Observing topological charges and dynamical bulk-surface correspondence with ultracold atoms

TL;DR

This paper experimentally demonstrates the dynamical bulk-surface correspondence in a 2D quantum anomalous Hall system using ultracold atoms, linking topological charges and invariants to quench dynamics.

Contribution

First experimental observation of dynamical bulk-surface correspondence in a 2D topological system via ultracold atoms, measuring topological charges and invariants dynamically.

Findings

Topological charges measured dynamically match the Chern index.

Dynamical topological invariant equals the post-quench band Chern number.

Observation of nontrivial topology in the emergent dynamical field.

Abstract

In quenching a topological phase across phase transition, the dynamical bulk-surface correspondence emerges that the bulk topology of $d$-dimensional ($d$D) phase relates to the nontrivial pattern of quench dynamics emerging on $(d-1)$D subspace, called band inversion surfaces (BISs) in momentum space. Here we report the first experimental observation of the dynamical bulk-surface correspondence through measuring the topological charges in a 2D quantum anomalous Hall model realized in an optical Raman lattice. The system can be quenched with respect to every spin axis by suddenly varying the two-photon detuning or phases of the Raman couplings, in which the topological charges and BISs are measured dynamically by the time-averaged spin textures. We observe that the total charges in the region enclosed by BISs define a dynamical topological invariant, which equals the Chern index of the post-quench band. The topological charges relate to an emergent dynamical field which exhibits nontrivial topology on BIS, rendering the dynamical bulk-surface correspondence. This study opens a new avenue to explore topological phases dynamically.

Equations

$$\mathcal{H}(\mathbf{q})=$$

$$+(m_z-2t_0^x\cos q_x-2t_0^y\cos q_y){\sigma }_z,$$

$$S_{y,x}(\mathbf{q})=\frac{1}{{\mathcal{N}}_q}\mathrm{sign}[h_z(\mathbf{q})]{\overline{⟨{\sigma }_z(\mathbf{q})⟩}}_{y,x}.$$

$$g_{y,x}(\mathbf{q})\equiv \frac{1}{{\mathcal{N}}_{\mathbf{q}}}{\partial }_{q_{\perp }}{\overline{⟨{\sigma }_z(\mathbf{q})⟩}}_{y,x},$$

$${\mathbf{E}}_x$$

$${\mathbf{E}}_y$$

$$H=\left[\frac{{\mathbf{k}}^2}{2m}+V_{\mathrm{latt}}(x,y)\right]\otimes 1+{\Omega }_x(x,y){\sigma }_x+{\Omega }_y(x,y){\sigma }_y+m_z{\sigma }_z,$$

$$V_{\mathrm{latt}}(x,y)=\frac{1}{3}\left(\frac{t_{D_2}^2}{{\Delta }_{D_2}}+\frac{t_{D_1}^2}{{\Delta }_{D_1}}\right)(|{\mathbf{E}}_{xz}{|}^2+|{\mathbf{E}}_{xy}{|}^2+|{\mathbf{E}}_{yz}{|}^2+|{\mathbf{E}}_{yx}{|}^2).$$

$${\Omega }_x=\frac{1}{6}\left(\frac{t_{D_2}^2}{2{\Delta }_{D_2}}-\frac{t_{D_1}^2}{{\Delta }_{D_1}}\right)({\hat{e}}_z\cdot {\mathbf{E}}_{xz})({\hat{e}}_{-}\cdot {\mathbf{E}}_{yx}),{\Omega }_y=\frac{1}{6}\left(\frac{t_{D_2}^2}{2{\Delta }_{D_2}}-\frac{t_{D_1}^2}{{\Delta }_{D_1}}\right)({\hat{e}}_{+}\cdot {\mathbf{E}}_{xy})({\hat{e}}_z\cdot {\mathbf{E}}_{yz}),$$

$${\mathbf{E}}_x$$

$${\mathbf{E}}_y$$

$$V_{\mathrm{latt}}(x,y)=V_{0x}{\cos }^2(k_{0}x-{\alpha }_L/2)+V_{0y}{\cos }^2(k_{0}y-{\beta }_L/2),$$

$$V_{0x/0y}=\frac{t_{D_1}^2}{3}\left(\frac{1}{|{\Delta }_{D_1}|}-\frac{2}{|{\Delta }_{D_2}|}\right)(|E_{xy/yx}{|}^2-|E_{xz/yz}{|}^2),$$

$${\Omega }_x={\Omega }_{0x}\sin (k_{0}x-{\alpha }_L/2)\cos (k_{0}y-{\beta }_L/2),{\Omega }_y={\Omega }_{0y}\cos (k_{0}x-{\alpha }_L/2)\sin (k_{0}y-{\beta }_L/2),$$

$${\Omega }_{0x/0y}=\frac{t_{D_1}^2}{6\sqrt{2}}\left(\frac{1}{|{\Delta }_{D_1}|}+\frac{1}{|{\Delta }_{D_2}|}\right)|E_{xz/xy}||E_{yx/yz}|.$$

$${\mathcal{H}}^{(\mathrm{i})}=[m_z-2t_0(\cos q_{x}a+\cos q_{y}a)]{\sigma }_z+2t_{\mathrm{so}}\sin q_{x}a{\sigma }_y+2t_{\mathrm{so}}\sin q_{y}a{\sigma }_x$$

$$t_0$$

$$t_{\mathrm{so}}$$

$${\mathbf{E}}_x$$

$${\mathbf{E}}_y$$

$$V_{\mathrm{latt}}(x,y)=V_{0x}^{\prime }{\cos }^2(k_{0}x-{\alpha }_L/2)+V_{0y}{\cos }^2(k_{0}y-{\beta }_L/2),$$

$${\Omega }_x=0,{\Omega }_y={\Omega }_{0x}\cos (k_{0}x-{\alpha }_L/2)\cos (k_{0}y-{\beta }_L/2)+{\Omega }_{0y}\cos (k_{0}x-{\alpha }_L/2)\sin (k_{0}y-{\beta }_L/2).$$

$${\mathcal{H}}^{(\mathrm{ii})}=[m_z-2(t_0^x\cos q_{x}a+t_0^y\cos q_{y}a)]{\sigma }_z+2t_{\mathrm{so}}^x\sin q_{y}a{\sigma }_x+m_y{\sigma }_y,$$

