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Observation and characterization of a non-Abelian gauge field's Wilczek-Zee phase by the Wilson loop

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

Published Oct 31, 2019• 4 citations
OpenAlex

Abstract

Quantum states can acquire a phase called the Berry phase after adiabatically traversing a closed loop in parameter space. This phase is geometric -- dependent on the path -- not the rate of motion. Wilczek and Zee extended this concept to include non-Abelian phases that can be characterized the Wilson loop, a gauge independent quantity. Here we quantum-engineer a non-Abelian SU(2) gauge field for an atomic Bose-Einstein Condensate (BEC), in a 5-dimensional synthetic parameter space derived from internal atomic degrees of freedom. By slowly encircling a topological monopole, we observed the adiabatic Wilczek-Zee phase that we characterize in terms of the Wilson loop.

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Quantum states can acquire a phase called the Berry phase after adiabatically traversing a closed loop in parameter space. This phase is geometric -- dependent on the path -- not the rate of motion. Wilczek and Zee extended this concept to include non-Abelian phases that can be characterized the Wilson loop, a gauge independent quantity. Here we quantum-engineer a non-Abelian SU(2) gauge field for an atomic Bose-Einstein Condensate (BEC), in a 5-dimensional synthetic parameter space derived from internal atomic degrees of freedom. By slowly encircling a topological monopole, we observed the adiabatic Wilczek-Zee phase that we characterize in terms of the Wilson loop.
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.
Non-Abelian gauge fields play a key role in describing the behavior of particles whose motion is coupled to internal degrees of freedom, such as their spin. Here, we experimentally realize a tuneable non-Abelian gauge field in an active mechanical lattice by using pairs of oscillators to encode a local pseudo-spin for each site, with inter-site spin-dependent couplings engineered via real-time measurement and feedback. We experimentally extract Wilson-loop observables in our set-up and hence demonstrate that we can create a genuinely non-Abelian gauge field. We then exploit the controllability of our mechanical lattice to engineer non-reciprocal hoppings to explore non-Hermitian non-Abelian gauge potentials. For a two-dimensional (2D) lattice, we demonstrate that the non-Hermiticity can manifest in direction-dependent Wilson loops for a single plaquette, while for a one-dimensional (1D) system, we show that a non-Abelian gauge potential can switch the localization of non-Hermitian skin modes between opposite ends of a chain. Our work establishes active mechanical lattices as a flexible and programmable platform for probing non-Abelian gauge fields and exploring their interplay with non-Hermitian dynamics.
Emergence of fundamental forces from gauge symmetry is among our most profound insights about the physical universe. In nature, such symmetries remain hidden in the space of internal degrees of freedom of subatomic particles. Here we propose a way to realize and study gauge structures in real space, manifest in external degrees of freedom of quantum states. We present a model based on a ring-shaped lattice potential, which allows for both Abelian and non-Abelian constructs. Non trivial Wilson loops are shown possible via physical motion of the system. The underlying physics is based on the close analogy of geometric phase with gauge potentials that has been utilized to create synthetic gauge fields with internal states of ultracold atoms. By scaling up to an array with spatially varying parameters, a discrete gauge field can be realized in position space, and its dynamics mapped over macroscopic size and time scales.
The method of synthetic gauge potentials opens up a new avenue for our understanding and discovering novel quantum states of matter. We investigate the topological quantum phase transition of Fermi gases trapped in a honeycomb lattice in the presence of a synthetic non-Abelian gauge potential. We develop a systematic fermionic effective field theory to describe a topological quantum phase transition tuned by the non-Abelian gauge potential and explore its various important experimental consequences. Numerical calculations on lattice scales are performed to compare with the results achieved by the fermionic effective field theory. Several possible experimental detection methods of topological quantum phase transition are proposed. In contrast to condensed matter experiments where only gauge invariant quantities can be measured, both gauge invariant and non-gauge invariant quantities can be measured by experimentally generating various non-Abelian gauges corresponding to the same set of Wilson loops.
We study Wilson loops as a necessary tool for unambiguous identification of non-Abelian synthetic gauge fields, with attention to certain crucial but often overlooked features, such as the requirement of at least three distinct loops. We devise a method to determine the complete Wilson loop matrix from the time-evolved amplitudes of the internal atomic states of laser-coupled ultracold atoms that does not require lattice confinement. The analysis is done in the context of a cyclic model that can realize both Abelian and non-Abelian structures within a single configuration with continuous variation possible between U(1) and U(2) gauge groups by varying the detuning of the laser fields.
Topological order is often quantified in terms of Chern numbers, each of which classifies a topological singularity. Here, inspired by concepts from high-energy physics, we use quantum simulation based on the spin degrees of freedom of atomic Bose-Einstein condensates to characterize a singularity present in five-dimensional non-Abelian gauge theories-a Yang monopole. We quantify the monopole in terms of Chern numbers measured on enclosing manifolds: Whereas the well-known first Chern number vanishes, the second Chern number does not. By displacing the manifold, we induce and observe a topological transition, where the topology of the manifold changes to a trivial state.