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International Journal of Modern Physics B

Characterized helical modulation of three-dimensional Skyrmions in the spin-orbit coupled condensate

Hao Tian Jiang, Chong Yu, Shu-Wei Song, Wei Gao, Wei-Feng Sun, Hanquan Wang

Published Jul 30, 2020• 0 citations
Publisher

Abstract

For the Bose–Einstein condensate in the non-Abelian gauge field, the atomic wavefunction obtains helical modulation, which would lead to nontrivial objects, such as three-dimensional Skyrmion. The spin–orbit coupled condensate is usually capable of supporting Skyrmion ground states, which can be well understood through the wavefunction modulation vector concept. In the present manuscript, the modulation vector is characterized both analytically and numerically with respect to the spin–orbit coupling strength. For the SU(2) symmetric condensate, the modulation vector of the wavefunction is linearly proportional to the spin–orbit coupling strength. In the presence of spin-dependent interaction, the modulus of the modulation vector is predicted to increase or decrease according to the sign of the interaction parameter, resulting in a shrunken or expanded Skyrmion. However, the Skyrmion is unstable for a certain area of the parameter space of the spin–orbit coupling and spin-dependent interaction. The condensate favors a single Skyrmion state or the Skyrmion lattice state according to the strength of the spin–orbit coupling.

