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Geometric phases of two- and three-level atomic systems related to group theory

The Hamiltonians of two- and three-level atomic systems interacting with nearly resonant electro-magnetic fields are described. The geometric phase related to the σ z operator of a two-level system is analysed and it is shown that this phase is well defined also for an open circuit. For a closed circuit this phase becomes equivalent to twice the Berry phase. For three-level atomic system the eigenenergies of the Hamiltonian are obtained by solving a cubic equation. Simplified forms for these eigenenergies and a condition for degeneracy are obtained. For non-degenerate three-level system geometric phases related to Cartan subalgebra are analysed under the adiabatic approximation and their relations with the three-level Berry phases are clarified. We show how to use the general theory of non-Abelian connections to the Hamiltonian of a three-level atomic system which includes a degeneracy.