Experimental Observation of Topologically Protected Bound States with Vanishing Chern Numbers in a Two-Dimensional Quantum Walk

Quantum walks (QWs) provide a powerful tool as a quantum simulator to study and understand topological phases. Using such a quantum simulator, some topological phenomena have been discussed. However, all the experimental observations on the topological phenomena in QWs have been restricted to evolution in one dimension (1D) so far. The existing 2D experimental platforms cannot be applied to study topological phenomena due to lack of full control in the position space. Thus, some interesting topological phenomena in the 2D QW that do not exist in the 1D case, e.g., the edge-state-enhanced transport, have not been demonstrated experimentally. Here we report the experimental realization of 2D QW using spatial positions and orbital angular momentum of light. Based on our constructed experimental platform, we have observed 2D topological bound states with vanishing Chern numbers and confirmed the robustness of these bound states with respect to perturbations and disorder, which go beyond what has been known in static systems and are unique to periodically driven systems. Our studies not only represent an important advance in the study of topological phases, but also open up an avenue to explore topological properties in multidimensional QWs.

Figure 1

Experimental scheme for 2D QWs. (a) State preparation. (b) Many steps of 2D QW. In our study of probing topological properties in 2D QWs, we create spatially inhomogeneous rotations in regions $-N_y<y<0$ and $0\le y<N_y$, which are realized by semicircular HWPs $R({\theta }_{2,+})$ and $R({\theta }_{2,-})$ in (b). (c) Detection. (d) $T_x$. (e) The list of all optical devices. (f) The schematic representation of first step in a 2D QW. Rotation angles: purple region, $R({\theta }_1)$, $R({\theta }_{2,+})$; yellow region, $R({\theta }_1)$, $R({\theta }_{2,-})$.

Figure 2

(a)–(c) Experimental probability distributions in a homogeneous 2D QW at the fourth step. The walker starts from the lattice site (0,0) and rotation angles are ${\theta }_1={\theta }_{2,+}={\theta }_{2,-}=(\pi /2)$. Error bars indicate the standard deviation. The initial polarizations of light are (a)$|V⟩$, (b) $|D⟩=1/\sqrt{2}(|H⟩+|V⟩)$, (c) $|L⟩=1/\sqrt{2}(|H⟩+i|V⟩)$. (d) The variances between 2D QWs and CRW with steps.

Figure 3

(a)–(c) Experimental observations of bound states. (a) The phase diagram for 2D QW. Green squares in $A$ (${\theta }_1=(\pi /2)$, ${\theta }_{2,-}=-(2\pi /15)$) and $B$ (${\theta }_1=(\pi /2)$, ${\theta }_{2,+}=(10/9)\pi$) represent rotation angles of coin operators used in (b) and (c). (b) The eigenvalues $\lambda =e^{-iϵ}$ of $U$. Red dots indicate $ϵ=0$ and $ϵ=\pi$ energy bound states. (c) Measured probability distribution at the fourth step. (d)–(f) Experimental observations of no bound states. (d) The phase diagram for 2D QW. Red triangles represent rotation angles of coin operators in (e) and (f). Left (${\theta }_1=(\pi /2)$, ${\theta }_{2,-}=-(\pi /3)$); right (${\theta }_1=(\pi /2)$, ${\theta }_{2,+}=(\pi /18)$). (e) The eigenvalues $\lambda =e^{-iϵ}$ of $U$. (f) Measured probability distribution at the fourth step. In (c) and (f), the walker starts from the lattice site (0,0), and the initial polarization of light is $|H⟩$. Error bars indicate the standard deviation.

Figure 4

(a) The localized probability $P_{\text{bound}}$ at the boundary in 2D QWs with step. The cases with and without bound states are obviously different with the step $n\ge 4$. (b) The phase diagram for the 2D QW. Brown circles, green squares and red diamonds are coin rotations in (c),(d), and (e). (c)–(d) Bound states are robust against small perturbations. (c) Measured probability distribution at the fourth step with rotation angles changing from the case in Fig. 3 to (${\theta }_1=(\pi /2)$, ${\theta }_{2,-}=-(\pi /3)$) and (${\theta }_1=(\pi /2)$, ${\theta }_{2,+}=(4\pi /3)$). (d) Measured probability distribution at the fourth step with initial polarization changing to $|D⟩$. Error bars indicate the standard deviation. Insets in (c) and (d), the eigenvalues $\lambda =e^{-iϵ}$ of $U$, red dots indicate bound states. (e)–(f) Bound states are robust against disorder. Measured probability distributions at the fourth step without (e) and with (f) disorder. (e) Parameters are (${\theta }_1=(\pi /2)$, ${\theta }_{2,-}=-(\pi /9)$) and (${\theta }_1=(\pi /2)$, ${\theta }_{2,+}=(10/9)\pi$); inset, the eigenvalues $\lambda =e^{-iϵ}$ of $U$, red dots indicate bound states. (f) Compared with the case in (e), the disorder is introduced to ${\theta }_{2\pm }+\delta \theta$, where $\delta \theta$ is chosen uniformly from the intervals [$-(2\pi /9)$, ($2\pi /9$)].