$$m_y={\Omega }_{0x}\int d\mathbf{r}{\varphi }_s(\mathbf{r})\cos (k_{0}x)\cos (k_{0}y){\varphi }_s(\mathbf{r}).$$

$${\mathbf{E}}_x$$

$${\mathbf{E}}_y$$

$$V_{\mathrm{latt}}(x,y)=V_{0x}{\cos }^2(k_{0}x-{\alpha }_L/2)+V_{0y}^{\prime }{\cos }^2(k_{0}y-{\beta }_L/2),$$

$${\Omega }_x={\Omega }_{0x}\sin (k_{0}x-{\alpha }_L/2)\cos (k_{0}y-{\beta }_L/2)-{\Omega }_{0y}\cos (k_{0}x-{\alpha }_L/2)\cos (k_{0}y-{\beta }_L/2),{\Omega }_y=0.$$

$${\mathcal{H}}^{(\mathrm{iii})}=[m_z-2(t_0^x\cos q_{x}a+t_0^y\cos q_{y}a)]{\sigma }_z+2t_{\mathrm{so}}^y\sin q_{x}a{\sigma }_y+m_x{\sigma }_x,$$

$$m_x={\Omega }_{0y}\int d\mathbf{r}{\varphi }_s(\mathbf{r})\cos (k_{0}x)\cos (k_{0}y){\varphi }_s(\mathbf{r}).$$

$$P_z(\mathbf{q},t)=\sum _{i=1}^3{A}_i(\mathbf{q})\cos (2\pi f_i(\mathbf{q})t+{\varphi }_i)e^{-\frac{t}{{\tau }_1}}+B(\mathbf{q})e^{-\frac{t}{{\tau }_2}}+D(\mathbf{q}).$$

$${\overline{⟨{\sigma }_z(\mathbf{q})⟩}}_{x,y,z}=\frac{1}{T}\sum _{t=0}^T\left(\sum _{i=1}^3{A}_i(\mathbf{q})\cos (2\pi f_i(\mathbf{q})t+{\varphi }_i)\right)+B(\mathbf{q})+D(\mathbf{q}).$$

$${\overline{⟨{\sigma }_z(\mathbf{k})⟩}}_i$$

Full text

Introduction.— Topological quantum matter Hasan and Kane (2010); Qi and Zhang (2011) has attracted intense interest due to the discovery of new fundamental phases König et al. (2007); Chang et al. (2013); Lv et al. (2015) and broad potential applications Bergholtz and Liu (2013); Nayak et al. (2008). Recent experimental advances in cold atoms highlight the realizations of various topological models, such as the one-dimensional (1D) Su-Schrieffer-Heeger model Atala et al. (2013), 1D chiral topological phase Song et al. (2018), and 2D Chern insulator Aidelsburger et al. (2013); Miyake et al. (2013); Jotzu et al. (2014); Aidelsburger et al. (2015); Wu et al. (2016); Sun et al. (2018a). The studies commonly faces an important question: how to measure topological indices for cold atom systems? The 1D winding number can be detected by measuring Zak phase via Ramsey interferometry Atala et al. (2013). In 2D spin-orbit (SO) coupled Chern phase Wu et al. (2016); Sun et al. (2018a), the Chern number can be determined by measuring Bloch states at highly symmetric momenta Liu et al. (2013). These methods are however not generic or lack sufficient accuracy.

Recently, a research focus has been drawn to non-equilibrium dynamics in topological quantum phases  Song et al. (2018); Fläschner et al. (2018, 2016); Jurcevic et al. (2017); Hu et al. (2016); Wilson et al. (2016). Several theoretical works Wang et al. (2017); Zhang et al. (2018); Yu (2019); Zhang et al. (2019a, b) proposed dynamical characterizations of topological phases by quantum quenches, with some predictions having been studied in experiment Sun et al. (2018b); Tarnowski et al. (2019); Yi et al. (2019). In particular, a dynamical bulk-surface correspondence was proposed Zhang et al. (2018), showing that the bulk topology of a $d$D topological phase universally corresponds to the nontrivial pattern of quench dynamics emerging on the $(d-1)$D momentum subspace called band inversion surfaces (BISs), analogy to the well-known bulk-boundary correspondence in real space Hasan and Kane (2010); Qi and Zhang (2011). A recent experiment observed the ring pattern of BISs dynamically Sun et al. (2018b). However, the topological invariant of quench dynamics emerging on BISs was not observed, so the experimental verification of the essential dynamical bulk-surface correspondence is yet to be pursued.

In this letter, we report the experimental observation in ultracold atoms of the dynamical bulk-surface correspondence Zhang et al. (2018) following a new scheme proposed in Ref. Zhang et al. (2019a), and characterize the topological phases by dynamically detecting topological charges of monopoles in momentum space. The central idea of the new scheme is that through a sequence of quantum quenches along all spin axes in the topological system, the topology can be detected by measuring the quantum dynamics for only $z$-component spin polarization in each quench. We implement the study in a 2D quantum anomalous Hall (QAH) model in an optical Raman lattice, and quench the system along different spin axes by quickly varying the two-photon detuning or phases of Raman couplings based on the new scheme Zhang et al. (2019a). The complete information of bulk topology, including the topological charges and BISs, is extracted by measuring the $z$-component spin dynamics. We observe that the total charges in the region enclosed by BISs equal the Chern number of the post-quench phase, and are related to an emergent dynamical field which exhibits nontrivial topological pattern on the BIS, rendering the dynamical bulk-surface correspondence in momentum space.

