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arXiv:2010.12154 · Physical Review Research

Damped point-vortex model for polar-core spin vortices in a ferromagnetic spin-1 Bose-Einstein condensate

L. A. Williamson, P. B. Blakie

Published Feb 17, 2021• 12 citations
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Abstract

Ferromagnetic spin-1 Bose-Einstein condensates in the broken-axisymmetric phase support polar-core spin vortices (PCVs), which are intimately linked to the nonequilibrium dynamics of the system. For a purely transversely magnetized system, the Turner point-vortex model predicts that PCVs behave like massive charged particles interacting via a two-dimensional Coulomb potential. We test the accuracy of the Turner model for two oppositely charged PCVs, via comparisons with numerical simulations. While the bare Turner model shows discrepancies with our numerical results, we find that a simple rescaling of the PCV mass gives much better agreement. This can be explained via a phenomenological damping arising from coupling to modes extrinsic to the point-vortex phase space. We also identify the excitations produced following PCV annihilation, which help elucidate recent phase ordering results. We extend the Turner model to cases where the system is magnetized both transversally and axially, identifying a crossover to scalar vortex dynamics for increasing external Zeeman field.

DOI
10.1103/physrevresearch.3.013154

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Ferromagnetic spin-1 Bose-Einstein condensates in the broken-axisymmetric phase support polar-core spin vortices (PCVs), which are intimately linked to the nonequilibrium dynamics of the system. For a purely transversely magnetized system, the Turner point-vortex model predicts that PCVs behave like massive charged particles interacting via a two-dimensional Coulomb potential. We test the accuracy of the Turner model for two oppositely charged PCVs, via comparisons with numerical simulations. While the bare Turner model shows discrepancies with our numerical results, we find that a simple rescaling of the PCV mass gives much better agreement. This can be explained via a phenomenological damping arising from coupling to modes extrinsic to the point-vortex phase space. We also identify the excitations produced following PCV annihilation, which help elucidate recent phase ordering results. We extend the Turner model to cases where the system is magnetized both transversally and axially, identifying a crossover to scalar vortex dynamics for increasing external Zeeman field.
A ferromagnetic spin-1 condensate supports polar-core spin vortices (PCVs) in the easy plane phase. We derive a model for the dynamics of these PCVs using a variational Lagrangian approach. The PCVs behave as massive charged particles interacting under the two-dimensional Coulomb interaction, with the mass arising from interaction effects within the vortex core. We compare this model to numerical simulations of the spin-1 Gross-Pitaevskii equations and find semiquantitative agreement. In addition, the numerical results suggest that the PCV core couples to spin waves, and this affects the PCV dynamics even far from the core. We identify areas of further research that could extend the model of PCV dynamics presented here.
In the easy-plane phase, a ferromagnetic spin-1 Bose-Einstein condensate is magnetized in a plane transverse to the applied Zeeman field. This phase supports polar-core spin vortices (PCVs), which consist of phase windings of transverse magnetization. Here we show that spin-changing collisions cause a PCV to accelerate down density gradients in an inhomogeneous condensate. The dynamics is well-described by a simplified model adapted from scalar systems, which predicts the dependence of the dynamics on trap tightness and quadratic Zeeman energy. In a harmonic trap, a PCV accelerates radially to the condensate boundary, in stark contrast to the azimuthal motion of vortices in a scalar condensate. In a trap that has a local potential maximum at the centre, the PCV exhibits oscillations around the trap centre, which persist for a remarkably long time. The oscillations coincide with the emission and reabsorption of axial spin waves, which reflect off the condensate boundary.
