Bargmann invariants and off-diagonal geometric phases for multilevel quantum systems: A unitary-group approach

We investigate the geometric phases and the Bargmann invariants associated with multilevel quantum systems. In particular, we show that a full set of ‘‘gauge-invariant’’ objects for an n-level system consists of n geometric phases and 1/2 (n-1)(n-2) algebraically independent four-vertex Bargmann invariants. In the process of establishing this result, we develop a canonical form for U(n) matrices that is useful in its own right. We show that the recently discovered ‘‘off-diagonal’’ geometric phases [N. Manini and F. Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analyzed in terms of the basic building blocks developed in this work. This result liberates the off-diagonal phases from the assumption of adiabaticity used in arriving at them