Experimental setup.— Our experiment is based on a 2D QAH model Liu et al. (2014) with $C_{4}$ symmetry proposed in theory Wang et al. (2018) and realized for ultracold ${}^{87}\textrm{Rb}$ bosons trapped in a square optical Raman lattice Wu et al. (2016); Xu et al. (2018); Sun et al. (2018a). The laser beam ${\bm{E}}_{x}$ (${\bm{E}}_{y}$) with wavelength of $\lambda={787}nm$ is incident from $x$ ($y$) direction and reflected by mirror. The two beams pass through $\lambda/2$ wave plates and each splits into two orthogonally polarized components ${\bm{E}}_{x}=\bm{E}_{xz}+\bm{E}_{xy}$ and ${\bm{E}}_{y}=\bm{E}_{yz}+\bm{E}_{yx}$ [Fig. 1(a)], with ${\bm{E}}_{xy}=\hat{y}\left|E_{xy}\right|\cos(k_{0}x)$, ${\bm{E}}_{xz}=\hat{z}\left|E_{xz}\right|e^{i\varphi_{1}}\cos(k_{0}x-\varphi_{1})$, ${\bm{E}}_{yx}=\hat{x}\left|E_{yx}\right|\cos(k_{0}y)$, and ${\bm{E}}_{yz}=\hat{z}\left|E_{yz}\right|e^{i\varphi_{2}}\cos(k_{0}y-\varphi_{2})$, from which the scalar and vector potentials are generated and give the lattice and Raman potentials, respectively Sun et al. (2018a); Wang et al. (2018). Here $k_{0}=2\pi/\lambda$ and $\varphi_{1}$ ($\varphi_{2}$) is the relative phase between $\bm{E}_{xz}$ ($\bm{E}_{yz}$) and $\bm{E}_{xy}$ ($\bm{E}_{yx})$. As proposed in Ref. Zhang et al. (2019a), the key technique here is that we apply electro-optic modulators (EOMs) to manipulate the relative symmetries between lattice and Raman potentials by tuning $\varphi_{1,2}$. The spin-$1/2$ system, selected from the hyperfine states $\left|\uparrow\right\rangle=\left|F=1,m_{F}=-1\right\rangle$ and $\left|\downarrow\right\rangle=\left|1,0\right\rangle$, are coupled by the Raman transitions [see Fig. 1(b)]. In tight-binding regime the realized Bloch Hamiltonian ${\cal H}({\bm{q}})={\bm{h}}({\bm{q}})\cdot{\bm{\sigma}}$ reads (see more details in supplementary materials sup )

[TABLE]

where ${\bm{q}}=(q_{x},q_{y}),m_{i},\sigma_{i}$, and $t^{i}_{0}$ ($t_{\rm so}^{i}$) are respectively the Bloch momentum, Zeeman constants, Pauli matrices acting on spin space, and the spin-conserved (flip) hopping coefficients, with $i=x,y,z$. The $m_{z}$-term is related to the two-photon detuning by $m_{z}=\delta/2$. The Zeeman constants $m_{x,y}$ and spin-flip hopping coefficients $t_{\rm so}^{x,y}$ are induced by the Raman couplings and controlled by $\varphi_{1,2}$. In this experiment, the quenches along different spin axes will be performed by quickly modulating $m_{x,y,z}$ sup ; Zhang et al. (2019a).

The two Raman potentials are generated by two pairs of beams ($\bm{E}_{xz},\bm{E}_{yx}$) and ($\bm{E}_{yz},\bm{E}_{xy}$), respectively. For the setting with $(\varphi_{1},\varphi_{2})=(\pi/2,\pi/2)$, the former (or latter) potential $\propto\sin(k_{0}x)\cos(k_{0}y)$ (or $\sin(k_{0}y)\cos(k_{0}x)$) is symmetric with respect to every lattice site in $y$ (or $x$) direction, but antisymmetric in $x$ (or $y$) direction sup . This is the configuration of 2D optical Raman lattice to realize QAH model Wang et al. (2018); Wu et al. (2016); Sun et al. (2018a); Xu et al. (2018), in which the nearest-neighbor spin-flip hopping is induced by each Raman potential along the antisymmetric direction, while on-site spin-flip transition is forbidden. We then reach the Hamiltonian (S10) with $m_{x}=m_{y}=0$. Further, for the other setting with $(\varphi_{1},\varphi_{2})=(0,\pi/2)$ (or $(\pi/2,0)$), the former (or latter) Raman potential $\propto\cos(k_{0}x)\cos(k_{0}y)$, which is symmetric with respect to each lattice site in both $x$ and $y$ directions, and induces on-site spin-flip transition, but no nearest-neighbor spin-flip hopping. Then a nonzero $m_{y}$ (or $m_{x}$) is induced, while $t^{x}_{\rm so}\approx 0$ (or $t^{y}_{\rm so}\approx 0$) Zhang et al. (2019a); sup , and the Hamiltonian (S10) can describe a trivial system with large transverse Zeeman field.

Quenches and measurements.— We implement the quenches with respect to all spin quantization axes. For simplicity we consider that $t^{x}_{\rm so}=t^{y}_{\rm so}=t_{\rm so}$ and $t^{x}_{0}=t^{y}_{0}=t_{0}$ by setting the lattice depths ($V_{0}$) and the Raman potential amplitudes ($\Omega_{0}$) to be isotropic for post-quench Hamiltonian. Quenching $h_{z}$ is realized by tuning the two-photon detuning $\delta$ Sun et al. (2018b); sup , while quenching $h_{x,y}$ is performed by suddenly tuning $(\varphi_{1},\varphi_{2})$ between the aforementioned two settings [see Fig. 1(c)]. In each quench, the post-quench parameters are fixed. Quenching $h_{x,y}$ is performed as follows. First, the ${}^{87}\textrm{Rb}$ atoms are prepared slightly above the critical temperature of Bose-Einstein condensation, and then adiabatically loaded into the Raman lattice with ($\varphi_{1},\varphi_{2})=(0,\pi/2)$ or $(\pi/2,0)$ by ramping up the laser intensity in ${100}ms$ sup . The atoms are then populated almost in the lowest trivial energy band of the pre-quench Hamiltonian [Fig. 1(d)]. Second, using the EOM in the ${x}$ (or ${y}$) direction we suddenly tune the phase to shift as ($\varphi_{1},\varphi_{2})\to(\pi/2,\pi/2$) within ${2}\mu s$, corresponding to quenching $h_{y}$ (or $h_{x}$). The atoms are driven out of equilibrium and evolve under the post-quench Hamiltonian with the topologically nontrivial bands [Fig. 1(d)]. Third, we hold the system for a certain time $t$, and measure the atom density $n_{\uparrow,\downarrow}({\bm{q}},t)$ of each spin component in momentum space by time-of-flight (TOF) expansion. The evolution of spin polarization $\langle\sigma_{z}({\bm{q}},t)\rangle=[n_{\uparrow}({\bm{q}},t)-n_{\downarrow}({\bm{q}},t)]/[n_{\uparrow}({\bm{q}},t)+n_{\downarrow}({\bm{q}},t)]$ is obtained at each Bloch momentum.

