Fig. 1. (a) For any fixed momentum k, the cross section can be viewed as a circle where the azimuthal angle represents the time t. After gluing and (saffron circles), the topology of the momentum-time manifold becomes ; (b) if there are two fixed points and , the corresponding circle contracts to a point, then the momentum-time manifold reduces to a series of spheres ^[21]. (a) 每个截面对应固定动量, 截面内的极角对应于时间. 橘黄色的环代表 和 , 它们粘合起来组成了 ; (b) 如果动量空间中存在一些不动点 , , 截面的时间可连续收缩为一个点, 动量时间流形约化成一系列球面^[21]

Fig. 2. (a) In SSH model, the initial state of is topologically trivial, evolution of entanglement spectrum for different post-quenched are shown with different colors. If and only if the post-quenched Hamiltonian is topologically nontrivial, the entanglement spectrum can cross at ; (b) in Extended SSH model, the third-nearest-neighbor hopping carries a phase factor, and the Hamiltonian belongs to class AIII. The blue curve shows the dynamics of entanglement spectrum evolved by flattened Hamiltonian, and the red curve shows the dynamics evolved by entanglement spectrum of real Hamiltonian. It can be seen that the band dispersion opens the gap of entanglement spectrum. (a) SSH模型, 初态 为拓扑平庸的, 末态 取不同的值. 仅当末态为拓扑非平庸时, 纠缠谱在 处有交叉; (b)扩展的SSH模型, 次次次近邻跃迁具有相位, 系统属于AIII类. 蓝色的线代表用平带化的哈密顿量进行动力学演化, 红色的线代表由真实末态哈密顿量进行演化. 可以看出能带的色散打开了纠缠谱的能隙

Fig. 3. The scheme of experiment control sequence. The initial state is prepared at the state-initialization period by control quantity for a fixed momentum k. Then for a quantum quench, by controlling and , we adjust the direction of the rotation axis. (b), (c) the evolution of Bloch vectors for different momenta. The red points and yellow rings are experimental and numerical datas. (b) pre-quenched parameter , post-quench parameter . (c) pre-quenched parameter , post-quench parameter . (a)实验过程序列示意图. 对每一个动量k, 初始时刻通过脉冲 制备初态, 而后通过改变外加的脉冲的 和 实现淬火动力学. (b), (c) 不同动量k对应的Bloch矢量的演化. 红色的星为实验的数据, 黄色的环为数值计算的数据. (b)初态 , 末态 . (c) 初态 , 末态 ^[22]

Table 1. Topological classification of parent Hamiltonian. TRS, PHS and CS represent the time reversal symmetry, particle hole symmetry and chorial symmetry, respectively. The definition of s, t, d, and additional symmetry P can be found in Ref.[37]. Original class represents the topological classification without additional symmetry. is the K group. Dynamical realization means the topological classes which can be realized in quench dynamics. Stable against dispersion means entanglement spectrum crossing which is stable against band dispersion. 母哈密顿量 的拓扑分类. TRS、PHS、CS分别是时间反演对称性, 粒子空穴对称性和手征对称性. s, t, d, , 额外对称性P参见文献[37]; original class是没有额外对称性时系统的拓扑分类. 是系统的K群. Dynamical realization表示在淬火动力学中存在的拓扑分类. Stable against dispersion指能带存在色散时纠缠谱交叉能够稳定存在的拓扑分类