Topological atom optics and beyond with knotted quantum wavefunctions

The paper situates the work within the broader impact of topology across physics and focuses on knot theory as a framework for understanding non-trivial structures. Prior realizations of knotted and linked configurations have appeared in field theory, atomic systems, optical fields, liquid crystals, and fluids. The research question is whether analogous and controllable knotted structures can be engineered and observed in coherent atomic matter waves, specifically spinor Bose-Einstein condensates (BECs), by coupling internal spin degrees of freedom with external orbital angular momentum (OAM). The purpose is to realize spin-orbit invariant atomic wavefunctions exhibiting discrete internal symmetries that, when mapped to parameter space, form torus knots, Möbius strips, and Solomon’s links. This advances atom optics by transferring and extending structured-light topologies to matter waves, leveraging unique atomic capabilities such as tunable interactions and higher-spin manifolds.

The authors review how knot topology has influenced many areas, including the Skyrme–Faddeev model and analogous knotted structures in atomic systems, light, and liquid crystals. Innovations in structured light fields have enabled the creation of singular electromagnetic field lines in the form of links and knots and complex topologies in polarization, including Möbius strips, ribbons, and knots in polarization rotations. These optical structures reflect light’s vector nature and exotic symmetries. Previous atomic and fluid studies commonly realized knots as real-space singular lines, whereas condensed matter and field-theory models consider knots as mappings between order-parameter and real space. This work adopts the latter viewpoint in atomic systems, aiming to realize long-lived knotted matter-wave structures that persist during ballistic expansion. The literature establishes both the feasibility and significance of transferring topological concepts between optics and atomic media.

System and state preparation: The experiment uses quantum-degenerate 87Rb spinor BECs in the electronic ground state manifolds F=1 and F=2. Initial spin-polarized states |F, mF⟩ = |1, −1⟩ or |2, 2⟩ are prepared in a magnetic trap. After release (t=0), the cloud undergoes 9 ms free fall and ballistic expansion to σ≈50 μm to suppress interactions during the imprinting sequence. Raman imprinting and spin–orbit coupling: Knotted, spin–orbit invariant states are engineered via coherent two-photon Raman coupling in an effective three-level Λ system with ground states |ψ⟩ and |φ⟩ and excited state |e⟩. Two Raman beams with Rabi frequencies ΩA, ΩB and phases ϕA, ϕB drive the transitions, with polarizations (σ+, π, σ−) enabling ΔmF=±1,0. A bias magnetic field B≈11 G lifts Zeeman degeneracy. In the large detuning limit Δ, |e⟩ is adiabatically eliminated, yielding a two-level unitary evolution U(t)=exp[i(Ωt/2) σ·n/2], where Ω∝(ΩA2+ΩB2)/4Δ, n depends on the relative intensities (via α) and relative phase ϕ=ϕA−ϕB. Square Raman pulses of 5–10 μs duration are used; parameters are controlled with AOMs. Orbital angular momentum imprinting: One Raman beam is a Laguerre–Gaussian (LG) mode (OAM charge ℓ=1 or 2) while the other is Gaussian. This produces a ring-shaped transfer to the target spin state with an azimuthal phase twist set by the LG charge and beam polarizations, leaving a non-rotating core in the initial state. Interferometric control flips mode handedness when needed. Multipulse Raman sequences, combined with coherent rf pulses (100–150 μs) between adjacent Zeeman sublevels, generalize coupling to multiple mF states, controlling populations, relative phases, and OAM (l) for each component to realize specific discrete internal symmetries and fractional spin rotations. Target magnetic phases and sequences: - Spin-1 polar phase (Möbius topology): Start in |1, −1⟩. Use a (σ+,σ−) Raman pair with ℓ1=0, ℓ2=1 to transfer population to |1, 1⟩ with OAM l=1, leaving a non-rotating |1, −1⟩ core. Regions with equal |1,±1⟩ densities form the polar phase with a half-quantum-vortex-like spin–orbit invariance. - Spin-2 cyclic phase (trefoil-capable): Start in |2, 2⟩. Use rf to move population to |2, 1⟩, then a (π,σ−) Raman pair with (l,l′)=(1,0) to imprint a phase winding in |2, 2⟩ with a non-rotating |2, 1⟩ core; then a (σ,σ) Gaussian Raman pulse transfers the core to |2, −1⟩, establishing the cyclic phase relation between |2, −1⟩ and |2, 2⟩. Adjusting OAM values realizes different knot types, including trefoils when |l−l′|=2. - Spin-2 biaxial-nematic (BN) phase (Solomon’s link): Start in |2, 2⟩. A single (σ,σ) Raman beam pair with vortex charges (lx=1, ly=0), tuned to favor a four-photon process while detuning the two-photon |2,2⟩→|2,0⟩ path, transfers population to |2, −2⟩ with two units of OAM, leaving |2, 2⟩ as a non-rotating core, creating (l,l′)=(0,2). Variants with larger OAM differences (e.g., l′−l=8) directly realize Solomon’s knot structures. Imaging and reconstruction: After Raman interaction, a time-of-flight Stern–Gerlach sequence applies a brief inhomogeneous magnetic field at t=13 ms, followed by additional 13 ms time-of-flight. Absorption imaging along the quantization axis records spatially resolved density for each mF component. Known Raman phase relations (calibrated via matter-wave interferometry) and measured amplitudes reconstruct local spin states. Topological visualization: The spinor is represented via spherical harmonics Z(θ,ϕ)=Σm cm Yl,m(θ,ϕ); surfaces of |Z|2 are colored by arg Z to show internal symmetry and phase. The azimuthal angle ϕ and the spin-rotation angle θ parameterize S1×S1 on a torus; the spin–orbit invariance imposes θ=λϕ with fractional λ. Lobe-tip paths of the spherical harmonics traced over ϕ map to curves on a 3D torus, revealing torus knots K_{m,n} (or links) associated with the coupled rotations. Möbius band surfaces are visualized by tracking nematic axes for polar and BN phases.

