Geometry of 2-Fold Degenerated 2-Level System
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It is well-known  that during the cyclic adiabatic evolution multi-level (quantum) system acquires the geometric phase factor, or Berry’s phase. B.Simon has shown  that it is precisely the holonomy in a Hermitian line bundle since the adiabatic theorem naturally defines a connection in such a bundle. In the case of degenerated systems this factor is non-Abelian .