We have constructed the geometric phases emerging from the non-trivial topology of a space-dependent magnetic field B(r), interacting with the spin magnetic moment of a neutral particle. Our basic tool, adapted from a previous work on Berry’s phases, is the space-dependent unitary transformation , which leads to the identity, , at each point r. In the ‘rotated’ Hamiltonian , is replaced by the non-Abelian covariant derivative where can be written as A1(r)Sx + A2(r)Sy + A3(r)Sz. The Abelian differentials Ak(r)·dr are given in terms of the Euler angles defining the orientation of B(r). The non-Abelian field transforms as a Yang–Mills field; however, its vanishing ‘curvature’ reveals its purely geometric character. We have defined a perturbation scheme based upon the assumption that in the longitudinal field A3(r) dominates the transverse field A1, 2(r) contributions, evaluated to second order. The geometry embedded in both the vector field A3(r) and the geometric magnetic field is described by their associated Aharonov–Bohm phase. As an illustration we study the physics of cold 171Yb atoms dressed by overlaying two circularly polarized stationary waves with orthogonal directions, which form a 2D square optical lattice. The frequency is tuned midway between the two hyperfine levels of the (6s6p)3P1 states to protect the optical B(r) field generated by the lattice from the dressed atom instability. The geometric field B3(r) is computed analytically in terms of the Euler angles. The magnitude of the second-order corrections due to the transverse fields can be reduced to the per cent level by a choice of light intensity that keeps the dressed atom loss rate ⩽5 s−1. A second optical lattice can be designed to confine the atoms inside 2D domains where . We extend our analysis to the case of a triangular lattice.
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