1.Department of Physics, Taiyuan Normal University, Jinzhong 030619, China 2.Basic Courses, Shanxi Institute of Energy, Jinzhong 030600, China 3.Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Abstract: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. Keywords:Bose-Einstein condensate/ toroidal trap/ spin-orbit coupling/ rotation
3.研究结果与讨论首先固定环形外势的势阱高度、宽度参数以及旋转频率, 讨论自旋-轨道耦合效应对体系基态的影响. 图1显示不同自旋-轨道耦合强度下基态粒子数密度和相位分布. 第1, 2, 3列表示自旋 ${m_F} = 1$, ${m_F} = 0$和${m_F} = - 1$分量的粒子数密度分布, 第4, 5, 6列分别对应不同自旋组分的相位分布. 从相位图中可以看到相位值大小的变化, 从蓝色逐渐变化到红色, 描述相位值从$ - {\text{π}} $到${\text{π}} $的增大. 当考虑弱自旋-轨道耦合$\kappa = 0.4$, 对于铁磁BEC系统, 自旋-轨道耦合与自旋交换相互作用之间的双重效应导致体系呈现平面波相, 平面波相对应的相位变化趋势如图1(a)相位图中逆时针方向. 此外由于有旋转势的存在, 凝聚体各个自旋组分在平面波的背景下都会出现涡旋, 如图1(a)前三列所示. ${m_F} = 1$和${m_F} = - 1$分量表现为非轴对称涡旋, ${m_F} = 0$分量表现为轴对称涡旋. 如果将每个分量相同位置处的一个涡旋看作一个元胞组合, 这个元胞组合在自旋纹理中实际上都对应一个half-skyrmion, 因此我们也称此时的基态为环形排列的对称half-skyrmion, 下文将详细讨论. 进一步增大自旋-轨道耦合强度$\kappa = 0.8$, 如图1(b)所示, 体系中half-skyrmion数量增多并呈现八角对称排列, 这一结果不同于在简谐外势中的情况. 之前的研究证实简谐外势中增加自旋-轨道耦合强度, 体系内的half-skyrmion分布对称性几乎没有变化[40]. 我们的结果证实在环形势阱中调节自旋-轨道耦合强度能够改变体系中half-skyrmion对称性. 继续将自旋-轨道耦合增强到$\kappa = 1.2$, 发现凝聚体中half-skyrmion数量进一步变多, 此时呈现十角对称排列分布, 如图1(c)所示. 最后在强自旋-轨道耦合$\kappa = 3$作用下, 凝聚体中half-skyrmion数量依然保持增多, 而此时half-skyrmion呈现径向多层的对称排列, 如图1(d)所示. 在环形势阱中, 通过改变自旋-轨道耦合强度, 可以调控体系中half-skyrmion数量以及对称性, 实现了体系中half-skyrmion分布模式从环形单层式对称排列到环形多层式规则排列的转变. 这一结果从物理角度不难理解, 一方面随着原子自旋和原子质心运动的耦合增强, 体系内自旋结构将发生频繁翻转, 导致体系内half-skyrmion数量逐渐增多; 另一方面, 自旋-轨道耦合和环形势阱的共同作用会改变体系内half-skyrmion分布的对称性. 图 1 环形势阱中不同自旋-轨道耦合强度下铁磁87Rb BEC基态粒子数密度分布(第1, 2, 3列)和相位分布(第4, 5, 6列). 改变自旋-轨道耦合强度, 可以调控体系中half-skyrmion数量以及环形排列的对称性 (a) $\kappa = 0.4$; (b) $\kappa = 0.8$; (c) $\kappa = 1.2$; (d) $\kappa = 3$. 该图其余模拟参数选为${\lambda _0} = 3200$, ${\lambda _2} = - 32$, $\varOmega = 0.2$, ${V_0} = 300$, $\sigma = 2$和ω = 2π × 250 Hz Figure1. Ground state of the rotating ferromagnetic BEC of 87Rb for the different spin-orbit coupling strengths under the toroidal trap. The first, second and third columns show the particle number densities. The fourth, fifth and sixth columns show phase distributions. Changing the strength of spin-orbit coupling can control the number of half-skyrmion in the system and the symmetry of half-skyrmion with circular distribution. The parameters are set as follows: (a) $\kappa = 0.4$; (b) $\kappa = 0.8$; (c) $\kappa = 1.2$; (d) $\kappa = 3$. And the rest of parameters are ${\lambda _0} = 3200$, ${\lambda _2} = - 32$, $\varOmega = 0.2$, ${V_0} = 300$, $\sigma = 2$ and ω = 2π × 250 Hz.