The phase diagram of lowest-energy vortices in the polar phase of spin-1 Bose-Einstein condensates is investigated theoretically. Singly quantized vortices are categorized by the local ordered state in the vortex core and three types of vortices are found as lowest-energy vortices, which are elliptic AF-core vortices, axisymmetric F-core vortices, and N-core vortices. These vortices are named after the local ordered state, ferromagnetic (F), antiferromagnetic (AF), broken-axisymmetry (BA), and normal (N) states apart from the bulk polar (P) state. The N-core vortex is a conventional vortex, in the core of which the superfluid order parameter vanishes. The other two types of vortices are stabilized when the quadratic Zeeman energy is smaller than a critical value. The axisymmetric F-core vortex is the lowest-energy vortex for ferromagnetic interaction, and it has an F core surrounded by a BA skin that forms a ferromagnetic-spin texture, as exemplified by the localized Mermin-Ho texture. The elliptic AF-core vortex is stabilized for antiferromagnetic interaction; the vortex core has both nematic-spin and ferromagnetic orders locally and is composed of the AF-core soliton spanned between two BA edges. The phase transition from the N-core vortex to the other two vortices is continuous, whereas that between the AF-core and F-core vortices is discontinuous. The critical point of the continuous vortex-core transition is computed by the perturbation analysis of the Bogoliubov theory and the Ginzburg-Landau formalism describes the critical behavior. The influence of trapping potential on the core structure is also investigated.
Weinvestigate the vortex structures excited by Ioffe–Pritchard magnetic field and Dresselhaus-type spin–orbit coupling in F = 2 ferromagnetic Bose–Einstein condensates. In the weakly interatomic interacting regime, an external magnetic field can generate a polar-core vortex in which the canonical particle current is zero. With the combined effect of spin–orbit coupling and magnetic field, the ground state experiences a transition from polar-core vortex to Mermin–Ho vortex, in which the canonical particle current is anticlockwise. For fixed spin–orbit coupling strengths, the evolution of phase winding, magnetization, and degree of phase separation with magnetic field are studied. Additionally, with further increasing spin–orbit coupling strength, the condensate exhibits symmetrical density domains separated by radial vortex arrays. Our work paves the way to explore exotic topological excitations in high-spin systems.
We theoretically investigate vortex-lattice phases of rotating spinor Bose-Einstein condensates (BEC's) with a ferromagnetic spin interaction by numerically solving the Gross-Pitaevskii equation. The spinor BEC under slow rotation can sustain a rich variety of exotic vortices due to the multicomponent order parameters, such as Mermin-Ho and Anderson-Toulouse coreless vortices (the two-dimensional Skyrmion and meron) and nonaxisymmetric vortices with shifting vortex cores. Here, we present the spin texture of various vortex-lattice states at higher rotation rates and in the presence of an external magnetic field. In addition, the vortex phase diagram is constructed in the plane of the total magnetization MMM and the external rotation frequency $\ensuremath{\Omega}$ by comparing the free energies of possible vortices. It is shown that the vortex phase diagrams in the $M\text{\ensuremath{-}}\ensuremath{\Omega}$ plane may be divided into two categories: (i) the coreless vortex lattice formed by the several types of Mermin-Ho vortices and (ii) the vortex lattice filling in the cores with the pure polar (antiferromagnetic) state. In particular, it is found that the type-(ii) state forms composite lattices of coreless and polar-core vortices. The difference between type (i) and type (ii) results from the existence of the singularity of the spin textures, which may be experimentally confirmed by the spin imaging within polarized light recently proposed by Carusotto and Mueller. We also discuss the stability of triangular and square lattice states for rapidly rotating condensates.
We demonstrate how conservation of longitudinal magnetization can have pronounced effects on both stability and structure of vortices in the atomic spin-1 Bose-Einstein condensate by providing a systematic characterization of nonsingular and singular vortex states. Constructing spinor wave functions for vortex states that continuously connect ferromagnetic and polar phases, we systematically derive analytic models for nonrotating cores of different singular vortices and for composite defect states with distinct small- and large-distance topology. We explain how the conservation law provides a stabilizing mechanism when the coreless vortex imprinted on the condensate relaxes in the polar regime of interatomic interactions. The resulting structure forms a composite defect: The inner ferromagnetic coreless vortex deforms toward an outer singly quantized polar vortex. We also numerically show how other even more complex hierarchies of vortex-core topologies may be stabilized. Moreover, we analyze the structure of the coreless vortex also in a ferromagnetic condensate and show how reducing magnetization leads to a displacement of the vortex from the trap center and eventually to the deformation and splitting of its core where a singular vortex becomes a lower-energy state. For the case of singular vortices, we find that the stability and the core structure are notably less influenced by the conservation of magnetization.