Fig. 1. (a) The NIST scheme to realize 1D spin-orbit coupling; (b) the tripod scheme to realize 2D synthetic gauge field; (c) the Raman optical lattice scheme to realize 2D spin-orbit coupling(a) 一维自旋轨道耦合的实验实现方案； (b) 利用tripod方案实现二维人造规范 势的方案；(c) 利用Raman晶格实现二维人造自旋轨道耦合的实验方案

Fig. 2. (a) A quasi-one-dimensional Fermi gas, which is coupled to a two-mode optical cavity, is under both transverse (along ) and longitudinal (along ) pumping; (b) the level scheme of atom^[76](a)准一维费米气体与双模光腔耦合的示意图. 在腔轴向(沿 轴)和径向(沿 轴)均有泵浦光; (b)原子能级和光耦合的示意图^[76]

Fig. 3. TSR state in a quasi-one-dimensional lattice with open boundary conditions: (a) The cavity field varies with across the TSR transition; (b) on the central six sites. The dotted curve corresponds to the before the TSR phase transition, where . The solid and dash-dotted curves to the after the TSR phase transition, where . The transition point is around . Because of the spontaneous symmetry breaking, the cavity field of the TSR phase acquires a positive (negative) real part, corresponding to solid (dash-dotted) curve; (c) when the system crosses the phase boundary, a pair of edge states emerge in the superradiance-induced bulk gap. (d) the wave functions of the edge states in (c) with . In our calculation, we consider a half-filled lattice of 80 sites, with the parameters , , , , , and . For atoms, these parameters can be satisfied by choosing MHz, MHz, MHz, GHz, and nK^[76]开边界条件下准一维晶格中TSR态的一些特征　(a) 腔场强度 随有效泵浦 的变化; (b) 序参量 在TSR相变前(点线表示, )和相变后(实线和点划线表示, )在中心六个格点中的变化情况. 临界点位于 处. 在TSR相中, 由于自发对称性破缺, 腔场 可以取正值或负值, 对应序参量由实线或点划线表示; (c) 当系统穿过相边界进入TSR态后, 系统会由于超辐射相变打开一个体能隙, 同时出现一对零能的边缘态 (d) 当 时, 图(c)中的边缘态所对应的实空间波函数. 本图中考虑一个拥有80个格点的半满晶格体系, 体系参数选取为: , , , , , . 对于 原子, 通过选取 MHz, MHz, MHz, GHz和 nK可以满足上述参数条件^[76]

Fig. 4. The phase diagram of steady-state with . The solid curve corresponds to the TSR phase boundary, and the topological phase boundary between the TSR and the trivial SR states corresponds to dotted curve. The thin dashed curve at is the boundary between the M and the I states, and the dash-dotted curve is the conventional SR phase boundary. At the tetracritical point (dot) with and , the various boundaries merge. Other parameters are the same as those used in Fig. 2. Inset: change of bulk gap before (solid), after (dashed), and right (dotted) at the phase boundaries labeled by arrows^[76]有限温度 时系统的稳态相图. 图中实线为TSR相边界, 点线为TSR态和普通SR态间的拓扑相边界. 在 处的细虚线为金属态(M)和绝缘态(I)间的边界. 点划线为普通SR相与绝缘相的边界. 不同的相边界汇聚于 , 处(如图中四相点所示). 图中其他参数与图(2)一致. 内嵌图展示了与大图中箭头对应的相变前后体能隙的变化. 图中实线为相变前的能带, 虚线为相变后的能带, 点线为相边界上的情况^[76]

Fig. 5. A quasi-1D atomic gas under Raman lasers; (b) Raman level schemes in the clock-states manifold. The green curve corresponds to the interorbital spin-exchange interaction. By using spin-dependent laser shifts, the four nuclear spin states from and manifolds can be separated from the other nuclear spins^[128](a) 处在拉曼光中的准一维超冷原子气体; (b) 通过拉曼过程耦合的原子能级示意图. 图中绿色曲线指示了不同轨道态之间的自旋交换相互作用. 通过利用与自旋相关的激光频移, 可以将图中的四个核自旋态与其他核自旋态分离开来进行操控^[128]

Fig. 6. (a) The entanglement spectrum ; (b) in a chain with lattice sites and under open boundary conditions, the second-order Rényi entropy and the von Neumann entropy vary with ; (c) in a chain with lattice sites and under the periodic boundary condition, the bulk energy gap varies with . Inset: The bulk gap as a function of at the critical point, and the red solid line is a linear fit with in the large-N limit. (d) in a chain with lattice sites and at the critical point , the von Neumann entropy of a subchain of length varied with . The solid line is the linear fit with and . The central charge is 6 times the slope of the linear fit. All calculations are performed at half filling and with the fixed parameters , , and ^[128](a) 本征值最小的四个纠缠谱 随自旋交换相互作用的变化; (b) 开边界条件下, 在格点数 的光晶格链中, 二阶Rényi熵 和von Neumann熵 随 的变化情况; (c) 周期边界条件下, 在格点数 的光晶格链中, 体能隙 的变化情况. 内嵌图为体能隙在临界点处随 的变化情况. 图中线性拟合的红色实线给出大 极限下 ; (d) 临界点 处, 长度为 且格点数 的子链中von Neumann熵随 的变化. 通过线性拟合 , 可以得到中心荷(central charge) . 图中所有计算均在半满状态下进行, 且固定参数 , , ^[128]

Fig. 7. The phase diagram of a lattice with sites at half filling with ^[128]格点数 且处于半满状态下的体系相图, 其中 ^[128]

