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arXiv:1703.03795 · Journal of Physics B Atomic Molecular and Optical Physics

Skyrmions with arbitrary topological charges in spinor Bose–Einstein condensates

R Zamora-Zamora, Vı́ctor Romero-Rochı́n

Published Dec 20, 2017• 11 citations
arXivHTMLPDF

Abstract

Abstract Topological Skyrmion defects with arbitrary topological charges are shown to appear in spinor F = 1 Bose–Einstein condensates, under the presence of external gauge or magnetic fields with cylindrical symmetry. We show that, depending on the magnetic field boundary conditions, the spin texture, at the planes perpendicular to the cylindrical axis, can be forced to show Skyrmions with topological charge zero, half-integer, or any desired arbitrary value between − 1 / 2 and 1/2. Our findings are obtained by numerically solving the corresponding fully coupled Gross–Pitaevskii equations without any symmetry assumptions and by analyzing the Skyrmion topological charge. We study, both, polar 23 Na and ferromagnetic 87 Rb condensates.

DOI
10.1088/1361-6455/aaa324

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Abstract Topological Skyrmion defects with arbitrary topological charges are shown to appear in spinor F = 1 Bose–Einstein condensates, under the presence of external gauge or magnetic fields with cylindrical symmetry. We show that, depending on the magnetic field boundary conditions, the spin texture, at the planes perpendicular to the cylindrical axis, can be forced to show Skyrmions with topological charge zero, half-integer, or any desired arbitrary value between − 1 / 2 and 1/2. Our findings are obtained by numerically solving the corresponding fully coupled Gross–Pitaevskii equations without any symmetry assumptions and by analyzing the Skyrmion topological charge. We study, both, polar 23 Na and ferromagnetic 87 Rb condensates.
Abstract We derive the most general expression for the Skyrmion topological charge for a two-dimensional spin texture, valid for any type of boundary conditions or for any arbitrary spatial region within the texture. It reduces to the usual one for the appropriate boundary conditions. The general expression resembles the Gauss-Bonet theorem for the Euler-Poincaré characteristic of a 2D surface, but it has definite differences, responsible for the assignment of the proper signs of the Skyrmion charges. Additionally, we show that the charge of a single Skyrmion is the product of the value of the normal component of the spin texture at the singularity times the Index or winding number of the transverse texture, a generalization of a Poincaré theorem. We illustrate our general results analyzing in detail a two-component spinor Bose–Einstein condensate (BEC) in 2D in the presence of an external magnetic field, via the Gross-Pitaevskii equation. The condensate spin textures present Skyrmions singularities at the spatial locations where the transverse magnetic field vanishes. We show that the ensuing superfluid vortices and Skyrmions have the same value for their corresponding topological charges, in turn due to the structure of the magnetic field.
We present the creation and time evolution of two-dimensional Skyrmion excitations in an antiferromagnetic spinor Bose-Einstein condensate. Using a spin rotation method, the Skyrmion spin textures were imprinted on a sodium condensate in a polar phase, where the two-dimensional Skyrmion is topologically protected. The Skyrmion was observed to be stable on a short time scale of a few tens of ms but to dynamically deform its shape and eventually decay to a uniform spin texture. The deformed spin textures reveal that the decay dynamics involves breaking the polar phase inside the condensate without having topological charge density flow through the boundary of the finite-sized sample. We discuss the possible formation of half-quantum vortices in the deformation process.
Since their experimental discovery in 1995, dilute Bose–Einstein condensates have proven to be excellent platforms for studying interacting many-body quantum systems. The gaseous condensates can be conveniently controlled in space and time with lasers and magnetic fields. Theoretically, these low-temperature systems can be described starting from first principles, and their dynamics is accurately captured by the mean-field approach. Dilute condensates with internal spin degrees of freedom may host various topological defects including vortices, monopoles, and skyrmions. Precise external control fields can be used to engineer and study these topological states. \n \nIn this thesis, we investigate the creation, stability, and dynamics of various topological defects in spinor Bose–Einstein condensates. A combination of numerical and analytical methods is used and a part of the research is carried out in collaboration with an experimental research group. The majority of the research involves numerically solving the mean-field Gross–Pitaevskii equation for a spin-1 condensate. The computations are accelerated with graphics processing units. \n \nWe find that the ground state of a ferromagnetic spin-1 condensate supports a Dirac monopole in its synthetic magnetic field in the presence of a quadrupole magnetic field. The energetics of various stationary states of spin-orbit coupled condensates are analyzed and a robust method to observe these exotic states in experiments is proposed. The creation of topological skyrmions is simulated and a very good quantitative agreement with a recent experiment is obtained. Importantly, the first experimental observations of both, Dirac monopoles and topological point defects in a system governed by a quantum field are presented. Subsequently, the decay dynamics of the point defect is studied, revealing the decay into a ferromagnetic state supporting the Dirac monopole. \n \nThe impact of the research conducted in this thesis is not limited to gaseous Bose–Einstein condensates, but due to the universality of quantum mechanics and topological defects, our work provides knowledge, ideas, and inspiration across research fields.
We investigate the spontaneous generation of crystallized topological defects via the combining effects of fast rotation and rapid thermal quench on spin-1 Bose-Einstein condensates (BECs). By solving the stochastic projected Gross-Pitaevskii equation, we show that, when the system reaches equilibrium, a hexagonal lattice of skyrmions and a square lattice of half-quantized vortices can be formed in a ferromagnetic and antiferromagnetic spinor BEC, respectively, which can be imaged by using the polarization-dependent phase-contrast method.
We propose a new scheme for creating three-dimensional Skyrmions in a ferromagnetic spin-1 Bose-Einstein condensate by manipulating a multipole magnetic field and a pair of counter-propagating laser beams. The result shows that a three-dimensional Skyrmion with topological number Q = 2 can be created by a sextupole magnetic field and the laser beams. Meanwhile, the vortex ring and knot structure in the Skyrmion are found. The topological number can be calculated analytically in our model, which implies that the method can be extended to create Skyrmions with arbitrary topological number. As the examples, three-dimensional Skyrmions with Q = 3, 4 are also demonstrated and are distinguishable by the density distributions with a specific quantization axis. These topological objects have the potential to be realized in ferromagnetic spin-1 Bose-Einstein condensates experimentally.
We investigate the spontaneous generation of crystallized topological defects via the combining effects of fast rotation and rapid thermal quench on spin-1 Bose-Einstein condensates (BECs). By solving the stochastic projected Gross-Pitaevskii equation, we show that, when the system reaches equilibrium, a hexagonal lattice of skyrmions and a square lattice of half-quantized vortices can be formed in a ferromagnetic and antiferromagnetic spinor BEC, respectively, which can be imaged by using the polarization-dependent phase-contrast method.