Measuring second Chern number from non-adiabatic effects

TL;DR

This paper proposes a dynamical method to measure the non-Abelian Berry curvature and second Chern number in quantum systems, enabling experimental access to higher-dimensional topological invariants.

Contribution

It introduces a technique to measure the second Chern number via non-adiabatic effects, applicable to various physical systems like qubits, ions, and cold atoms.

Findings

Demonstrates measurement of non-Abelian Berry curvature components

Shows how to extract the second Chern number from dynamical data

Applicable to realistic quantum systems such as superconducting qubits

Abstract

The geometry and topology of quantum systems have deep connections to quantum dynamics. In this paper, I show how to measure the non-Abelian Berry curvature and its related topological invariant, the second Chern number, using dynamical techniques. The second Chern number is the defining topological characteristic of the four-dimensional generalization of the quantum Hall effect and has relevance in systems from three-dimensional topological insulators to Yang-Mills field theory. I illustrate its measurement using the simple example of a spin-3/2 particle in an electric quadrupole field. I show how one can dynamically measure diagonal components of the Berry curvature in an over-complete basis of the degenerate ground state space and use this to extract the full non-Abelian Berry curvature. I also show that one can accomplish the same ideas by stochastically averaging over random initial states in the degenerate ground state manifold. Finally I show how this system can be manufactured and the topological invariant measured in a variety of realistic systems, from superconducting qubits to trapped ions and cold atoms.