Fig. 8. Scheme for creating SU(3) spin-orbit coupling in spinor BECs: (a) Laser geometry. The cloud of atoms is illuminated by three laser beams with different frequencies and polarizations, intersecting at an angle of (b) Each of the three Raman lasers dresses one hyperfine Zeeman level from eman level from , and of the ⁸⁷Rb . , and are the detuning in the Raman transitions^[162]旋量BEC中产生SU(3)自旋轨道耦合的原理图　(a) 激光作用. 三束有不同频率和偏振的激光, 以 的角度作用于原子气体; (b) 能级图. 三个拉曼过程分别缀饰⁸⁷Rb中饰⁸⁷Rb中 基态的超精细塞曼能级 , , 和 . , 和 与拉曼过程的失谐对应^[162]

Fig. 9. Two distinct phases of SU(3) spin-orbit-coupled BECS: (a)−(d) The topologically nontrivial lattice phase with antiferromagnetic spin interaction ( ). (a) The heights and colors correspond to the density and phase of respectively, (b) the positions of vortices (white circles) and antivortices (black circles) in the phase within one unit cell, (c) the corresponding momentum distributions, (d) the structural schematic drawing of the phase separation; (e), (f) the threefold-degenerate magnetized phase for ferromagnetic spin interaction ( ). (e) the density and phase distributions of , (f) the corresponding momentum distributions^[162]有SU(3)自旋轨道耦合的BEC中的两种不同相　(a)−(d) 存在反铁磁自旋相互作用时( )的拓扑非平庸晶格相. 图(a)中的高度和颜色分别代表 的密度和相位. 在图(b)中, 一个单胞中呈现出涡旋(白色圆圈)和反涡旋(黑色圆圈)的位置. 图(c)和图(d)分别展示了晶格相的动量分布和相分离结构的示意图; (e), (f) 存在铁磁自旋相互作用时( )的三重简并磁化相. 图(e)和图(f)分别展示了 在实空间和动量空间的分布^[162]

Fig. 10. (a) Energy comparison between the lattice and stripe phases. The solid (lattice state) and dashed (stripe state) lines correspond to the energy difference between the numerical simulation and the variational calculation; (b)−(d) the ground-state density, phase and momentum distributions of the stripe phase with the parameters and ^[162](a) 晶格相和条纹相的能量对比; (b)−(d) 参数 和 时, 条纹相基态的密度、相位和动量的分布^[162]

Fig. 11. Vortex arrangement among the three components in antiferromagnetic spinor BECs with SU(3) spin-orbit coupling. There are three types of vortices, including a polar-core vortex with winding combination (blue line) and two ferromagnetic-core vortices with winding number combinations (green line) and (red line)^[162]有SU(3)自旋轨道耦合的反铁磁旋量BEC中的涡旋结构. 图中描绘了三个自旋分量中的涡旋排列. 三种类型的涡旋包括: 一个不同自旋分量的环绕数组合为 的极化核心涡旋(蓝线所示), 两个环绕数组合分别为 (绿线所示)和 (红线所示)的铁磁核心涡旋^[162]

Fig. 12. The double-quantum spin vortex in antiferromagnetic spinor BECs with SU(3) spin-orbit coupling: (a) Spatial maps of the transverse magnetization. The colors correspond to the magnetization orientation; (b) longitudinal magnetization; (c) amplitude of the total magnetization . The big and small circles represents the two kinds of topological defects: double-quantum spin vortex and half-Skyrmion.^[162]有SU(3)自旋轨道耦合的反铁磁自旋BEC中的自旋双涡旋 (a)横向磁化的空间分布, 其中颜色表示磁化方向; (b), (c)分别描绘了纵向磁化和总磁化幅度 的分布. 图中分别用大圆圈和小圆圈标记了自旋双涡旋和half-Skyrmion这两种类型的拓扑缺陷^[162]

Fig. 13. Chiral supersolid induced by Rashba spin-orbit coupling and soft-core long-range interactions. The brightness and color represent the density and phase distributions respectively. The soft-core long-range interactions in (a) and (b) is , and in (c) and (d) is . The directions of the arrows in the color wheel indicate the elevation of the respective quantities. Other parameters are fixed at , and ^[200]由Rashba类型SOC和软核长程相互作用诱导产生的手性超固体. 图中的亮度和颜色分别表示密度和相位分布. 软核长程相互作用在图(a)和图(b)中为 , 在图(c)和图(d)中为 . 色标圆盘中的箭头方向表示相应物理量增加的方向. 其他固定参数分别为 , 和 . 这里使用的软核长程相互作用强度 在实验中可以实现^[200]

Fig. 14. Particle currents and longitudinal magnetizations of the spin induced by Rashba spin-orbit coupling ( (a), (b)) and Dresselhaus spin-orbit coupling ((c), (d) ). and are represented by the color map and black arrows, respectively. The colors ranging from blue to red represent the values from the minimum to the maximum. The parameters used here are same as those in Fig. 12^[200]由Rashba类型SOC((a), (b))和Dresselhaus类型SOC((c), (d))诱导产生的粒子流 和自旋的径向磁化 . 其中颜色从蓝到红代表 从小到大, 黑色箭头代表环流方向 . 本图中参数与图12相同^[200]

Fig. 15. (a) The phase diagram by varying the soft-core long-range interaction strengths and ; (b) the phase diagram by varying the Rashba spin-orbit-coupling strength and the soft-core long-range interaction strength . The spin-orbit-coupling strength in (a) is fixed at , and the soft-core long-range interaction strength in (b) is fixed at . Other parameters are taken as and ^[200](a) 通过改变软核长程相互作用强度 和 的体系相图; (b) 通过改变SOC强度 和软核长程相互作用强度 的体系相图. 图(a)中SOC强度固定为 , 图(b)中软核长程相互作用强度固定为 . 其他参数为 和 ^[200]