The research progress of topological properties in spinor Bose-Einstein condensates

Wang Li; Liu Jing-Si; Li Ji; Zhou Xiao-Lin; Chen Xiang-Rong; Liu Chao-Fei; Liu Wu-Ming

doi:10.7498/aps.69.20191648

Abstract

Most of the atoms that realize Bose-Einstein condensation have internal spin degree of freedom. In the optical potential trap, the internal spin of the atom is thawed, and the atom can be condensed into each hyperfine quantum state to form the spinor Bose-Einstein condensate. Flexible spin degrees of freedom become dynamic variables related to the system, which can make the system appear novel topological quantum states, such as spin domain wall, vortex, magnetic monopole, skymion, and so on. In this paper, the experimental and theoretical study of spinor Bose-Einstein condensation, the types of topological defects in spinor Bose-Einstein condensate, and the research progress of topological defects in spinor two-component and three-component Bose-Einstein condensate are reviewed.

Keywords:

Authors and contacts

1.
College of Physics, Sichuan University, Chengdu 610065, China
2.
Beijing Jingshan School Chaoyang Branch School, Beijing 100012, China
3.
College of Physics, Taiyuan Normal University, Jinzhong, 030619, China
4.
School of Physics and Electronic engineering, Sichuan Normal University, Chengdu 610101, China
5.
School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China
6.
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
7.
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China

Corresponding author: Liu Chao-Fei, liuchaofei0809@163.com ; Liu Wu-Ming, wmliu@iphy.ac.cn

Funds: Project supported by the NKRDP, China (Grant No. 2016YFA0301500) and the National Natural Science Foundation of China (Grant Nos. 11434015, 61835013, 11875149, 61565007)

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Cited By

图 1 光势阱中F = 1 ²³Na凝聚体的超精细态^[16]. (a) 250 ms时光势阱中钠原子的吸收图像; (b) 340 ms时光势阱中钠原子的吸收图像

Figure 1. Optical trapping of ²³Na condensates in all F = 1 hyperfine states: shown are absorption images after (a) 250 ms and (b) 340 ms of optical confinement.

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图 2 铷原子云在Stern-Gerlach梯度磁场中自由膨胀10 ms后的吸收图像^[17]. 从下到上分别是F = 1, m_F = (–1, 0, 1)凝聚体的三个分量

Figure 2. Absorptive image of Rb atomic cloud after 10 ms free expansion in a Stern-Gerlach magnetic field gradient. Three distinct components are observed corresponding to F = 1, m_F = (–1, 0, 1) spin projections from bottom to top, respectively.

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图 3 赝自旋密度S_z, S_x, S_y的空间分布^[73]　(a)−(c)表示旋转角频率为0; (d)自旋纹理投影到x-y平面内的矢量表示

Figure 3. The pseudospin density distribution for (a) S_z, (b) S_x and (c) S_y for Ω = 0; (d) the vectorial representation of the spin texture projected onto the x-y plane.

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图 4 自旋1 BEC中半量子涡旋的近似解和相应的奇异自旋纹理^[77]　(a)和(b)对应$\left| {F = 1, {{{m}}_{\rm F}} = 0} \right\rangle $和$\left| {F = 1, {{{m}}_{\rm F}} = - 1} \right\rangle $分量的密度; (c)和(d)是对应的相; (e)为半量子涡旋的分布; (f)|S|自旋密度; (g) |S|自旋密度分布; (h)自旋纹理; (i)拓扑荷密度$q\left( {x, y} \right)$

Figure 4. Approximate half-quantum vortex solution in the spin-1 BEC and the corresponding singular spin texture: (a) and (b) are the densities of the $\left| {F = 1, {m_{\rm F}} = 0} \right\rangle $和$\left| {F = 1, {m_F} = - 1} \right\rangle $ components, respectively; (c) and (d) are the corresponding phases; (e) shows the profile of the half-quantum vortex; (f) spin density|S|; (g) the profile of the spin density|S|; (h) spin texture; (i) topological charge density $q\left( {x, y} \right)$.