进一步, 固定自旋-轨道耦合强度大小, 研究不同旋转频率对系统基态性质的影响. 当旋转频率很小时, 体系没有出现任何拓扑缺陷, 基态呈现平面波相[35], 如图2(a)所示. 尽管系统此时考虑了旋转效应, 但旋转频率小于能够产生拓扑缺陷的临界值, 此外旋转势的作用也小于自旋-轨道耦合对体系的作用, 所以仅有平面波相出现. 继续增大旋转频率, 基态结构呈现六角对称排列的half-skyrmion, 如图2(b)所示. 较大的旋转频率会导致half-skyrmion数目增多, 此外这些拓扑缺陷呈现双层径向排布, 外层的缺陷数目多于内层, 如图2(c)所示. 当旋转频率很大时, 如图2(d)所示, 越来越多的half-skyrmion出现在体系中, 同时它们形成三角格子排列方式, 这是因为拓扑缺陷以三角格子排布时对应体系能量最低态, 体系最稳定. 图 2 不同旋转频率对基态的影响. 随着旋转频率增大, 体系从平面波相转化为环形对称排列的half-skyrmion链相. 第1, 2, 3列描述粒子数密度分布; 第4, 5, 6列表示对应的相位分布 (a) $\varOmega = 0.1$; (b) $\varOmega = 0.2$; (c) $\varOmega = 0.6$; (d) $\varOmega = 0.9$. 该图其余模拟参数选为 ${\lambda _0} = 3200$, ${\lambda _2} = - 32$, ${V_0} = 300$, $\sigma = 2$, $\kappa = 0.4$和ω = 2π × 250 Hz Figure2. Effects of the different rotation frequency on ground state. With the increase of rotation frequency, the system transforms from plane wave phase to half-skyrmion chain phase with circular symmetry arrangement. The first, second and third columns are the particle number densities. The fourth, the fifth and the sixth columns are the corresponding phase distributions. The parameters are set as follows: (a) $\varOmega = 0.1$; (b) $\varOmega = 0.2$; (c) $\varOmega = 0.6$; (d) $\varOmega = 0.9$. And the other parameters are ${\lambda _0} = 3200$, ${\lambda _2} = - 32$, ${V_0} = 300$, $\sigma = 2$, $\kappa = 0.4$ and ω = 2π × 250 Hz.
