Observing Topological Charges and Dynamical Bulk-Surface Correspondence with Ultracold Atoms

Quantum dynamics induced in quenching a $d$-dimensional topological phase across a phase transition may exhibit a nontrivial dynamical topological pattern on the $(d-1)\mathrm{D}$ momentum subspace, called band inversion surfaces (BISs), which have a one-to-one correspondence to the bulk topology of the postquench phase. Here we report the experimental observation of such 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 number of the postquench band and also characterizes the topological pattern of a dynamical field emerging on the BISs, rendering the dynamical bulk-surface correspondence. This study opens a new avenue to explore topological phases dynamically.

Figure 1

Experimental scheme. (a) Experimental setup. Two laser beams ${\mathit{E}}_x$ and ${\mathit{E}}_y$ are incident on the atoms and then reflected by two mirrors $R_{x,y}$, simultaneously producing a 2D square lattice and two Raman coupling potentials. The $\lambda /2$ wave plates are used to generate two orthogonally polarized components ${\mathit{E}}_{xy,xz}$ or ${\mathit{E}}_{yx,yz}$ for each beam, and EOMs are placed in front of the mirrors to tune the phase shift between the two components. (b) Level structure and Raman transitions. Two hyperfine states $|1,-1⟩$ and $|1,0⟩$ are selected to form the double-$\mathrm{\Lambda }$-type coupling configuration. (c) The quench process corresponds to changing the symmetry of Raman potentials (sinusoidal curves) with respect to lattice sites (yellow ellipsoids) in a spatial direction. The relative (anti)symmetry selects the hopping type: Before quench, only on-site spin flipping is permitted, while spin-flip hopping is forbidden; after quench, the situation is reversed. (d) The $s$-band structure before and after quenching $h_x$ or $h_y$. The color indicates the distribution of spin polarization $⟨{\sigma }_z⟩$ with respect to eigenstates of the pre- or postquench Hamiltonian.

Figure 2

Time evolution of the spin polarization $⟨{\sigma }_z(\mathit{q}){⟩}_{y,z}$ after quenching (a) $h_y$ and (b) $h_z$, respectively. Quenching $h_y$ is realized by tuning the phases from $({\phi }_1,{\phi }_2)=(0,\pi /2)$ to the setting $(\pi /2,\pi /2)$. Quenching $h_z$ corresponds to varying the two-photon detuning from $\delta =-200E_r$ to $-0.2E_r$. In each case, the measured spin textures for different hold time $t$ (upper) are compared with numerical calculations (lower). Spin textures after the two quenches exhibit distinctive patterns during the time evolution. The postquench parameters are lattice depth $V_0=4.0E_r$, Raman coupling strength ${\mathrm{\Omega }}_0=1.0E_r$, and $\delta =-0.2E_r$ for each quench. Here $E_r$ is the recoil energy.

Figure 3

Dynamical measurement of topological charges and Chern number. (a) Experimental measurements (upper) of time-averaged spin textures ${\overline{⟨{\sigma }_z(\mathit{q})⟩}}_i$ by quenching $h_i$ ($i=x$, $y$, $z$), compared with numerical calculations (lower). Dynamical characterization of ${\overline{⟨{\sigma }_z(\mathit{q})⟩}}_z=0$ identifies the BIS (green), which also emerges in ${\overline{⟨{\sigma }_z(\mathit{q})⟩}}_{x,y}$. The BIS divides the BZ into two regions: ${\mathcal{V}}_{-}$ with $h_z<0$ and ${\mathcal{V}}_{+}$ with $h_z>0$. In both ${\overline{⟨{\sigma }_z(\mathit{q})⟩}}_x$ and ${\overline{⟨{\sigma }_z(\mathit{q})⟩}}_y$, other than the BIS, two additional curves are formed at momenta with vanishing spin polarization, denoted by $L_{x1,x2}$ (magenta) and $L_{y1,y2}$ (purple), respectively. $L_{x1}$ or $L_{y1}$ ($L_{x2}$ or $L_{y2}$) further divides ${\mathcal{V}}_{-}$ (${\mathcal{V}}_{+}$) into two subregions $S_{x3,x4}$ or $S_{y3,y4}$ ($S_{x1,x2}$ or $S_{y1,y2}$). The dotted small circles in $S_{x3,y3}$ denote extra surfaces with vanishing spin polarization caused by experimental errors (see Supplemental Material [33]). (b) The locations of topological charges are determined by the intersections $D_{1,2,3,4}$ of the curves $L_{x1,x2}$ and $L_{y1,y2}$. Three charges are in the region ${\mathcal{V}}_{-}$ and one in ${\mathcal{V}}_{+}$ [33]. The green curve is from ${\overline{⟨{\sigma }_z(\mathit{q})⟩}}_z$ in (a). (c) The charge value ${\mathcal{C}}_i=\pm 1$ at each intersection $D_i$ ($i=1$, 2, 3, 4) is determined by the signs of ${\overline{⟨{\sigma }_z(\mathit{q})⟩}}_{x,y}$ in the neighboring four subregions. The postquench parameters are $V_0=4.0E_r,{\mathrm{\Omega }}_0=1.0E_r$ and $\delta =-0.2E_r$.

Figure 4

Dynamical topological pattern emerging on the BIS. The field component $g_i(\mathit{q})$ ($i=x$, $y$) is obtained by the variation of time-averaged spin texture ${\overline{⟨{\sigma }_z(\mathit{q})⟩}}_i$ [shown in Fig. 3] across the BIS. The first (second) row is the experimental (numerical) results. The measured dynamical field $\mathit{g}(\mathit{q})$ exhibits a nonzero winding along the ring, which characterizes the bulk topology with Chern number $\text{Ch}=+1$.