Figure 2 shows the measured spin textures $\langle\sigma_{z}({\bm{q}},t)\rangle_{y,z}$ at different time after quenching $h_{y,z}$, which are compared with the numerical calculations. The spin evolution $\langle\sigma_{z}({\bm{q}},t)\rangle_{x}$ is also measured by quenching $h_{x}$ but not shown here for simplicity. The polarizations $\langle\sigma_{z}({\bm{q}},t)\rangle_{y,z}$ oscillate at frequencies equaling to local gap of the post-quench bands at momentum $\bm{q}$. In particular, for quenching $h_{y}$ in (a), there appears a spiral-like dynamical pattern, which gradually spreads to the whole BZ. For quenching $h_{z}$ in (b), the spin oscillations exhibit a ring structure with $C_{4}$ symmetry, which is the dynamical signature of the BIS emerging in quench dynamics. For each quench, we fit the spin oscillation at every momentum ${\bm{q}}$ sup , with which we further obtain the time-averaged spin textures $\overline{\left\langle\sigma_{z}(\bm{q})\right\rangle}_{x,y,z}$. These spin textures exhibit nontrivial patterns which characterize the topology of post-quench band, as elaborated below.

Measuring topological charges.— As shown in theory Zhang et al. (2018, 2019a), the bulk topology can be characterized by the total topological charges in the region enclosed by BISs. The information of topological charges and BISs is captured dynamically by the time-averaged spin textures $\overline{\left\langle\sigma_{z}(\bm{q})\right\rangle}_{i}$, which are shown in Fig.3(a). The momenta where all the polarizations $\overline{\left\langle\sigma_{z}(\bm{q})\right\rangle}_{x,y,z}$ being zero form the BIS, which shapes a ring around the $M$ point ($q_{x}=q_{y}=k_{0}$), and divide the whole BZ into ${\cal V}_{+}$ (with $h_{z}>0$) and ${\cal V}_{-}$ (with $h_{z}<0$) regions sup . Besides the BIS, there are also other momenta satisfying $\overline{\langle\sigma_{z}(\bm{q})\rangle}_{x}=0$ or $\overline{\langle\sigma_{z}(\bm{q})\rangle}_{y}=0$, and form the curves denoted as $L_{x1,x2}$ and $L_{y1,y2}$, respectively. As shown in Fig.3(b), the curves $L_{x1,x2}$ and $L_{y1,y2}$ intersect at four momentum points $D_{1,2,3,4}$ in the first BZ, which mark the locations of dynamical topological charges Zhang et al. (2019a); Top : three in ${\cal V}_{-}$ and one in ${\cal V}_{+}$. All our experimental measurements agree well with numerical calculations (Fig.3), which are based on the full-band model and take into account the finite temperature effect sup . A special case is that if the pre-quench trivial phase is fully polarized by a large initial Zeeman term with $|m_{i}|\gg|t_{0,\rm so}|$, the BIS refers to the 1D band-crossing ring with $h_{z}=0$, and topological charges are located at the nodes of SO coupling with $h_{x}=h_{y}=0$, which are $\Gamma,X_{1,2}$ and $M$ points in the BZ Zhang et al. (2019a).

The charge value ${\cal C}=\pm 1$ can be further characterized by the winding of the spin-texture field ${\bm{S}}({\bm{q}})=(S_{y},S_{x})$, with components given by Zhang et al. (2019a)

[TABLE]

Here ${\cal N}_{q}$ denotes the normalization factor. Since the two curves that intersect at a charge divide the neighboring region into four parts, the charge value ${\cal C}$ can be simply determined by the signs of spin textures $\overline{\langle\sigma_{z}(\bm{q})\rangle}_{x,y}$ in the four subregions, as illustrated in Fig. 3(c). The plus (minus) sign of $\overline{\langle\sigma_{z}({\bm{q}})\rangle}_{x,y}$ indicates that the field components $S_{x,y}$ point to the positive (negative) spin axes. The combined dynamical field ${\bm{S}}({\bm{q}})$ characterizes the charge ${\cal C}=+1$ if the field flows from or to the charge; ${\cal C}=-1$ if it is “repulsed” by the charge. We finally find that three dynamical topological charges are enclosed in the region ${\cal V}_{-}$ with one being ${\cal C}=-1$ and two ${\cal C}=+1$, whose summation characterizes the post-quench topological phase with the Chern number ${\rm Ch}=+1$.

Dynamical bulk-surface correspondence.— Similar to the well-known bulk-boundary correspondence in real space, there is also a dynamical bulk-surface correspondence emerging from quench dynamics in the momentum space Zhang et al. (2018). It claims that the bulk topology can be characterized by the dynamical topological pattern on the lower-dimensional BISs. In Ref. Sun et al. (2018b), we have observed the quench dynamics that occurs on BISs, and accordingly determined the topological phase diagram, but the topological pattern can not be measured by quenching only $h_{z}$. Here, using the time-averaged spin textures obtained by a sequence of quenches, we can observe the winding of an emergent dynamical field defined on the BIS, as an experimental observation of the dynamical bulk-surface correspondence.

According to Refs. Zhang et al. (2018, 2019a), the dynamical field $\bm{g}(\bm{q})=(g_{y},g_{x})$ can be defined by

[TABLE]

where $\mathcal{N}_{\bm{q}}$ is normalization factor, and $q_{\perp}$ denotes the momentum perpendicular to the BIS and points from ${\cal V}_{-}$ to ${\cal V}_{+}$. We calculate the field components $g_{i}(\bm{q})$ ($i=x,y$) by the variations of spin textures $\overline{\langle\sigma_{z}(\bm{q})\rangle}_{i}$ along the direction $q_{\perp}$, i.e., $g_{i}(\bm{q}_{0})=\lim_{q_{\perp}\to 0}[\overline{\langle\sigma_{z}(\bm{q}_{0}+q_{\perp}\hat{e}_{\perp})\rangle}_{i}-\overline{\langle\sigma_{z}(\bm{q}_{0}-q_{\perp}\hat{e}_{\perp})\rangle}_{i}]$ for every $\bm{q}_{0}$ on the BIS. The results are shown in Fig. 4. A combination of the two components gives the emergent dynamical field $\bm{g}(\bm{q})$, which is observed to exhibit a nonzero winding along the ring despite local fluctuations induced by experimental errors. The winding number of the field $\bm{g}(\bm{q})$, resembling the flux of magnetic monopoles, counts the total charges ${\cal C}=+1$ in the region ${\cal V}_{-}$, and characterizes the post-quench quantum phase with Chern number ${\rm Ch}=+1$.

