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

N. Mukunda·Arvind·S. Chaturvedi·R. Simon
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Abstract

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.

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N. Mukunda, Arvind, S. Chaturvedi, R. Simon. 2001-07-02. Bargmann invariants and off-diagonal geometric phases for multi-level quantum systems -- a unitary group approach. https://doi.org/10.1103/physreva.65.012102

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