紧接着固定自旋-轨道耦合强度和旋转频率, 研究不同自旋非相关和自旋相关相互作用对基态的影响. 图3(a1)和图3(a2)分别显示了不同自旋非相关相互作用下基态粒子数密度和相位的分布, 当相互作用强度较小时, 发现half-skyrmion沿着环形凝聚体以单层排列, 如图3(a1)所示. 当相互作用增强, 凝聚体表面分布面积增大, 由于在旋转条件下, 凝聚体中出现的拓扑缺陷数目与凝聚体分布面积成线性关系, 另一方面相互作用增强也可以改变体系内的磁序分布, 因此将导致凝聚体中拓扑缺陷的增多. 从图3(a2)可以看出, 围绕环形凝聚体分布的half-skyrmion数目增多, 并且以环形双层排列. 图3(b1)和图3(b2)分别显示不同自旋相关相互作用下基态粒子数密度和相位分布. 对比发现, 调节不同大小的自旋相关相互作用对half-skyrmion的数目和分布模式影响微弱, 仅仅使half-skyrmion的环形空间排布变得更规则. 图 3 不同自旋非相关作用和自旋相关作用对基态的影响. 随着自旋非相关作用强度增加, half-skyrmion分布从环形单层排列转化为环形双层排列. 不同自旋相关相互作用对half-skyrmion的数目影响微弱, 仅仅使half-skyrmion环形排列变得更规则 (a1) ${\lambda _0} = $ 1200, ${\lambda _2} = - 32$; (a2) ${\lambda _0} = 4600$, ${\lambda _2} = - 32$; (b1) ${\lambda _0} = 3200$, ${\lambda _2} = - 12$; (b2) ${\lambda _0} = 3200$, ${\lambda _2} = - 9{\rm{6}}$. 该图其余模拟参数选为 $\varOmega = 0.6$, ${V_0} = 300$, $\sigma = 2$, $\kappa = 0.4$和ω = 2π × 250 Hz Figure3. Effects of the different spin-independent and spin-dependent interactions on ground state. Increasing the strength of spin-independent interaction can induce the transition of half-skyrmion distribution from a circular monolayer arrangement to a circular bilayer arrangement. The influence of different spin-dependent interaction on the number of half-skyrmion is weak, which only makes the ring arrangement of half-skyrmion more regular. The parameters are set as follows: (a1) ${\lambda _0} = 1200$, ${\lambda _2} = - 32$; (a2) ${\lambda _0} = $ 4600, ${\lambda _2} = - 32$; (b1) ${\lambda _0} = 3200$, ${\lambda _2} = - 12$; (b2) ${\lambda _0} = 3200$, ${\lambda _2} = - 9{\rm{6}}$. And the other parameters are $\varOmega = 0.6$, ${V_0} = 300$, $\sigma = 2$, $\kappa = 0.4$ and ω = 2π × 250 Hz.
接下来分析不同势阱宽度和势阱高度对基态的影响. 当势阱宽度很小时, 系统内凝聚体靠近体系中心区域分布, 其中的half-skyrmion沿着中心径向排列, 如图4(a1)所示. 对于势阱宽度较大的情况, 如图4(a2)所示. 对应环形凝聚体中half-skyrmion形成环形单层排布模式, 但它们的数目并没有发生明显变化, 这是因为调节势阱宽度仅仅改变凝聚体形状, 而其中的拓扑缺陷只发生空间排布上的改变. 图4(b1)和图4(b2)分别描述不同势阱高度下基态粒子数密度和相位分布. 对比发现势阱高度的变化几乎不会导致half-skyrmion数目的改变. 图 4 不同势阱宽度和势阱高度对基态的影响. 改变势阱宽度和高度, 可以调控half-skyrmion链的环形分布. 基态粒子数密度分布如1, 2, 3列所示; 对应相位分布如4, 5, 6列所示 (a1) ${V_0} = 300$, $\sigma = 0.8$; (a2) ${V_0} = 300$, $\sigma = 4$; (b1) ${V_0} = 60$, $\sigma = 2$; (b2) ${V_0} = {\rm{6}}00$, $\sigma = 2$. 该图其余模拟参数选为${\lambda _0} = 3200$, ${\lambda _2} = - 32$, $\varOmega = 0.6$, $\kappa = 0.4$和ω = 2π × 250 Hz Figure4. Effects of the width and the central height of the toroidal potential on ground state. The ring distribution of half-skyrmion chain can be controlled by changing the width and height of potential well. The particle number densities of ground state are shown in the first, second and third columns. The corresponding phase distributions are shown in the fourth, fifth and sixth columns. The parameters are set as follows: (a1) ${V_0} = 300$, $\sigma = 0.8$; (a2) ${V_0} = 300$, $\sigma = 4$; (b1) ${V_0} = 60$, $\sigma = 2$; (b2) ${V_0} = {\rm{6}}00$, $\sigma = 2$. And the other parameters are ${\lambda _0} = 3200$, ${\lambda _2} = - 32$, $\varOmega = 0.6$, $\kappa = 0.4$ and ω = 2π × 250 Hz.