It is generally admitted that in quantum mechanics, the electromagnetic potentials have physical interpretations otherwise absent in classical physics as illustrated by the Aharonov-Bohm effect. In 1984, Berry interpreted this effect as a geometrical phase factor. The same year, Wilczek and Zee generalized the concept of Berry phases to degenerate levels and showed that a non-Abelian gauge field arises in these systems. In sharp contrast with the Abelian case, spatially uniform non-Abelian gauge fields can induce particle noninertial motion. We explore this intriguing phenomenon with a degenerated Fermionic atomic gas subject to a two-dimensional synthetic SU(2) non-Abelian gauge field. We reveal the spin Hall nature of the noninertial dynamic as well as its anisotropy in amplitude and frequency due to the spin texture of the system. We finally draw the similarities and differences of the observed wave packet dynamic and the celebrated Zitterbewegung effect of the relativistic Dirac equation.

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Experimental observation of Zitterbewegung is considered a holy grail of the modern physics since its debut in 1930 by Schrödinger. In this thesis we describe the first experimental observation of this effect in a two-dimensional ultracold atomic wave packet. In addition to that, we have also revealed the anisotropic of nature of Zitterbewegung, in the presence of non-Abelian Gauge field.To all my teachers who have prepared me for this adventurous journey of a lifelong student List of Figures2.1 A typical way to realize the Abelian gauge field via Raman scheme. Here L 1 and L 2 each addresses the transition |↑ → |e and |↓ → |e , respectively. The two-photon detuning is δ while both lasers L 1,2 are far off-detuned from the excited-state |e . . . . . . . . . . . . . . . . 2.2 Dispersion relation of a one-dimensional spin-orbit coupled system given by the Hamiltonian in Eq. 2.4. The left panel describes two shifted parabolas when the Raman coupling Ω and two-photondetuning δ are zero. In the middle panel, a non zero Raman coupling opens the gap at zero momentum since this term behaves like a Zeeman term in the Hamiltonian given by Eq. 2.4. The right panel shows the effect of non-zero two-photon detuning δ. The dispersion relation becomes tilted as δ acts as a constant shift in momentum in Eq.