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Topological phases of matter became a new standard to classify quantumsystems in many cases, yet key quantities like the quantum geometric tensorproviding local information about topological properties are stillexperimentally hard to access. In non-Abelian systems this accessibility togeometric properties can be even more restrictive due to the degeneracy of thestates. We propose universal protocols to determine quantum geometricproperties in non-Abelian systems. First, we show that for a weak resonantdriving of the local parameters the coherent Rabi oscillations are related tothe quantum geometric tensor. Second, we derive that in a Landau-Zener liketransition the final probability of an avoided energy crossing is proportionalto elements of the non-Abelian quantum geometric tensor. Our schemes suggest away to prepare eigenstates of the quantum metric, a task that is difficultotherwise in a degenerate subspace.

Geometrical Rabi oscillations and Landau-Zener transitions in non-Abelian systems