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Experimental observation of topologically protected bound states in two dimensional quantum walk

Bo Wang, Tian Chen, Xiangdong Zhang

Published Dec 18, 2019• 1 citation
Publisher

Abstract

Quantum walks (QWs) provide a powerful tool as a quantum simulator to study topological phases. However, some interesting topological phenomena in the two dimensional (2D) QW that do not exist in the one dimensional case, e.g., the edge-state-enhanced transport, have not been demonstrated experimentally. Here 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 open up an avenue to explore topological properties in multidimensional QWs.

DOI
10.1117/12.2547467

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Quantum walks (QWs) provide a powerful tool as a quantum simulator to study topological phases. However, some interesting topological phenomena in the two dimensional (2D) QW that do not exist in the one dimensional case, e.g., the edge-state-enhanced transport, have not been demonstrated experimentally. Here 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 open up an avenue to explore topological properties in multidimensional QWs.
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.
Periodically driven systems have displayed a variety of fascinating phenomena without analogies in static systems, which enrich the classification of quantum phases of matter and stimulate a wide range of research interests. Here, we employ discrete-time quantum walks to investigate a nontrivial topological effect unique to a two-dimensional periodically driven system: chiral edge states can exist at the interface of Floquet insulators whose Chern numbers vanish. Thanks to a resource-saving and flexible fiber-loop architecture, we realize inhomogeneous two-dimensional quantum walks up to 25 steps, over an effective 51×51 lattice with tunable local parameters. Spin-polarized chiral edge states are observed at the boundary of two distinct quantum walk domains. Our results contribute to establishing a well-controlled platform for exploring nontrivial topological phases.
Quantum walks on translation-invariant regular graphs spread quadratically faster than their classical counterparts. The same coherence that gives them this quantum speedup inhibits or even stops their spread in the presence of disorder. We ask how to create an efficient transport channel from a fixed source site (A)(A)(A) to fixed target site (B)(B)(B) in a disordered two-dimensional discrete-time quantum walk by cutting some of the links. We show that the somewhat counterintuitive strategy of cutting links along a single line connecting AAA to BBB creates such a channel. The efficient transport along the cut is due to topologically protected chiral edge states, which exist even though the bulk Chern number in this system vanishes. We give a realization of the walk as a periodically driven lattice Hamiltonian and identify the bulk topological invariant responsible for the edge states as the quasienergy winding of this Hamiltonian.
Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological invariant in a simple way that is consistent with a general definition for one-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondence. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.
Topological insulators host protected boundary states that are robust to disorder, making them attractive for applications and motivating their realization in engineered systems. Discrete-time quantum walks, which have been implemented in a variety of experimental platforms, can exhibit topologically nontrivial phases and their associated boundary states. Here, we introduce a class of topological quantum walks with variable step lengths that offer theoretical insights into the topological classification of such walks. For example, these walks can access higher winding numbers and support multiple edge states. They also provide context for the reduction of topological protection from a Z\mathbb{Z}Z- to Z2\mathbb{Z}_2Z2​-valued invariant when time-reversal symmetry is broken. The topologically protected edge states are investigated using a transfer-matrix approach that describes their spatial profiles and spin structures. These predictions are in excellent agreement with numerical results and, where applicable, with the Jackiw--Rebbi zero-mode solution of the Dirac equation. Taken together, our analysis provides a roadmap for designing quantum walks with control over the number, spatial profile, and spin structure of their topological edge states.