Fig. 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. 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

Fig. 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. 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

Fig. 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. 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

Fig. 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 和 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 . 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 和 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

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

Fig. 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 and , respectively. 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 和 的振幅

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

Fig. 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 in S² (inset). 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 的关系

Fig. 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]. 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

Fig. 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 in (a)−(d) are 0, 0.2, 0.8, 2, respectively^[107]. 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 值分别为0, 0.2, 0.8, 2

Fig. 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]. 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

Fig. 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 ; The rightmost three columns show images taken along the vertical axis^[80]. 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 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

Fig. 13. The effect of rotation frequency for spinor BEC of ²³Na with , , κ_x = κ_y = κ_z = 1, , 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]. 旋转频率对²³Na旋量BEC的影响^[118], 其中 , , κ_x = κ_y = κ_z = 1, , 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

Fig. 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]. 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

Table 1. Topological defect structures described by homotopy groups.