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Published September 30, 2021 | Submitted + Published
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Wilson loop and Wilczek-Zee phase from a non-Abelian gauge field


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.

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© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Received 26 February 2021; Accepted 01 September 2021; Published 30 September 2021. We acknowledge the support for this work provided by the AFOSRs Quantum Matter MURI, NIST, and the NSF through the PFC at JQI. S.S. acknowledges support from JST, PRESTO (JPMJPR1664), and JSPS (fellowship for research abroad). We thank Mingwu Lu and Chris Billington for carefully reading our manuscript. Data availability: The data are available from the corresponding author upon reasonable request. Author Contributions: S.S., F.S.-C., Y.Y., and A.P. measured the data. S.S. conceived the experiment and analyzed the data. I.B.S. supervised the project. All the authors contributed to the discussion and preparation of the manuscript. The authors declare no competing interests.

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