Nonadiabatic Berry’s phase for a quantum system with a dynamical semisimple Lie group

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Abstract
The nonadiabatic Berry’s phase is investigated for a quantum system with a dynamical semisimple Lie group within the framework of the generalized cranking approach. An expression for nonadiabatic Berry’s phase is given, which indicates that the nonadiabatic Berry’s phase is related to the expectation value of Cartan operators along the cranking direction in group space, and that it depends on (i) the geometry of the group space, (ii) the time evolution ray generated by the Hamiltonian (i.e., by the dynamics) in some irreducible representation Hilbert space, and (iii) the cranking rate. The expression also provides a simple algorithm for calculating the nonadiabatic Berry’s phase. The general formalism is illustrated by examples of SU(2) dynamic group.

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