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arXiv:1709.10287 · Physical Review Letters

Detecting Topological Invariants in Nonunitary Discrete-Time Quantum Walks

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

Published Sep 27, 2017• 208 citations
arXivHTMLPDF

Abstract

We report the experimental detection of bulk topological invariants in nonunitary discrete-time quantum walks with single photons. The nonunitarity of the quantum dynamics is enforced by periodically performing partial measurements on the polarization of the walker photon, which effectively introduces loss to the dynamics. The topological invariant of the nonunitary quantum walk is manifested in the quantized average displacement of the walker, which is probed by monitoring the photon loss. We confirm the topological properties of the system by observing localized edge states at the boundary of regions with different topological invariants. We further demonstrate the robustness of both the topological properties and the measurement scheme of the topological invariants against disorder.

DOI
10.1103/physrevlett.119.130501

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We report the experimental detection of bulk topological invariants in nonunitary discrete-time quantum walks with single photons. The nonunitarity of the quantum dynamics is enforced by periodically performing partial measurements on the polarization of the walker photon, which effectively introduces loss to the dynamics. The topological invariant of the nonunitary quantum walk is manifested in the quantized average displacement of the walker, which is probed by monitoring the photon loss. We confirm the topological properties of the system by observing localized edge states at the boundary of regions with different topological invariants. We further demonstrate the robustness of both the topological properties and the measurement scheme of the topological invariants against disorder.
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.
Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks, we implement the scattering scheme proposed by Tarasinski et al. [Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topological phase transitions. The ability to read-out the position and the coin state distribution, complemented by explicit interferometric sign measurements, allowed the reconstruction of the scattered reflection amplitudes and thus the computation of the associated bulk topological invariants. As predicted, we also find localized states at the edges between two bulks belonging to different topological phases. In order to analyze the impact of disorder, we have measured invariants of two different types of disordered samples in large ensemble measurements, demonstrating their constancy in one disorder regime and a continuous transition with increasing disorder strength for the second disorder sample.
Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks, we implement the scattering scheme proposed by Tarasinski et al. [Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topological phase transitions. The ability to read-out the position and the coin state distribution, complemented by explicit interferometric sign measurements, allowed the reconstruction of the scattered reflection amplitudes and thus the computation of the associated bulk topological invariants. As predicted, we also find localized states at the edges between two bulks belonging to different topological phases. In order to analyze the impact of disorder, we have measured invariants of two different types of disordered samples in large ensemble measurements, demonstrating their constancy in one disorder regime and a continuous transition with increasing disorder strength for the second disorder sample.
We present a protocol to realize topological discrete-time quantum walks, which comprise a sequence of spin-dependent flipping displacement operations and quantum coin tossing operations, with a single trapped ion. It is demonstrated that the information of bulk topological invariants can be extracted by measuring the average projective phonon number when the walk takes place in coherent state space. Interestingly, the specific chiral symmetry owned by our discrete-time quantum walks simplifies the measuring process. Furthermore, we prove the robustness of such bulk topological invariants by introducing dynamical disorder and decoherence. Our work provides a simple method to measure bulk topological features in discrete-time quantum walks, which can be experimentally realized in the system of single trapped ions.
Topological insulators are fascinating states of matter exhibiting protected edge states and robust quantized features in their bulk. Here, we propose and validate experimentally a method to detect topological properties in the bulk of one-dimensional chiral systems. We first introduce the mean chiral displacement, and we show that it rapidly approaches a multiple of the Zak phase in the long time limit. Then we measure the Zak phase in a photonic quantum walk, by direct observation of the mean chiral displacement in its bulk. Next, we measure the Zak phase in an alternative, inequivalent timeframe, and combine the two windings to characterize the full phase diagram of this Floquet system. Finally, we prove the robustness of the measure by introducing dynamical disorder in the system. This detection method is extremely general, as it can be applied to all one-dimensional platforms simulating static or Floquet chiral systems.
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.