Conclusion.— We have detected by quench dynamics the topological charges and further the dynamical bulk-surface correspondence in a 2D QAH model through a sequence of quantum quenches. We observed that the bulk topology can be generally characterized by the total charges enclosed by a lower-dimensional momentum subspace called band inversion surface (BIS). Such topological charges are related to the emergent topological pattern of a dynamical field defined on the BIS, showing an essential dynamical bulk-surface correspondence. We emphasize that the scheme of quenching all the spin axes enables to extract the complete topological information by measuring only a single spin component, hence has great advantages in detecting topological states. This study clearly verifies the insight that a higher-dimensional topological system (e.g. 2D Chern insulator) can be characterized by lower-dimensional dynamical invariants (1D ring topology or 0D monopole charges), which has broad applications to dynamical classification of generic topological phases Zhang et al. (2018, 2019a).

Note added.–In completing the manuscript, we notice a preprint which demonstrates the dynamical bulk-surface correspondence based on nitrogen-vacancy center Wang et al. (2019). In comparison, we first measured the topological charges with ultracold atoms, and provide an alternative powerful approach to verify the dynamical bulk-surface correspondence.

Acknowledgement.— We thank Jin-Long Yu for fruitful discussions. This work was supported by the National Key R&D Program of China (under grants 2016YFA0301601 and 2016YFA0301604), National Natural Science Foundation of China (under grants No. 11674301, 11574008, 11825401, and 11761161003), the Chinese Academy of Sciences and the Anhui Initiative in Quantum Information Technologies (AHY120000) and the Chinese Academy of Sciences, and the Strategic Priority Research Program of Chinese Academy of Science (Grant No. XDB28000000).

Chang-Rui Yi and Long Zhang contribute equally to this work.

I @slowromancapi@. Experimental Realization

Here we briefly explain the generation of lattice and Raman potentials (Sec. @slowromancapi@A), the derivation of the Bloch Hamiltonian for different settings (Sec. @slowromancapi@B), and the calibration of the two phases $\varphi_{1,2}$ (Sec. @slowromancapi@C). More details can be found in Refs. Zhang et al. (2019a); Wang et al. (2018); Sun et al. (2018a).

I.1 @slowromancapi@A. Raman lattice scheme

The experimental setup for 87Rb Bose gas is shown in Fig. 1a of the main text. Two magnetic sublevels $|\uparrow\rangle\equiv|F=1,m_{F}=-1\rangle$ and $|\downarrow\rangle\equiv|1,0\rangle$ are selected by ${10.2}$\mathrm{MHz} Zeeman splitting generated via a bias magnetic field of ${14.5}$G applied in $\hat{z}$ direction. The laser beam $\bm{E}_{x}$ ($\bm{E}_{y}$) with wavelength of $\lambda={787}$\mathrm{nm} injected from $\hat{x}$ ($\hat{y}$) direction passes through a high extinction-ratio polarization beam splitting and a $\lambda/2$ wave plates to generate two orthogonally polarized components $\bm{E}_{xy}$ ($\bm{E}_{yx}$) and $\bm{E}_{xz}$ ($\bm{E}_{yz}$). A electro-optic modulator (EOM) is placed in front of the mirror $R_{x}$ ($R_{y}$) to introduce a phase shift $\varphi_{1}$ ($\varphi_{2}$) between the two components $\bm{E}_{xy}$ and $\bm{E}_{xz}$ ($\bm{E}_{yx}$ and $\bm{E}_{yz}$). Hence, the laser beams $\bm{E}_{x}$ and $\bm{E}_{y}$ form the standing-wave fields for atoms:

[TABLE]

where $k_{0}=2\pi/\lambda$, $\alpha$ and $\beta$ denote the initial phases, and $\alpha_{L}$ ($\beta_{L}$) is the phase acquired by ${\bm{E}}_{x}$ (${\bm{E}}_{y}$) for an additional optical path from the atom cloud to mirror $R_{x}$ ($R_{y}$), then back to the atom cloud. As shown in Fig. S1, the lattice and Raman coupling potentials are contributed from both the $D_{2}$ ($5{{}^{2}S}_{1/2}\to 5{{}^{2}P}_{3/2}$) and $D_{1}$ ($5^{2}S_{1/2}\to 5^{2}P_{1/2}$) lines. The total Hamiltonian reads ($\hbar=1$)

[TABLE]

where

[TABLE]

is the square lattice potential, $\Omega_{x,y}$ are the Raman potentials determined by the two Raman couplings

[TABLE]

and $m_{z}=\delta/2$ measures the two-photon detuning of Raman coupling. Here $\hat{e}_{\pm}=(\hat{e}_{x}\mp i\hat{e}_{y})/\sqrt{2}$, $t_{D_{2}}\equiv|\langle J=1/2||e{\bm{r}}||J^{\prime}=3/2\rangle|$, $t_{D_{1}}\equiv|\langle J=1/2||e{\bm{r}}||J^{\prime}=1/2\rangle|$, and $t_{D_{2}}\approx\sqrt{2}t_{D_{1}}\approx 4.227ea_{0}$ Steck , with $a_{0}$ being the Bohr radius. One can tune the phases $\varphi_{1,2}$ by EOMs to manipulate the Raman and lattice potentials.

I.2 @slowromancapi@B. Symmetric and asymmetric settings

We consider three cases of the phases: (i) $(\varphi_{1},\varphi_{2})=(\pi/2,\pi/2)$; (ii) $(\varphi_{1},\varphi_{2})=(0,\pi/2)$; (iii) $(\varphi_{1},\varphi_{2})=(\pi/2,0)$.

