological structures. A pair of disjoint lobe-tip paths must be constructed to fully visualize the topology of the fractional spin state rotation. c The two disjoint paths are represented with solid and dashed curves. The 3D representation shows that these paths are interlinked. The associated knot is the torus link K2,4. d We reconstruct the local spin state and the torus link from experimental data. The color bar depicts the phase in all panels.
