# Optimal design strategy for non-Abelian geometric phases using Abelian gauge fields based on quantum metric

## TL;DR

This paper demonstrates how to realize non-Abelian geometric phases in photonic waveguides, using the quantum metric to optimize adiabatic evolution, and exemplifies this with a Wilson loop of an Abelian gauge field.

## Contribution

It introduces a method to implement non-Abelian geometric phases in photonic systems using Abelian gauge fields and quantum metrics for optimal adiabatic control.

## Key findings

- Realization of non-Abelian geometric phases in photonic waveguides.
- Use of quantum metric to define optimal adiabatic paths.
- Illustration of non-Abelian evolution via a Wilson loop.

## Abstract

Geometric phases, which are ubiquitous in quantum mechanics, are commonly more than only scalar quantities. Indeed, often they are matrix-valued objects that are connected with non-Abelian geometries. Here we show how generalized, non-Abelian geometric phases can be realized using electromagnetic waves travelling through coupled photonic waveguide structures. The waveguides implement an effective Hamiltonian possessing a degenerate dark subspace, in which an adiabatic evolution can occur. The associated quantum metric induces the notion of a geodesic that defines the optimal adiabatic evolution. We exemplify the non-Abelian evolution of an Abelian gauge field by a Wilson loop.