Wilson loop and Wilczek-Zee phase from a non-Abelian gauge field

Seiji Sugawa,Francisco Salces-Carcoba,Yuchen Yue,Andika Putra,I B Spielman

doi:10.1038/s41534-021-00483-2

Abstract

Quantum states can acquire a geometric phase called the Berry phase after adiabatically traversing a closed loop, which depends on the path not the rate of motion. The Berry phase is analogous to the Aharonov–Bohm phase derived from the electromagnetic vector potential, and can be expressed in terms of an Abelian gauge potential called the Berry connection. Wilczek and Zee extended this concept to include non-Abelian phases—characterized by the gauge-independent Wilson loop—resulting from non-Abelian gauge potentials. Using an atomic Bose–Einstein condensate, we quantum-engineered a non-Abelian SU(2) gauge field, generated by a Yang monopole located at the origin of a 5-dimensional parameter space. By slowly encircling the monopole, we characterized the Wilczek–Zee phase in terms of the Wilson loop, that depended on the solid-angle subtended by the encircling path: a generalization of Stokes’ theorem. This observation marks the observation of the Wilson loop resulting from a non-Abelian point source.

Highlights

The seemingly abstract geometry of a quantum system’s eigenstates finds application in fields ranging from condensed matter and quantum information science to high-energy physics
Our manuscript is organized as follows: (1) we introduce the essential physics of the Wilson loop (WL), (2)
Our experiments demonstrated essentially the full set of highfidelity SU(2) holonomic control in a subspace that was protected against environmental noise and imperfections

Summary

Introduction

The seemingly abstract geometry of a quantum system’s eigenstates finds application in fields ranging from condensed matter and quantum information science to high-energy physics. The Berry phase is the direct analog to the Aharonov–Bohm phase for motion along a closed loop with the enclosed Berry curvature playing the role of a magnetic field. Conical intersections, singular points in the energy landscapes of a range of physical systems[5,8,9,10,11], where the curvature diverges, play a crucial role in geometric effects, since particles encircling the singular point can acquire non-zero Berry phase. The Wilczek–Zee (W.-Z.) phase[12] extends these ideas to include non-Abelian “operator-valued” geometric phases possible for adiabatically evolving systems with a degenerate subspace (DS). Despite the universality of Wilczek and Zee’s concept and the tremendous theoretical and experimental interest in synthetic non-Abelian gauge fields[19,21,22,23,24,25,26,27,28,29,30,31], there has been no realistic cold-atom scheme for robust control of non-

Methods

Results

Conclusion