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

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