Berry phase effects in two-level and three-level atoms

Group-theoretical methods are developed for treating Berry phase effects, which are related to Cartan subalgebra. The theory is applied to two-level and three-level atoms interacting with perturbations that are described by the SU(2) or SU(3) algebra. By using fiber-bundle theories, it is found that a time development operator that depends on Cartan group generators can represent a fiber while a time development operator that depends on other generators of the group represents the base of the quantum manifold. The total time development operator is obtained by multiplication of these two parts and the fiber-bundle theory is applied for calculating Berry phase effects. Explicit expressions for Berry phases are obtained under the adiabatic approximation.