Bargmann invariants and off-diagonal geometric phases for multi-level quantum systems -- a unitary group approach

We investigate the geometric phases and the Bargmann invariants associated with a multi-level 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 4-vertex Bargmann invariants. In the process of establishing this result we develop a canonical form for U(n) matrices which 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 analysed 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.