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 $\frac{1}{2}\left(n-1\right)\left(n-2\right)$ algebraically independent four-vertex Bargmann invariants. In the process of establishing this result, we develop a canonical form for $\mathrm{U}\mathrm{}\left(n\right)\mathrm{}$ 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.