  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.
