Yang monopole

A story on: Second Chern number of a quantum-simulated non-Abelian Yang monopole (Science 360 (6396), 1429-1434)

Background

Quantum simulation platforms study phenomena in analogous systems built from the bottom up; for example, an array of identical atoms placed in an external periodic potential may be used to simulate the properties of a crystalline material. The key property of such a system is that it allows for reconfiguration and a limited degree of control; so it may be possible to probe more complicated phases, or even drive it into more exotic ones that do not occur naturally.

Topological classification of matter on the other had uses the global properties of a class of systems which may extend beyond their symmetries and conservation laws. Topology has been widely employed both in condensed matter physics and high energy physics, a connection which is far from being a coincidence thanks to the underlying presence of gauge field theories describing emergent phases of matter.

Here, we explore an aspect of topological characterization of a simple quantum simulator using the tools of coherent quantum control. We draw an analogy to the effective field theory underlying the topological order and illustrate its properties using the language of gauge fields and high-order topology.

Experiment

We work with 4 of the 8 internal spin states in the hyperfine ground manifold of ultracold 87-Rb. After preparing an initial state superposition, we drive it coherently along a parametric trajectory (i.e. described by the parameters of an external field) such that the final state acquires matrix-valued geometric phases. These matrix-valued phase factors do not commute with each other (i.e. they are non-Abelian) and may be used to reveal the structure of a gauge field in the parameter space, as they relate to the “flux” of a geometric gauge field. Integrating the “flux” counts the number of “sources”, an integer number known as “second Chern number”. The importance of this number is that it characterizes the topology of the non-Abelian gauge field, finding more relevance in less abstract settings where the parameter space is related to the crystalline band structure of an exotic material.

Key finding

The second Chern number is measured for the first time, and the analogy with non-Abelian Yang fields sourced by a “magnetic” monopole is manifest in the experiment.