  Topology, geometry, and gauge fields play key roles in quantum physics as
exemplified by fundamental phenomena such as the Aharonov-Bohm effect, the
integer quantum Hall effect, the spin Hall, and topological insulators. The
concept of topological protection has also become a salient ingredient in many
schemes for quantum information processing and fault-tolerant quantum
computation. The physical properties of such systems crucially depend on the
symmetry group of the underlying holonomy. We study here a laser-cooled gas of
strontium atoms coupled to laser fields through a 4-level resonant tripod
scheme. By cycling the relative phases of the tripod beams, we realize
non-Abelian SU(2) geometrical transformations acting on the dark-states of the
system and demonstrate their non-Abelian character. We also reveal how the
gauge field imprinted on the atoms impact their internal state dynamics. It
leads to a new thermometry method based on the interferometric displacement of
atoms in the tripod beams.
