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Physical Review A

Vortex chain in anisotropic spin-orbit-coupled spin-1 Bose-Einstein condensates

Chao-Fei Liu, Yan-mei Yu, Shih-Chuan Gou, Wu-Ming Liu

Published Jun 24, 2013• 53 citations
Publisher

Abstract

We investigate the rotating spin-1 Bose-Einstein condensates with anisotropic spin-orbit coupling by using the damped projected Gross-Pitaevskii equation. The anisotropic spin-orbit coupling can cause some vortices to form the vortex chain, where the vortices link up with each other in a line. The formation of the vortex chain starts from very small rotation frequency and coexists with more remanent vortices with the increasing rotation frequency. By increasing the anisotropy of the spin-orbit coupling, the creation of the vortex chain is enhanced, accompanied by the emergence of several types of vortex chain and other vortex configurations. When the vortices of the ${m}_{F}=\ensuremath{-}1$ and ${m}_{F}=+1$ components are of synchronous phases, such vortex configurations do not cause any topological excitation in the spin texture.

DOI
10.1103/physreva.87.063630

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We investigate the rotating spin-1 Bose-Einstein condensates with anisotropic spin-orbit coupling by using the damped projected Gross-Pitaevskii equation. The anisotropic spin-orbit coupling can cause some vortices to form the vortex chain, where the vortices link up with each other in a line. The formation of the vortex chain starts from very small rotation frequency and coexists with more remanent vortices with the increasing rotation frequency. By increasing the anisotropy of the spin-orbit coupling, the creation of the vortex chain is enhanced, accompanied by the emergence of several types of vortex chain and other vortex configurations. When the vortices of the mF=\ensuremath−1{m}_{F}=\ensuremath{-}1mF​=\ensuremath−1 and mF=+1{m}_{F}=+1mF​=+1 components are of synchronous phases, such vortex configurations do not cause any topological excitation in the spin texture.
Abstract We consider the ground-state properties of a rotating spin-orbit-coupled Bose—Einstein condensate under extreme elongation in a harmonic plus quartic potential. The effects of spin-orbit coupling and rotation on the ground-state vortex structures are investigated. In the absence of spin-orbit coupling, new nucleated vortices gradually form vortex lines and annular vortex structures with the increase of the rotation frequency. In the presence of spin-orbit coupling, part of the vortices arrange in a line and form a stable vortex chain, and the remanent vortices coexist in pairs aside such vortex chain. More specially, the remanent vortices of each component repel each other and form vortex pair for isotropic spin-orbit coupling, while attract each other and locate in the same positions for anisotropic spin-orbit coupling.
We investigate the anisotropic spin–orbit coupled spin-2 Bose–Einstein condensates with Ioffe–Pritchard magnetic field. With nonzero magnetic field, anisotropic spin–orbit coupling will introduce several vortices and further generate a vortex chain. Inside the vortex chain, the vortices connect to each other, forming a line along the axis. The physical nature of the vortex chain can be explained by the particle current and the momentum distribution. The vortex number inside the vortex chain can be influenced via varying the magnetic field. Through adjusting the anisotropy of the spin–orbit coupling, the direction of the vortex chain is changed, and the vortex lattice can be triggered. Moreover, accompanied by the variation of the atomic interactions, the density and the momentum distribution of the vortex chain are affected. The realization and the detection of the vortex chain are compatible with current experimental techniques.
Realistic methods to create vortices in spin-orbit-coupled Bose-Einstein condensates are discussed. It is shown that, contrary to common intuition, rotation of the trap containing a spin-orbit condensate does not lead to an equilibrium state with static vortex structures but gives rise instead to nonequilibrium behavior described by an intrinsically time-dependent Hamiltonian. We propose here the following alternative methods to induce thermodynamically stable static vortex configurations: (i) to rotate both the lasers and the anisotropic trap and (ii) to impose a synthetic Abelian field on top of synthetic spin-orbit interactions. Effective Hamiltonians for spin-orbit condensates under such perturbations are derived for most currently known realistic laser schemes that induce synthetic spin-orbit couplings. The Gross-Pitaevskii equation is solved for several experimentally relevant regimes. The new interesting effects include spatial separation of left- and right-moving spin-orbit condensates, the appearance of unusual vortex arrangements, and parity effects in vortex nucleation where the topological excitations are predicted to appear in pairs. All these phenomena are shown to be highly nonuniversal and depend strongly on a specific laser scheme and system parameters.
We study the vortex-lattice formation in a rotating Rashba spin-orbit (SO) coupled quasi-two-dimensional (quasi-2D) hyper-fine spin-1 spinor Bose-Einstein condensate (BEC) in the x-y plane using a numerical solution of the underlying mean-field Gross-Pitaevskii equation. In this case, the non-rotating Rashba SO-coupled spinor BEC can have topological excitation in the form of vortices of different angular momenta in the three components, e.g. the (0, +1, +2)- and (-1, 0, +1)-type states in ferromagnetic and anti-ferromagnetic spinor BEC: the numbers in the parenthesis denote the intrinsic angular momentum of the vortex states of the three components with the negative sign denoting an anti-vortex. The presence of these states with intrinsic vorticity breaks the symmetry between rotation with vorticity along the z and -z axes and thus generates a rich variety of vortex-lattice and anti-vortex-lattice states in a rotating quasi-2D spin-1 spinor ferromagnetic and anti-ferromagnetic BEC, not possible in a scalar BEC. For weak SO coupling, we find two types of symmetries of these states - hexagonal and 'square'. The hexagonal (square) symmetry state has vortices arranged in closed concentric orbits with a maximum of 6, 12, 18… (8, 12, 16…) vortices in successive orbits. Of these two symmetries, the square vortex-lattice state is found to have the smaller energy.
By exploring the spin-orbit-coupled rotating pseudo-spin-1/2 Bose-Einstein condensates with the stochastic projected Gross-Pitaevskii equations, we prove the existence of the circular-hyperbolic skyrmion, which possesses two extreme values of Sz{S}_{z}Sz​. The nonequal intraspecies interactions and interspecies interactions cause the two components to be wholly disproportionate when the system reaches the equilibrium state. The circular-hyperbolic skyrmion results from the vortex-dipole structure. For the miscible condensates, the increase of spin-orbit coupling enhances the creation of the circular-hyperbolic skyrmion, and makes them link one after another locally. In particular, the circular-hyperbolic skyrmion even can form a chain when the spin-orbit coupling is only in one direction. For the immiscible condensates, the hyperbolic skyrmion occurs at the center, while the circular-hyperbolic skyrmion occurs at the outskirts of the condensates. The increase of the spin-orbit coupling restricts the creation of the hyperbolic skyrmion and enhances the creation of the circular-hyperbolic skyrmion.
Three types of vortex-pair are identified in two-component Bose–Einstein condensates (BEC) of different kinds of spin-orbit coupling. One type holds the two vortices in one component of the twocomponent condensates. Both the other two types hold a vortex in each component of the twocomponent condensates, and exhibit meron-pair textures that have either null or unit topological charge, respectively. The cores of the two vortices are connected by a string of the relative phase jump. These vortex pairs can be generated from a vortex-free wave packet by incorporating different non- Abelian gauge field into the BEC. When a Rabi coupling is introduced, the distance between the two cores is effectively controlled by the Rabi coupling strength and a transition of vortex configurations is observed.