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arXiv:1810.00594 · Physical Review A

Higher winding number in a nonunitary photonic quantum walk

Lei Xiao, Xingze Qiu, Kunkun Wang, Zhihao Bian, Xiang Zhan, Hideaki Obuse, Barry C. Sanders, Wei Yi, Peng Xue

Published Dec 28, 2018• 44 citations
arXivHTMLPDF

Abstract

Topological matter exhibits exotic properties yet phases characterized by large topological invariants are difficult to implement, despite rapid experimental progress. A promising route toward higher topological invariants is via engineered Floquet systems, particularly in photonics, where flexible control holds the potential of extending the study of conventional topological matter to novel regimes. Here we implement a one-dimensional photonic quantum walk to explore large winding numbers. By introducing partial measurements and hence loss into the system, we detect winding numbers of three and four in multistep nonunitary quantum walks, which agree well with theoretical predictions. Moreover, by probing statistical moments of the walker, we identify locations of topological phase transitions in the system and reveal the breaking of pseudounitary near topological phase boundaries. As the winding numbers are associated with nonunitary time evolution, our investigation enriches understanding of topological phenomena in nonunitary settings.

DOI
10.1103/physreva.98.063847

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We experimentally investigate topological phenomena in one-dimensional discrete-time photonic quantum walks using a combination of methods. We first detect winding numbers of the quantum walk by directly measuring the average chiral displacement, which oscillates around quantized winding numbers for finite-step quantum walks. Topological phase transitions can be identified as changes in the center of oscillation of the measured chiral displacement. The position of topological phase transition is then confirmed by measuring the moments of the walker probability distribution. Finally, we observe localized edge states at the boundary of regions with different winding numbers. We also confirm the robustness of edge states against chiral-symmetry-preserving disorder.
Emerged as the quantum counterpart of classical random walks, quantum walks are established precious resources in a variety of quantum sciences. Recent studies have shown that quantum walks may be characterized by topological invariants, in close analogy to condensed matter systems exhibiting topological order. Exploiting these features, quantum walks are currently used to simulate topological systems and to probe their exotic features. Here we present the implementation of a one-dimensional quantum walk protocol based on the orbital angular momentum of light, manifesting the topological phases that characterize time-periodic systems (Floquet topological insulators) showing chiral symmetry. By considering the orbital angular momentum spectrum of a light beam undergoing this quantum evolution, we show that the associated statistical moments have marked differences in distinct phases and contain information on the system topology. While varying a control parameter determining the value of the invariants, these moments in the large step-number limit exhibit a sharp variation at the phase changes. We show that these phenomena arise from the singular behavior of the dispersion relation at the transition points. The extension of our results to systems featuring different symmetries, or characterized by higher spatial dimensions, may unveil novel intriguing features associated with these complex systems.
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Floquet Topological Phases Driven by PT\mathcal{PT}PT Symmetric Nonunitary Time Evolution
We study Floquet topological phases driven by PT\mathcal{PT}PT symmetric nonunitary time evolution in one dimension, based on an experimental setup of discrete-time quantum walks. We develop, for nonunitary time-evolution operators, a procedure to calculate topological invariants for Floquet topological phases and find that the bulk-edge correspondence gives correct predictions of the emergent edge states. These edge states make exponential growth of wavefunction amplitudes at specific positions with time controllable. Hereby, we propose that these phenomena inherent in open quantum systems are feasibly observed by present experiments of quantum walks in both classical and quantum regimes.
Winding number and Zak phase, as topological invariants, play crucial roles in characterizing the topological phases of one-dimensional Floquet topological insulators. It is difficult to directly detect Floquet topological invariants with a unified scheme, especially for low driving frequencies. Here, by defining the dynamic winding number based on the time-averaged spin textures, we propose to use it to characterize one-dimensional Floquet topological phases with arbitrarily varying frequencies. In the high-frequency regime, both the conventional winding number and Zak phase can be fully determined by measuring the dynamic winding number of time-averaged spin textures, regardless of initial states and initial phases. In the low-frequency regime, only conventional winding number can accurately describe the system's topological phase due to the presence of large topological numbers, the Zak phase only captures the parity of the winding number and fails to characterize the number of topological edge states. However, one can still correctly characterize the Floquet topological phases through our dynamic winding number approach, in which stroboscopic measurement of spin textures is needed and the initial phases should guarantee chiral symmetry of the time-evolution operator. Our paper not only clarifies the necessity of stroboscopic measurements, but also provides a general dynamic approach for the detection of Floquet topological insulators.