Measuring a dynamical topological order parameter in quantum walks

Xiao-Ye Xu,Qin-Qin Wang,Munsif Jan,Zhe Chen,Wei-Wei Pan,Guang-Can Guo,Jin-Shi Xu,Markus Heyl,Yong-Jian Han,Jan Carl Budich,Kai Sun,Chuan-Feng Li

doi:10.1038/s41377-019-0237-8

Abstract

Quantum processes of inherent dynamical nature, such as quantum walks, defy a description in terms of an equilibrium statistical physics ensemble. Until now, identifying the general principles behind the underlying unitary quantum dynamics has remained a key challenge. Here, we show and experimentally observe that split-step quantum walks admit a characterization in terms of a dynamical topological order parameter (DTOP). This integer-quantized DTOP measures, at a given time, the winding of the geometric phase accumulated by the wavefunction during a quantum walk. We observe distinct dynamical regimes in our experimentally realized quantum walks, and each regime can be attributed to a qualitatively different temporal behavior of the DTOP. Upon identifying an equivalent many-body problem, we reveal an intriguing connection between the nonanalytic changes of the DTOP in quantum walks and the occurrence of dynamical quantum phase transitions.

Highlights

Coherence in quantum dynamics is at the heart of fascinating phenomena beyond the realm of classical physics, such as quantum interference effects[1], entanglement production[2,3] and geometric phases[4,5,6]
In this experiment, we have achieved a dynamical characterization of split-step quantum walks using a dynamical topological order parameter (DTOP)—an integer-valued quantum number that measures the winding of the geometric phase in the latticemomentum space
Our results clearly show that as a global quantum number of the system’s dynamical topological phase, the DTOP is robust with respect to decoherence in our experimental platform on the time scales studied in this work

Summary

Introduction

Coherence in quantum dynamics is at the heart of fascinating phenomena beyond the realm of classical physics, such as quantum interference effects[1], entanglement production[2,3] and geometric phases[4,5,6]. As opposed to classical random walks, quantum walks are characterized by quantum superpositions of amplitudes rather than classical probability distributions. We report the direct observation of a dynamical topological order parameter (DTOP) that provides a dynamical characterization of quantum walks. To this end, we realize a split-step quantum walk in a photonic system using the framework of time multiplexing. Using a previously developed technique, we achieve full-state tomography of the time-evolved quantum state for up to 10 complete time steps This measurement provides us with the full complex amplitude information of the quantum walk state. This information is essential for our central goal of a dynamical classification of a quantum walk using the DTOP, since the DTOP measures the phase winding number ωD(t) in momentum space, namely, of the so-called Pancharatnam

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