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

Jia-Ming Cheng, Ming Gong+3 more|arXiv (Cornell University)|Jul 4, 2019

Cold Atom Physics and Bose-Einstein Condensates58 references4 citations

TL;DR

This paper proposes a realization of non-Abelian spin-orbital-angular-momentum (SOAM) coupling in spinor Bose-Einstein condensates using time-dependent hedge-hog-type magnetic fields and high-order Hermite-Gaussian laser beams to create a spherical surface trap. It demonstrates tunable ground-state degeneracy, supporting meta-ferromagnetic and meta-polar phases with quantized total angular momentum, and stable spin vortex lattices with nontrivial topology, offering a platform for studying strongly correlated physics in curved geometry.

ABSTRACT

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 $|\langle \mathbf{J} angle|$. 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.

Motivation & Objective

• To realize non-Abelian SOAM coupling in ultracold atomic systems, which has been experimentally elusive despite its theoretical importance in quantum Hall physics and atomic fine structure.
• To construct a spherical surface trap using high-order Hermite-Gaussian beams and time-dependent magnetic fields to enable curvature-induced quantum effects in spinor condensates.
• To explore the emergence of topologically nontrivial spin vortex states and degenerate ground states with quantized total angular momentum in the presence of tunable SOAM coupling.
• To investigate the interplay between spin-orbit coupling, spin exchange interactions, and geometric confinement in shaping exotic quantum phases such as meta-ferromagnetic and polar states.

Proposed method

• Implement non-Abelian SOAM coupling via a time-dependent hedge-hog-type magnetic field gradient, inducing effective spin-orbital coupling in spin-1 Bose-Einstein condensates.
• Use high-order Hermite-Gaussian laser beams to create a curved, spherical surface trap that confines atoms and enables spatially varying SOAM coupling strength.
• Construct an effective Hamiltonian based on the non-Abelian $σ \cdot \mathbf{L}$ coupling, where $\mathbf{L}$ is orbital angular momentum and $\mathbf{F}$ is total spin, leading to a $\mathbf{L} \cdot \mathbf{F}$ interaction.
• Solve the single-particle Schrödinger equation on the sphere to obtain eigenstates with quantized total angular momentum $j = |l - f|$, revealing degeneracy patterns and spin textures.
• Analyze many-body ground states under weak interactions using mean-field theory, identifying meta-ferromagnetic and meta-polar phases with non-zero $|\langle \mathbf{J} \rangle|$.
• Map spin vortex defect configurations using the Poincaré index $Q = \pm 1$, showing stable lattice patterns consistent with the Thomson problem on a sphere.

Experimental results

Research questions

• RQ1 Can non-Abelian SOAM coupling $\mathbf{L} \cdot \mathbf{F}$ be experimentally realized in ultracold atomic systems using magnetic gradient fields and laser beams?
• RQ2 How does the curvature of a spherical surface trap influence the degeneracy and topology of ground states in spin-1 Bose-Einstein condensates?
• RQ3 What types of topological spin vortex structures—such as coreless vortices or polar-core vortices—emerge in the presence of non-Abelian SOAM coupling and weak interactions?
• RQ4 How do meta-ferromagnetic and meta-polar phases with quantized total angular momentum arise in this system, and what is their stability under spin-exchange interactions?
• RQ5 Can the defect patterns of spin vortices on the sphere be described by known geometric models such as the Thomson problem?

Key findings

• Non-Abelian SOAM coupling $\mathbf{L} \cdot \mathbf{F}$ is realized via time-dependent magnetic fields and high-order Hermite-Gaussian beams, enabling tunable coupling strength and controlled geometric confinement.
• The system supports degenerate ground states with quantized total angular momentum $|\langle \mathbf{J} \rangle| = |\langle \mathbf{L} + \mathbf{F} \rangle|$, where the degeneracy can be tuned by adjusting the SOAM coupling strength.
• For $\lambda \in (4,6)$, the ground state is meta-ferromagnetic with $\psi^{2,1}_{1,1}$, exhibiting maximum spin and orbital angular momentum, and two mFM-centered coreless vortices at the poles.
• In the weak interaction limit, spin vortices with Poincaré index $Q = \pm 1$ form stable configurations; $Q = +1$ vortices are coreless and centered on mFM or FM regions, while $Q = -1$ vortices exhibit polar cores with $\vec{\mathcal{F}} = 0$ at the center.
• When spin-exchange interaction is strong, the system supports $Z_2$-symmetric polar states with nontrivial topological invariants, indicating robustness against perturbations.
• Defects of spin vortices form stable lattice configurations matching the Thomson problem on a sphere, with $N_+ - N_- = 2$ for $Q = +1$ vortices, confirming geometric stability.