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We have constructed the geometric phases emerging from the non-trivialtopology of a space-dependent magnetic field, interacting with the spinmagnetic moment of a neutral particle. Our basic tool is the local unitarytransformation which recasts the magnetic spin interaction under a diagonalform. Rewriting the kinetic term in the "rotated" frame requires theintroduction of non-Abelian covariant derivatives, involving the gradients ofthe Euler angles which define the orientation of the local field. Within therotated frame, we have built a perturbation scheme,assuming that thelongitudinal non-Abelian field component dominates the transverse ones, to beevaluated to second-order. The geometry embedded in the longitudinal gaugevector field and its curl, the geometric magnetic field, is described by theassociated Aharonov-Bohm phase. As an illustration, we study the physics ofcold 171Yb atoms dressed by two sets of circularly polarized beams, formingsquare or triangular 2D optical lattices. The geometric field is computedexplicitly from the Euler angles. The magnitude of 2nd-order corrections due totransverse fields can be reduced to the percent level by a choice of lightintensity which keeps the dressed atom loss rate below 5 s^{-1}. An auxiliaryoptical lattice confines the atoms within 2D domains where the geometric fieldis pointing upward.

journal of physics. a, mathematical and theoretical

Geometric Phases generated by the non-trivial spatial topology of static  vector fields coupled to a neutral spin-endowed particle. Application to  171Yb atoms trapped in a 2D optical lattice