Observing Topological Invariants Using Quantum Walks in Superconducting Circuits

E Flurin,I Siddiqi,N Y Yao,S Hacohen-Gourgy,V V Ramasesh,L S Martin

doi:10.1103/physrevx.7.031023

E Flurin, I Siddiqi + Show 4 more

Open Access

https://doi.org/10.1103/physrevx.7.031023

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Abstract

The direct measurement of topological invariants in both engineered and naturally occurring quantum materials is a key step in classifying quantum phases of matter. Here we motivate a toolbox based on time-dependent quantum walks as a method to digitally simulate single-particle topological band structures. Using a superconducting qubit dispersively coupled to a microwave cavity, we implement two classes of split-step quantum walks and directly measure the topological invariant (winding number) associated with each. The measurement relies upon interference between two components of a cavity Schr\"odinger cat state and highlights a novel refocusing technique which allows for the direct implementation of a digital version of Bloch oscillations. Our scheme can readily be extended to higher dimensions, whereby quantum walk-based simulations can probe topological phases ranging from the quantum spin Hall effect to the Hopf insulator.

Highlights

Topological phases elude the Landau-Ginzburg paradigm of symmetry breaking [1]
While tremendous theoretical progress has been made toward the full classification of topological phases of matter [4,5], a general experimental platform for the direct measurement of topological invariants is lacking
We demonstrate that time-dependent quantum walks comprise a powerful class of unitary protocols capable of digitally simulating single-particle topological band structures and directly observing the associated nonlocal invariants

Summary

Introduction

Topological phases elude the Landau-Ginzburg paradigm of symmetry breaking [1]. Unlike conventional phases, they do not exhibit order parameters that can be locally measured. We demonstrate that time-dependent quantum walks comprise a powerful class of unitary protocols capable of digitally simulating single-particle topological band structures and directly observing the associated nonlocal invariants. The quantum walk is comprised of two unitary operations [see Fig. 1(a)]: a coin toss, denoted RðθÞ, which rotates the spin state, and a spindependent translation, denoted T ↑↓, which translates the particle’s position by a single lattice site in a direction determined by the internal spin state.

Results

Conclusion