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Physical Review A

Characterization of topological phases and selection of topological interface modes in the parity-time-symmetric quantum walk

Tian Chen, Bo Wang, Xiangdong Zhang

Published May 18, 2018• 26 citations
Publisher

Abstract

We present a method to detect topological invariants in nonunitary parity-time-symmetric (PT-symmetric) quantum walks (QWs). The topological phases of a PT-symmetric QW can be characterized by the mean position of the walk, which is an experimentally observable quantity. This method can also be applied to reveal topological phases in the unitary split-step QW and lattice model. We also explore the selection of topological interface modes in the PT-symmetric QW. The introduction of nonunitarity into the PT-symmetric QW can render the eigenvalue of the topological interface mode larger than the bulk region modes, which leads to the selective excitation of the corresponding topological interface mode during the evolution. Contrary to the Su--Schrieffer--Heeger (SSH) lattice model, in the PT-symmetric QW, the transition between the amplification and the suppression of the topological interface mode may not occur. Our study of the selection of topological interface modes in the PT-symmetric QW provides a route to realize the lasing of specific modes within the system.

DOI
10.1103/physreva.97.052117

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We present a method to detect topological invariants in nonunitary parity-time-symmetric (PT-symmetric) quantum walks (QWs). The topological phases of a PT-symmetric QW can be characterized by the mean position of the walk, which is an experimentally observable quantity. This method can also be applied to reveal topological phases in the unitary split-step QW and lattice model. We also explore the selection of topological interface modes in the PT-symmetric QW. The introduction of nonunitarity into the PT-symmetric QW can render the eigenvalue of the topological interface mode larger than the bulk region modes, which leads to the selective excitation of the corresponding topological interface mode during the evolution. Contrary to the Su--Schrieffer--Heeger (SSH) lattice model, in the PT-symmetric QW, the transition between the amplification and the suppression of the topological interface mode may not occur. Our study of the selection of topological interface modes in the PT-symmetric QW provides a route to realize the lasing of specific modes within the system.
Discrete-time quantum walks (DTQWs) provide a convenient platform for a realization of many topological phases in noninteracting systems. They often offer more possibilities than systems with a static Hamiltonian. Nevertheless, researchers are still looking for DTQW symmetries protecting topological phases and for definitions of appropriate topological invariants. Although the majority of DTQW studies on this topic focus on the so-called split-step quantum walk, two distinct topological phases can be observed in more basic models. Here we infer topological properties of the basic DTQWs directly from the mapping of the Brillouin zone to the Bloch Hamiltonian. We show that for translation-symmetric systems they can be characterized by a homotopy relative to special points. We also propose a topological invariant corresponding to this concept. This invariant indicates the number of edge states at the interface between two distinct phases.
Topological edge states arise in non-Hermitian parity-time ([Formula: see text])-symmetric systems, and manifest themselves as bright or dark edge states, depending on the imaginary components of their eigenenergies. As the spatial probabilities of dark edge states are suppressed during the non-unitary dynamics, it is a challenge to observe them experimentally. Here we report the experimental detection of dark edge states in photonic quantum walks with spontaneously broken [Formula: see text] symmetry, thus providing a complete description of the topological phenomena therein. We experimentally confirm that the global Berry phase in [Formula: see text]-symmetric quantum-walk dynamics unambiguously defines topological invariants of the system in both the [Formula: see text]-symmetry-unbroken and -broken regimes. Our results establish a unified framework for characterizing topology in [Formula: see text]-symmetric quantum-walk dynamics, and provide a useful method to observe topological phenomena in [Formula: see text]-symmetric non-Hermitian systems in general.
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.
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.
Identifying and measuring topological invariants remains a fundamental challenge due to the non-local nature of momentum eigenstates in real space. Established approaches based on Bloch oscillations, quantum walks, charge pumping and quench dynamics, typically fail to provide direct momentum-dependent geometric information of each Bloch state. Here, we propose a novel framework for implementing direct band-topology tomography via momentum-selective measurements, and experimentally demonstrate it on a superconducting processor featuring a highly connected coupling architecture. Utilizing a ``star-like" geometry in which an auxiliary probe couples to a one-dimensional topological chain with spatially modulated coupling strengths, we selectively address the desired momentum states of the chain. This approach enables the direct and robust extraction of the winding phase that contributes to the associated topological invariant. Leveraging non-local couplers bridged by long tantalum airbridges, we realize an extended Su-Schrieffer-Heeger (SSH) model and map its full topological phase diagram. Through a nested Fourier analysis over time and space, we directly extract the winding phase across Brillouin zone and resolve quantized winding numbers of ν={0,±1,2}ν=\{0,\pm1,2\}ν={0,±1,2} characterized by phase winding {0,±2π,4π}\{0, \pm 2π,4π\}{0,±2π,4π}. Our work establishes a versatile tool for probing topological characteristics and opens a new avenue for robust quantum computing architectures based on topological protection.
Non-Hermitian topological edge states have many intriguing properties, however, to date, they have mainly been discussed in terms of bulk-boundary correspondence. Here, we propose using a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient was found to show unique features with the topological phase transitions driven by parity-time (PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite the fact that artificial boundaries are not constructed by an inhomogeneous quantum walk. For a Hermitian system, a turning point and abrupt change appears in the diffusion coefficient when the system is approaching the topological phase transition, while it remains stable in the trivial topological state. For a non-Hermitian system, except for the feature associated with the topological transition, the diffusion coefficient in the PT-symmetric-broken phase demonstrates an abrupt change with a peak structure. In addition, the Shannon entropy of the quantum walk is found to exhibit a direct correlation with the diffusion coefficient. The numerical results presented herein may open up a new avenue for studying the topological state in non-Hermitian quantum walk systems.