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图 5 两种常见的二维skyrmions的矢量场构型^[79]　(a) 豪猪型skyrmion; (b) 螺旋型skyrmion

Figure 5. Two common vector field configurations of two-dimensional skyrmions: (a) The hedgehog type skyrmion; (b) the spiral type skyrmion.

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图 6 稳定的三维skyrmions在x-y和z-x平面的空间分布^[83]　(a)中的箭头和颜色分别表示贋自旋方向和OP的U(1)相分布. 彩图(b)和(c)分别表示$\left| {{\varPsi _ \uparrow }\left( {\rm{r}} \right)} \right|$和$\left| {{\Psi _ \downarrow }\left( {\rm{r}} \right)} \right|$的振幅

Figure 6. The spatial profile of the stable 3D skyrmions in the x-y and z-x planes: The arrows and their colors in (a) indicate the pseudospin direction and the U(1) phase of the OP, respectively; the color maps of (b) and (c) give the amplitudes $\left| {{\varPsi _ \uparrow }\left( {\rm{r}} \right)} \right|$ and $\left| {{\varPsi _ \downarrow }\left( {\rm{r}} \right)} \right|$, respectively.

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图 7 四极场作用下球形光势阱中扭结产生的动力学过程^[85]. 上一行表示${{\hat d}} = {\left( {0, 0, - 1} \right)^{\rm{T}}}$和${{\hat d}}$ = (1, 0, 0)^T的图像快照, 下一行表示x-y平面上m = –1分量的密度截面

Figure 7. Dynamics of the creation of knots in a spherical optical trap under a quadrupole magnetic field. Snapshots of the preimages of ${{\hat d}}$ = (0, 0, –1)^T and ${{\hat d}}$ = (1, 0, 0)^T(top), and the cross sections of the density for the m = –1 components on the x–y plane (bottom).

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图 8 扭结孤子的结构及其产生方法^[89]　(a)和(b)为扭结形成之前和形成过程中磁感应线的示意图, 绿色椭圆为对应的凝聚体; (c)和(d)显示扭结形成时, 最初的z方向的向列相矢量(黑色箭头)沿着局部磁场(青色线)的方向进动, 以实现最终的结构(彩色箭头). 灰色虚线表示d_z = 0, 白线表示孤子核(d_z = –1), 深灰色线表示体积V (d_z = 1)的边界; (e)表示实空间中扭结孤子的构型及其与S²中向列矢量${{\hat d}}$的关系

Figure 8. Structure of the knot soliton and the method of its creation: Schematic magnetic field lines before (a) and during (b) the knot formation, with respect to the condensate (green ellipse); (c), (d) as the knot is tied, the initially z-pointing nematic vector (black arrows) precesses about the direction of the local magnetic field (cyan lines) to achieve the final configuration (coloured arrows); the dashed grey line shows where d_z = 0, the white line indicates the soliton core (d_z = –1), and the dark grey line defines the boundary of the volume V (d_z= 1); (e) the knot soliton configuration in real space and its relation to the nematic vector ${{\hat d}}$ in S² (inset).

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图 9 Skyrmions的类型(λ = 0.5)^[99]　(a)−(h)表示自旋矢量的模式: (a)径向-向外skyrmion, (b)径向-向内skyrmion, (c)环形skyrmion, (d)双曲skyrmion, (e)双曲-径向向外skyrmion, (f)双曲-径向向内skyrmion, (g)环形-双曲skyrmion-I, (h)环形-双曲skyrmion-II

Figure 9. Configuration of the skyrmion where λ = 0.5: The (a)−(h) figures indicate the mode of the spin vectors: (a) radial-out skyrmion, (b) radial-in skyrmion, (c) circular skyrmion, (d) hyperbolic skyrmion, (e) hyperbolic-radial(out) skyrmion, (f) hyperbolic-radial (in) skyrmion, (g) circular-hyperbolic skyrmion-I, and (h) circular-hyperbolic skyrmion-II^[99].

