Wilson loop

A story on: Wilson loop and Wilczek-Zee phase from a non-Abelian gauge field (npj Quantum Information volume 7, Article number: 144 (2021))

Background

Berry phases, an example of geometric phases which only depend on the traversed path in some parameter space, are analogous to the Aharonov-Bohm phase from the electromagnetic vector potential. Berry phases are scalars, and can be related to an Abelian gauge potential called Berry connection. For a single quantum state far from degenerate subspaces, the Berry phase and Berry connection are a manifestation of the U(1) gauge field living in the parameter space of the state evolution. But what if we had a degenerate subspace? Then, other gauge fields may give rise to other analogous geometric and topological properties.

Experiment

In this paper led by S. Sugawa, we characterize the Wilczek-Zee phase (a matrix-valued, and in general non-Abelian, phase term in quantum states) in terms of the Wilson loop, a quantity depending on the solid angle subtended by an encircling (loop) path in the (wavefunction’s) parameter space; a kind of generalization of Stokes’ theorem. Using the degenerate subspaces of a dressed Bose-Einstein condensate (BEC) spin states, we drive a set of initial states through nearly adiabatic evolution and perform quantum state tomography using projective measurements to reconstruct the non-Abelian Wilczek-Zee phase. Then, we evaluate the Wilson loop along a closed trajectory.

Key finding

This experiment marks the observation of a Wilson loop resulting from a non-Abelian point source giving rise to an SU(2) gauge field in the parameter space of our BEC’s internal states.