Extracting non-Abelian quantum metric tensor and its related Chern numbers

TL;DR

This paper proposes a universal protocol to experimentally measure the non-Abelian quantum metric tensor in degenerate quantum systems, enabling the extraction of topological invariants like Chern numbers in higher-dimensional parameter spaces.

Contribution

It introduces a practical scheme to directly measure all components of the non-Abelian quantum metric tensor using transition probabilities after parameter quenches, applicable in any dimensional parameter space.

Findings

Protocol successfully extracts non-Abelian quantum metric tensor components.

Measurement of the quantum metric yields real Chern and second Chern numbers.

Numerical simulations confirm the feasibility of the proposed scheme.

Abstract

The complete geometry of quantum states in parameter space is characterized by the quantum geometric tensor, which contains the quantum metric and Berry curvature as the real and imaginary parts, respectively. When the quantum states are degenerate, the quantum metric and Berry curvature take non-Abelian forms. The non-Abelian (Abelian) Berry curvature and Abelian quantum metric have been experimentally measured. However, an experimentally feasible scheme to extract all the components of the non-Abelian quantum metric tensor is still lacking. Here we propose a generic protocol to directly extract the non-Abelian quantum metric tensor in arbitrary degenerate quantum states in any dimensional parameter space, based on measuring the transition probabilities after parameter quenches. Furthermore, we show that the non-Abelian quantum metric can be measured to obtain the real Chern number of a generalized Dirac monopole and the second Chern number of a Yang monopole, which can be simulated in three and five-dimensional parameter space of artificial quantum systems, respectively. We also demonstrate the feasibility of our quench scheme for these two applications with numerical simulations.