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图 10 不同自旋-轨道耦合强度下梯度磁场中两分量⁸⁷RbBEC基态粒子数密度分布(第1、2列)和相位分布(第3、4列)^[107]　(a)−(d)的${\tilde {\rm{\kappa}} }$值分别为0, 0.2, 0.8, 2

Figure 10. Particle number densities (the first and second columns) and phase distributions (the third and fourth columns) of ground state of the two-component BEC of ⁸⁷Rb for the different spin-orbit coupling strengths: the parameters of ${\tilde {\rm{\kappa}} }$ in (a)−(d) are 0, 0.2, 0.8, 2, respectively^[107].

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图 11 涡旋的动力学形成^[110]. 涡旋形成于凝聚体的所有分量中, 在ψ_–1分量中占99%以上, 在ψ₀分量中动态涡旋和拓扑涡旋共存

Figure 11. Dynamical formation of vortices: vortices are formed in all components, more than 99% of total population is in ψ_–₁ component. In the ψ₀ component, dynamical and topological vortices coexist^[110].

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图 12 狄拉克磁单极子的实验产生^[80]　(a)−(f)每一行都包含单个凝聚体的图像. 最左边的列显示了三种自旋状态$\left\{ {\left| 1 \right\rangle, \left| 0 \right\rangle, \left| { - 1} \right\rangle } \right\}$沿水平轴的柱状密度彩色图像; 最右边三列显示沿纵轴拍摄的图像

Figure 12. Experimental creation of Dirac monopoles. Each row (a)−(f) contains images of an individual condensate. The leftmost column shows colour composite images of the column densities taken along the horizontal axis for the three spin states $\left\{ {\left| 1 \right\rangle, \left| 0 \right\rangle, \left| { - 1} \right\rangle } \right\}$; The rightmost three columns show images taken along the vertical axis^[80].

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图 13 旋转频率对²³Na旋量BEC的影响^[118], 其中${\mu _{j, 0}}\left( {j = 0, \pm 1} \right) = 3.6\;\hbar {\rm{\omega }}$, ${\text{μ}} = 25\;\hbar {\rm{\omega }}$, κ_x = κ_y = κ_z = 1, ${a_0} = 50\;{a_{\rm{B}}}$, and a₂ = 55 a_B　(a) Ω = 0; (b) Ω = 0.2 ω; (c) Ω = 0.5 ω. 第四列显示了相应的自旋纹理和涡旋的位置

Figure 13. The effect of rotation frequency for spinor BEC of ²³Na with ${\mu _{j, 0}}\left( {j = 0, \pm 1} \right) = 3.6\;\hbar {\rm{\omega }}$, ${\rm{\mu }} = 25\;\hbar {\rm{\omega }}$, κ_x = κ_y = κ_z = 1, ${a_0} = 50\;{a_{\rm{B}}}$, and a₂ = 55 a_B: (a) Ω = 0; (b) Ω = 0.2 ω; (c) Ω = 0.5 ω. The fourth column shows the corresponding spin textures and the positions of the vortices^[118].

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图 14 具有Mermin-Ho涡旋的磁单极子^[125]　(a)等值面的粒子数密度; (b)粒子数密度等深线段(y ≤ 0), 节点线(Dirac线)的位置用红色箭头突出显示; (c) z=0平面上的位相分布. 单涡旋(m_F = 0)和双涡旋(m_F = –1)具有相同的环流, 由红圈突出显示

Figure 14. The monopoles with the Mermin-Ho vortex: (a) Isosurface of particle densities; (b) segments of isosurface of particle densities (y ≤ 0). the position of the nodal line (Dirac string) is highlighted by the red arrow; (c) phase distributions in the z = 0 planes. the single vortex (m_F = 0) and double vortex (m_F = –1) have the same circulations, as highlighted by the red circles^[125].

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表 1 同伦群描述的拓扑缺陷结构

Table 1. Topological defect structures described by homotopy groups.

π_n 缺陷孤子

π₀ 磁畴壁暗孤子
π₁ 涡旋非奇异磁畴壁
π₂ 磁单极二维skyrmions
π₃ skyrmions, 扭结
π₄ 瞬子

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