- Demonstration of spin–orbit invariant atomic wavefunctions in 87Rb spinor BECs by coordinating internal spin rotations with external OAM using Raman imprinting. - Mapping of coupled spin–phase rotations onto torus coordinates to realize knot topologies in parameter space, yielding long-lived structures that persist during ballistic expansion. - Spin-1 polar phase: realization of a Möbius-strip topology with edge given by the torus (un)knot K_{1,2}; symmetry corresponding to half-quantum vortex behavior. Hopf link K_{2,2} is accessible with suitable OAM configuration (l=−2, l′=0). - Spin-2 cyclic phase: creation of three-fold symmetric knotted states K_{−1,3} and, with |l−l′|=2, true trefoil knots K_{±2,3}, exhibiting handedness. - Spin-2 biaxial-nematic phase: realization of a torus link K_{2,4}, topologically equivalent to Solomon’s knot K_{4,2}, visualized as two interlinked Möbius-type paths; experimental reconstruction confirms the linked topology. - Direct realization of a doubly-linked Solomon’s knot structure K_{4,2} using configurations with larger OAM differences (e.g., l′−l=8), yielding two disjoint, interlinked paths corresponding to the twice-linked Solomon’s link. - Experimental protocol specifics: Raman pulse durations 5–10 μs; bias field ≈11 G; free fall 9 ms; Stern–Gerlach kick at 13 ms plus 13 ms TOF; OAM charges ℓ=0,1,2 (and effectively higher via multi-photon processes).

The results address the core question of whether non-trivial knotted topologies can be engineered and observed in atomic matter waves by coupling internal and external degrees of freedom. By establishing spin–orbit invariant states with fractional spin rotations, the work demonstrates a direct atomic analogue—and extension—of structured optical fields with knotted polarization. The mapping to torus knots clarifies how discrete internal symmetries (two-fold polar, three-fold cyclic, four-fold biaxial-nematic) combine with condensate phase to yield Möbius strips, trefoils, and Solomon’s links. The atomic platform affords long lifetimes and controllable preparation/readout, enabling precise visualization via spherical harmonics and torus mapping. The connection to optical polarization Lissajous figures and an effective torus-knot angular momentum Jz=Lz+γSz underscores a unifying framework for spin–orbit-invariant structures across light and matter. These findings open pathways to emulate and go beyond optical phenomena in multicomponent quantum gases, with implications for studying braid groups, non-Abelian vortex algebra, and topological excitations in higher-spin condensates.

This work realizes topological atom optics by creating and characterizing knotted quantum wavefunctions in spinor BECs. Using Raman-induced spin–orbit coupling and OAM imprinting, the authors demonstrate spin-1 polar Möbius-strip structures (K_{1,2}), spin-2 cyclic knots including trefoils (K_{±2,3}), and spin-2 biaxial-nematic torus links equivalent to Solomon’s knot (K_{2,4}≈K_{4,2}), including direct twice-linked realizations. The experiments provide a robust atomic platform for studying knot topologies in parameter space, with clear visualization methods via spherical harmonics and torus mappings. Future work will extend to higher-spin manifolds (e.g., spin-3 supporting K_{5,5}), and explore analytic and geometric properties of torus knots, braid groups, and non-Abelian vortex algebra to probe local and global dynamical consequences of knot topology in quantum fluids.