**Published: **

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. Introduction. In 1984, M. V. Berry published the remarkable discovery that cyclic parallel transport of quantum states causes the appearance of geometrical phases factors [1]. His discovery, along with precursor works [2, 3], unified seemingly different phenomena within the framework of gauge theories [4, 5]. This seminal work was rapidly generalized to non-adiabatic and noncyclic evolutions [5] and, most saliently for our concern here, to degenerate states by F. Wilczek and A. Zee [6]. In this case, the underlying symmetry of the degenerate sub-space leads to a non-Abelian gauge field structure. These early works on topology in quantum physics have opened up tremendous interest in condensed matter [7--11] and more recently in ultracold gases [12--20] and photonic devices [21--23]. Moreover, it has been noted that geometrical qubits are resilient to certain noises, making them potential candidates for fault-tolerant quantum computing [24-- 27]. So far, beside some recent proposals [28, 29], experimental implementations have been performed for a 2-qubit gate on NV-centers in diamond [30] and for a non-Abelian single qubit gate in superconducting circuits [31]. These experiments were performed following a non-adiabatic protocol allowing for high-speed manipulation [29, 32, 33]. Recently, coherent control of ultracold spin-1 atoms confined in optical dipole traps was used to study the geometric phases associated with singular loops in a quantum system [34]. If non-adiabatic manipulation are promising methods for quantum computing, they prevent the study of external dynamic of quantum system in a non-abelian gauge field, where non-trivial coupling occurs between the internal qubit state and the center-of-mass wave function of the particle. We report here on non-Abelian adiabatic geometric transformations implemented on a noninteracting cold