(i) When $(\varphi_{1},\varphi_{2})=(\pi/2,\pi/2)$, the light fields read

[TABLE]

which leads to the lattice potential

[TABLE]

with

[TABLE]

and the Raman coupling potentials

[TABLE]

with

[TABLE]

In the tight-binding limit and only considering $s$-bands, the Hamiltonian (S2) takes the form in the momentum space $H=\sum_{\bm{q}}{\Psi}^{\dagger}_{\bm{q}}{\cal H}{\Psi}_{\bm{q}}$ Zhang et al. (2019a); Wang et al. (2018), where ${\Psi}_{\bm{q}}=({c}_{{\bm{q}}\uparrow},{c}_{{\bm{q}}\downarrow})$ and

[TABLE]

is exactly the 2D QAH model. Here $a$ is the lattice constant with $a=\lambda/2$, ${\bm{q}}$ is the Bloch momentum, and $t_{0}$ and $t_{\rm so}$ are, respectively, the spin-conserved and spin-flipped hopping coefficients

[TABLE]

where $\phi_{s}(x,y)$ denotes the Wannier function and $\Omega_{0x}=\Omega_{0y}=\Omega_{0}$.

(ii) When $(\varphi_{1},\varphi_{2})=(0,\pi/2)$, the light fields become

[TABLE]

The lattice potential is

[TABLE]

with $V_{0y}$ being defined as in Eq. (S7) and $V^{\prime}_{0x}=\frac{t_{D_{1}}^{2}}{3}\left(\frac{1}{|\Delta_{D_{1}}|}-\frac{2}{|\Delta_{D_{2}}|}\right)(|E_{xy}|^{2}+|E_{xz}|^{2})$. The Raman potentials are

[TABLE]

which generate spin-flipped hopping only in $\hat{y}$-direction, accompanied with on-site spin flipping. In the tight-binding limit, the Bloch Hamiltonian reads Zhang et al. (2019a)

[TABLE]

where

[TABLE]

(iii) When $(\varphi_{1},\varphi_{2})=(\pi/2,0)$, the light fields become

[TABLE]

The lattice potential is

[TABLE]

with $V_{0x}$ being defined as in Eq. (S7) and $V^{\prime}_{0y}=\frac{t_{D_{1}}^{2}}{3}\left(\frac{1}{|\Delta_{D_{1}}|}-\frac{2}{|\Delta_{D_{2}}|}\right)(|E_{yx}|^{2}+|E_{yz}|^{2})$. The Raman potentials are

[TABLE]

which generate spin-flipped hopping only in $\hat{x}$-direction, and also on-site spin flipping. In the tight-binding limit, the Bloch Hamiltonian reads Zhang et al. (2019a)

[TABLE]

where

[TABLE]

I.3 @slowromancapi@C. Calibration of the phases $\varphi_{1,2}$

In experiment, we tune the phases $\varphi_{1,2}$ by the voltage applied on EOMs (-200\text{\,}\mathrm{V}$\sim$200\text{\,}\mathrm{V}). The instability of $\varphi_{1,2}$, which is limited to approximately $\pi/60$ radians per 10 hours, is controlled by the thermoelectric cooler attached to EOMs. We calibrate the phases $\varphi_{1,2}$ by measuring the spin polarization of the Bose-Einstein condensate (BEC).

We first prepare about $3\times 10^{5}$ 87Rb atoms in the Raman lattice with some values of the phases $\varphi_{1,2}$, which are cooled to condense at the ground state of the lowest band ($q_{x}=q_{y}=0$). We then release the BEC atoms for $25\text{\,}\mathrm{ms}$ and take the spin-resolved TOF imaging to obtain the spin and momentum distribution of the atomic cloud. We calculate the spin polarization of the BEC $P=\frac{N_{\uparrow}-N_{\downarrow}}{N_{\uparrow}+N_{\downarrow}}$, where $N_{\uparrow}$ ($N_{\downarrow}$) denotes the total atom number in the spin-up (-down) state, and accordingly calibrate the values of $\varphi_{1,2}$. We show an example in Fig. S2, where the spin polarization $P$ is measured as a function of the phase $\varphi_{1}$ with $\varphi_{2}$ being fixed at $\pi/2$. The measured variation of $P$ reflects the generation of the transverse magnetization $m_{y}$: when $P$ takes the maximum, the magnetization $m_{y}\simeq 0$, corresponding to $\varphi_{1}=\pi/2$; when $P$ is at the minimum, $m_{y}$ takes its largest value, corresponding to $\varphi_{1}=0$.

II @slowromancapii@. Data processing

Here, we explain the obtainment of the time-averaged spin textures from the time evolution of spin polarization (Sec. @slowromancapii@A) and the sign of $h_{z}$ (Sec. @slowromancapii@B).

II.1 @slowromancapii@A. Data fitting

After the measurement, we fit the time-evolved spin polarization $\langle\sigma_{z}(\bm{q},t)\rangle$ at each momentum $\bm{q}$ into the combination function

[TABLE]

Here the first $A_{i}(\bm{q})$-terms denote the damped oscillations with the characteristic damping time $\tau_{1}$, the second $B(\bm{q})$-term represents a pure decay with the characteristic time $\tau_{2}$, and the last $D(\bm{q})$-term is an offset. Both the decay and damping are internal relaxation induced by the fluctuation of magnetic field and the atom-atom interaction Sun et al. (2018b); Kosugi et al. (2005). In the derivation of time-averaged spin textures, we remove those non-ideal effects from Eq. (S22) and obtain $\overline{\langle\sigma_{z}(\bm{q})\rangle}_{x,y,z}$ by

[TABLE]

Here $T$ is taken much longer than the oscillation period. As an example, the measured time-evolved spin polarization at ${\bm{q}}_{1}=(0.66k_{0},0.66k_{0})$ is shown in Fig. S3 after quenching $h_{y}$. Experimental data are fitted into Eq. (S22) (red curve). According to the fitting parameters, we have $\overline{\langle\sigma_{z}(\bm{q}_{1})\rangle}_{x}\approx-0.02$.

The time average of all momentum points in the first BZ are shown in Fig. 3(a) for three quenches. The deviation from numerical results comes from experimental errors induced by mechanical shaking and magnetic field fluctuation, as well as interaction induced decay or dephasing.

