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

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.

Measuring Quantum Geometric Tensor of Non-Abelian System in Superconducting Circuits

“…[39][40][41][42] Furthermore, the Wilczek-Zee connection, which leads to the Holonomic properties in the generated system, allows the geometric gate to be widely used in quantum circuits. [43][44][45][46] This subject is extensively and thoroughly studied for the optimization of quantum gates. [47][48][49][50] Therefore, measuring the generalized non-Abelian quantum geometric tensor has theoretical and practical significance.…”