DOI
10.1142/s0217979220501817

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For the Bose–Einstein condensate in the non-Abelian gauge field, the atomic wavefunction obtains helical modulation, which would lead to nontrivial objects, such as three-dimensional Skyrmion. The spin–orbit coupled condensate is usually capable of supporting Skyrmion ground states, which can be well understood through the wavefunction modulation vector concept. In the present manuscript, the modulation vector is characterized both analytically and numerically with respect to the spin–orbit coupling strength. For the SU(2) symmetric condensate, the modulation vector of the wavefunction is linearly proportional to the spin–orbit coupling strength. In the presence of spin-dependent interaction, the modulus of the modulation vector is predicted to increase or decrease according to the sign of the interaction parameter, resulting in a shrunken or expanded Skyrmion. However, the Skyrmion is unstable for a certain area of the parameter space of the spin–orbit coupling and spin-dependent interaction. The condensate favors a single Skyrmion state or the Skyrmion lattice state according to the strength of the spin–orbit coupling.
We demonstrate that the three-dimensional Skyrmion, which has remained elusive so far, spontaneously appears as the ground state of SU(2) symmetric Bose-Einstein condensates coupled with a non-Abelian gauge field. The gauge field is a three-dimensional analogue of the Rashba spin-orbit coupling. Upon squashing the SO(3) symmetric gauge field to one- or two-dimensional shapes, we find that the ground state continuously undergoes a change from a three-dimensional to a one- or two-dimensional Skyrmion, which is identified by estimating winding numbers and helicity. All of the emerged Skyrmions are physically understandable with the concept of the helical modulation in a unified way. These topological objects might potentially be realizable in two-component Bose-Einstein condensates experimentally.
Motivated by recent studies of Bose Einstein condensates (BECs) under synthetic gauge fields, we study the textures of F = 2 BECs with a Rashba like spin-orbit coupling (SOC). By analyzing the SOC term, we demonstrate that the SOC favors a helical modulation of order parameters (OPs). In addition, we find the stable textures obtained by the numerical minimization of the Gross-Pitaevskii energy functional and interpret them as a one- or two-dimensional helical modulation. In the parameter region where the polar OP is favored, the emergence of the skyrmion of the uniaxial polar OP and its energetics relative to one-dimensional helical modulation of biaxial polar OP are discussed in details. Through the comparison of the lattice texture in the cyclic favored parameter region and uniaxial polar skyrmion, we show the instability of the lattice of the uniaxial polar skyrmion.
We study the ground-state phases of a spin-orbit-coupled Bose-Einstein condensate loaded in a two-dimensional optical lattice. We find three types of helical supersolid phases, which are dependent on the anisotropy of the spin-orbit coupling with respect to the relative strength of the yyy and xxx directions. The period of density modulation increases with the strength of the spin-orbit coupling.
The ground state properties of the rotating Bose–Einstein condensates (BECs) with SU(3) spin–orbit coupling (SOC) in a two-dimensional harmonic trap are studied. The results show that the ferromagnetic and antiferromagnetic systems present three half-skyrmion chains at an angle of 120° to each other along the coupling directions. With the enhancement of isotropic SU(3) SOC strength, the position of the three chains remains unchanged, in which the number of half-skyrmions increases gradually. With the increase of rotation frequency and atomic density–density interaction, the number of half-skyrmions on the three chains and in the regions between two chains increases gradually. The relationships of the total number of half-skyrmions on the three chains with the increase of SU(3) SOC strength, rotation frequency and atomic density–density interaction are also given. In addition, changing the anisotropic SU(3) SOC strength can regulate the number and morphology of the half-skyrmion chains.
Realization of spinor Bose-Einstein condensate in an optical trap has made it possible to create a variety of topological nontrivial structures, due to the vector character of the order parameter. Recently, artificial spin-orbit coupling in the spinor Bose-Einstein condensate, owing to coupling between the spin and the center-of-mass motion of the atom, provides an unprecedented opportunity to search for novel quantum states. As is well known, the potential well in the Bose-Einstein condensate is adjustable. The toroidal trap is an important model potential because of its simplicity and richness in physics. In particular, the spinor Bose-Einstein condensate under the toroidal trap has brought an ideal platform for studying fascinating properties of a superfluid, such as persistent flow and symmetry-breaking localization. For the case of the spin-orbit-coupled Bose-Einstein condensate, the previous studies of the toroidal trap mainly focused on the two-component or antiferromagnetic case. However, in the presence of a toroidal trap, there remains an open question whether the combined effects of the spin-orbit coupling and rotation can produce previously unknown types of topological excitations in the ferromagnetic Bose-Einstein condensate. In this work, by using quasi two-dimensional Gross-Pitaevskii equations, we study the ground state structure of spin-orbit coupled rotating ferromagnetic Bose-Einstein condensate in the toroidal trap. We concentrate on the effects of the spin-orbit coupling and the rotation on the ground states. The numerical results show that in the presence of a toroidal trap, the ground state structure is displayed as half-skyrmion chain with circular distribution. Adjusting the strength of spin-orbit coupling not only changes the number of half-skyrmion in the system, but also controls the symmetry of half-skyrmion with circular distribution. As the rotation frequency increases, the system undergoes the transitions from the plane wave to the half-skyrmion chain with circular distribution, and eventually developing the half-skyrmion phase of triangular lattice. Next, we examine the effect of spin-independent interaction on spin-orbit coupled rotating spinor Bose-Einstein condensate. As the spin-independent interaction increases, the topological defects in the condensate increase due to the variation of the local magnetic order. We also discuss the influence of well shape on the ground state structure. These topological structures can be detected via the time-of-flight absorption imaging technique. The spin-orbit coupled spinor Bose-Einstein condensate in the toroidal trap is an important quantum platform, which not only opens up a new avenue for exploring the exotic topological structures, but also is crucial for realizing the transitions among different ground states. This work paves the way for futureexploring the topological defects and the corresponding dynamical stability in quantum system subjected to the toroidal trap.
Attractive Bose-Einstein condensates of dilute alkali gases are unstable against collapse in two- and three-dimensional free space. Nevertheless, we demonstrate that the spin-orbit coupling of the two-component condensates counteracts the tendency of collapse and makes the system preferable to an extended spatial distribution in the three-dimensional case. Furthermore, stable topological objects can be formed in the condensates, which are shown to be the lowest energy states. Two configurations of the density profiles, called three-dimensional skyrmion and three-dimensional dimeron, respectively, are identified depending on the strength of the spin-orbit coupling.