II.2 @slowromancapii@B. The sign of $h_{z}$

The sign of $h_{z}=m_{z}-2t_{0}(\cos(q_{x})+\cos(q_{y}))$ can be determined by the size of the BISs from the experiments. In the theory [Fig. S4], the region $\mathcal{V_{-}}$ as well as the size of the BISs shrinks when the detuning increases from $\delta=-0.6$ to $\delta=0.6$, from which we obtain $h_{z}<0$ ($h_{z}>0$) in the region $\mathcal{V_{-}}$ ($\mathcal{V_{+}}$). In the experiment, we measure the time evolution of spin polarization with different detuning and fixed lattice depth and Raman coupling strength in topological nontrivial regime. The measured results are found in Ref. Sun et al. (2018b). The size of the BISs shrinks when the detuning increases, which is consistent with theoretical calculation. Therefore, $h_{z}<0$ in the region $\mathcal{V_{-}}$ and $h_{z}>0$ in the region $\mathcal{V_{+}}$.

III @slowromancapiii@. Numerical calculations

We compare our experimental observations to the numerical calculations. In the calculations, the post-quench unitary dynamics is described by the time evolution operator $U(\bm{q},t)=\exp(-iHt)$, where $H$ is the full-band Hamiltonian (S2) with $\varphi_{1}=\varphi_{2}=\pi/2$. The time-dependent density matrix is then $\rho(\bm{q},t)=U(\bm{q},t)\rho(\bm{q},t=0)U(\bm{q},t)^{\dagger}$. Here the initial state reads $\rho(\bm{q},0)=\sum_{n}p_{n}|n\rangle\langle n|$, where $|n\rangle$ denotes the $n$-th eigenstate of $H$ with pre-quench parameters and $p_{n}$ is the population probability at the state $|n\rangle$ determined by the Bose-Einstein statistics $f(E_{n};T,\mu)$, with the temperature $T$ being $150$nk ($100$nk) for quenching $h_{x,y}$ ($h_{z}$), and the chemical potential $\mu$ being fixed by the particle density. The time-evolved spin textures are then given by $\langle\sigma_{z}(\bm{q},t)\rangle={\rm Tr}[\rho(\bm{q},t)\sigma_{z}]$, and the time averages $\overline{\langle\sigma_{z}(\bm{q})\rangle}_{x,y,z}$ are obtained by summing over a long enough time so that they approaches the ones averaged over infinite time.

The time-averaged spin textures in topologically trivial region are also measured, as shown in Fig.S4 with $V_{0}=4.0E_{\rm r},\Omega_{0}=1.0E_{\rm r},\delta=-1.0E_{\rm r}$. One can see that there is no momentum satisfying $\overline{\langle\sigma_{z}({\bm{q}})\rangle}_{x,y,z}=0$, which characterizes the trivial phase with Chern number $\mathcal{C}=0$. The observations agree with the numeral calculations with the same experimental parameters (Fig.S4).

IV @slowromancapiv@. Theoretical background

Here we present briefly the theoretical background of our experiment. Details can be found in Refs. Zhang et al. (2018, 2019a); Top . In our dynamical classification theory, two essential concepts are introduced: One is band inversion surfaces (BISs) defined as the surfaces with $h_{z}({\bm{k}})=0$; the other is topological charges located at nodes of the spin-orbit (SO) field ${\bm{h}}_{\rm so}({\bm{k}})\equiv(h_{y},h_{x})=0$. We find that the bulk topology is classified by either the total charges enclosed by BISs, or the winding of the SO field ${\bm{h}}_{\rm so}({\bm{k}})$ on BISs. We make use of the time-averaged spin textures after quench, which is defined by ($i=x,y,z$)

[TABLE]

with $E=\sqrt{h_{x}^{2}+h_{y}^{2}+h_{z}^{2}}$, to dynamically characterize the BISs, the SO field, the charges, and also the topology. Here $\rho_{i}(0)$ denotes the initial state for quenching $h_{i}$ and $\mathcal{H}$ is the post-quench Bloch Hamiltonian. In the following, we first give the description of dynamical characterization by deep quenches Zhang et al. (2018, 2019a), and then discuss the shallow quench situation that our experiment belongs to.

IV.1 A. Deep quenches

When the quench starts from the extremely deep trivial regime $m_{i}\to-\infty$, the initial state $\rho_{i}(0)$ yields ${\rm Tr}\left[\rho_{i}(0){\cal H}\right]=h_{i}$ and the time-averaged spin texture reads

[TABLE]

One can see that no matter which axis is quenched, the spin polarization $\overline{\langle\sigma_{z}({\bm{k}})\rangle}_{i}$ always vanishes on BISs where $h_{z}({\bm{k}})=0$. Hence, a BIS can be dynamically determined by vanishing time-averaged spin polarization independent of the quench axis:

[TABLE]

In particular, $\overline{\langle{\sigma_{z}}({\bm{k}})\rangle}_{z}=0$ occurs only on BISs, so one can find out these surfaces simply by quenching $h_{z}$. Besides, the spin texture $\overline{\langle\sigma_{z}({\bm{k}})\rangle}_{x/y}$ also vanish on the surfaces with $h_{x/y}({\bm{k}})=0$ [see Eq. (S25)]. Accordingly, the nodes of the SO field can be found out by $\overline{\langle\sigma_{z}({\bm{k}})\rangle}_{x,y}=0$ but $\overline{\langle\sigma_{z}({\bm{k}})\rangle}_{z}\neq 0$. Furthermore, near a node ${\bm{k}}={\bm{q}}_{c}$, the time-averaged spin texture directly reflects the SO field: \overline{\langle\sigma_{z}({\bm{k}})\rangle}_{i}\big{|}_{{\bm{k}}\to{\bm{q}}_{c}}\simeq h_{i}({\bm{k}})/h_{z}({\bm{q}}_{c}). Thus, we define a dynamical spin-texture field ${\bm{S}}({\bm{k}})$, whose components are

[TABLE]

with ${\cal N}_{\bm{k}}$ being a factor. With this result, the topological charge can be dynamically determined by Zhang et al. (2019a)

[TABLE]

where $J_{{\bm{S}}}({\bm{k}})\equiv\det\left[(\partial S_{i}/\partial k_{j})\right]$ is Jacobian determinant.

