  Topological phases of matter became a new standard to classify quantum
systems in many cases, yet key quantities like the quantum geometric tensor
providing local information about topological properties are still
experimentally hard to access. In non-Abelian systems this accessibility to
geometric properties can be even more restrictive due to the degeneracy of the
states. We propose universal protocols to determine quantum geometric
properties in non-Abelian systems. First, we show that for a weak resonant
driving of the local parameters the coherent Rabi oscillations are related to
the quantum geometric tensor. Second, we derive that in a Landau-Zener like
transition the final probability of an avoided energy crossing is proportional
to elements of the non-Abelian quantum geometric tensor. Our schemes suggest a
way to prepare eigenstates of the quantum metric, a task that is difficult
otherwise in a degenerate subspace.

Quantum Physics;Mesoscale and Nanoscale Physics