Moreover, we define a dynamical directional derivative field ${\bm{g}}({\bm{k}})=(g_{y},g_{x})$, whose components are given by

[TABLE]

where $k_{\perp}$ denotes the momentum perpendicular to BIS, and ${\cal N}_{k}$ is the normalization factor. It can be shown that on the BIS ${\bm{g}}({\bm{k}})=\hat{\bm{h}}_{\rm so}({\bm{k}})$, where $\hat{\bm{h}}_{\rm so}({\bm{k}})\equiv{\bm{h}}_{\rm so}({\bm{k}})/|{\bm{h}}_{\rm so}({\bm{k}})|$ is the unit SO field. Thus the topological invariant can be characterized dynamically by

[TABLE]

where the summation is over all the BISs. The result manifests itself a highly nontrivial dynamical bulk-surface correspondence Zhang et al. (2018).

IV.2 B. Shallow quenches

When the quench starts from a shallow trivial regime, i.e., $m_{i}$ is finitely large, we find that the dynamical characterization is not valid for any shallow quenches Top . Here we give the validity condition for our quench experiment. For an shallow quench, the time-averaged spin texture read

[TABLE]

The BISs can still be identified by the vanishing spin polarizations via Eq. (S26). The directional derivative on BISs becomes

[TABLE]

We denote $\widetilde{h}_{i}\equiv{\rm Tr}\left[\rho_{i}(0){\cal H}\right]$, and consider the situation that a quench corresponds to adding an additional constant magnetization $\delta m_{i}$ in $h_{i}$ and tuning $\delta m_{i}$ from $\delta m_{i}<-\max\{|h_{i}({\bm{k}})|\}$ to zero.

To obtain the validity condition, we introduce local transformations $U_{i}({\bf k})=e^{-i{\bf u}_{i}({\bf k})}$, with ${\bf u}_{i}({\bf k})=\sum_{j=x,y,z}u_{j}^{(i)}\sigma_{j}$, for each quench, which rotates the initial state $\rho_{i}$ around the axis $\hat{\bf u}_{i}\equiv{\bf u}_{i}/|{\bf u}_{i}|$ by an angle $2|{\bf u}_{i}|$ to the fully polarized one $\rho^{\prime}_{i}=U_{i}({\bf k})\rho_{i}({\bf k})U^{\dagger}_{i}({\bf k})$, with $\sigma_{i}\rho^{\prime}_{i}=\rho^{\prime}_{i}$ and $\sigma_{j\neq i}\rho^{\prime}_{i}=0$. We further set $\hat{\bf u}_{i}$ to be normal to the plane spanned by the $\sigma_{i}$ axis and the pre-quench vector field ${\bf h}$ such that $0\leq 2|{\bf u}_{i}|<\pi/2$ and $u^{(i)}_{i}=0$. It can be checked that this transformation does not change the topology Top . Note that after each quench, we return to the same post-quench Hamiltonian ${\cal H}_{\rm post}=\sum_{j}h_{j}\sigma_{j}$. When quenching $h_{i}$, we have the pre-quench Hamiltonian ${\cal H}_{\rm pre}^{(i)}={\cal H}_{\rm post}+\delta m_{i}\sigma_{i}$. We notice that the pre-quench vector field should be in the $\sigma_{i}$ axis after rotation, which gives

[TABLE]

The relation $(h_{i}+\delta m_{i})^{2}/\cos^{2}2|{\bf u}_{i}|=(h_{i}+\delta m_{i})^{2}+\sum_{j\neq i}h_{j}^{2}$ leads to

[TABLE]

We use the equality

[TABLE]

and obtain the rotated post-quench Hamiltonian

[TABLE]

Thus, we have ${\rm Tr}[\rho_{i}{\cal H}_{\rm post}]={\rm Tr}[\rho^{\prime}_{i}{\cal H}^{(i)^{\prime}}_{\rm post}]=\widetilde{h}_{i}$, where we denote

[TABLE]

On the BIS where $h_{z}({\bf k})=0$, we have

[TABLE]

It can be easily checked that the first terms $\{h_{x}\cos 2|{\bf u}_{x}|,h_{y}\cos 2|{\bf u}_{y}|\}$ preserve the topological pattern of ${\bf h}_{\rm so}$ on BISs while the second terms $\{-\sin 2|{\bf u}_{x}||h_{y}|,-\sin 2|{\bf u}_{y}||h_{x}|\}$ can change it. Here we assume $\delta m_{x}=\delta m_{y}=\delta m$, $h_{x}=2t_{\rm so}\sin q_{x}$, $h_{y}=2t_{\rm so}\sin q_{y}$, and $h_{z}=m_{z}-2t_{0}\cos q_{x}-2t_{0}\cos q_{y}$ with $|m_{z}|<4t_{0}$. In order to ensure that our dynamical characterization is valid, we should have $0\leq 2|{\bf u}_{x}|,2|{\bf u}_{y}|<\pi/4$ and $\delta m<-2t_{\rm so}(|\sin q_{x}|+|\sin q_{y}|)$. Since $\cos q_{x}+\cos q_{y}=m_{z}/(2t_{0})$ on the BIS, we have

[TABLE]

In our experiment, we have typically $t_{\rm so}=0.5t_{0}$ and $m_{x,y}=-14t_{0}$, which satisfies Eq. (S39) very well.

The locations of charges are dynamically determined by $\overline{\langle\sigma_{z}({\bm{k}})\rangle}_{x,y}=0$ but $\overline{\langle\sigma_{z}({\bm{k}})\rangle}_{z}\neq 0$, which gives

[TABLE]

These charges are obviously different from those monopole charges defined by ${\bm{h}}_{\rm so}({\bm{k}})=0$, and hence dubbed as dynamical topological charges Top . Moreover, the dynamical spin-texture field ${\bm{S}}({\bf k})$ defined in Eq. (S27) characterizes these dynamical topological charges. We have proved that the winding of $(\widetilde{h}_{y},\widetilde{h}_{x})$ along BISs classifies the bulk topology if the condition (S39) is satisfied. It is then expected that under the same condition, the total dynamical topological charges enclosed by BISs can be used for the dynamical characterization.