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The European Physical Journal D

Evolution of Grassmannian invariant-angle coherent states and nonadiabatic Hannay's angle

Mustapha Maamache, Omar Cherbal

Published May 1, 1999• 4 citations
Publisher
DOI
10.1007/s100530050294

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The ``geometrical phase factor,'' found by Berry in his study of the quantum adiabatic theorem, has a classical analogue discovered recently by Hannay and Berry himself. This classical analogue consists of an extra angle swept by the system in its classical evolution during an adiabatic excursion. As in Berry's phase, Hannay's angle reflects a nontrivial holonomy of the system and it has a purely geometric character. In this paper we calculate Hannay's angles for the classical Grassmannian system associated to a quantum spin in a magnetic field. We also show their close relation to the corresponding quantum-mechanical phase of Berry for the same system.

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We show how to derive the Hannay angles of Grassmannian classical mechanics from the evolution of Grassmannian action—angle quantum states. Just as in the commutative case, this evolution defines a geometric transport, which can also be obtained from a quantum canonical transformation or a variational principle. As examples, we explicitly construct the quantum states for the classical counterparts of a first-and second-quantized N-level system. In the latter case, these states reduce to standard fermionic coherent states and the classical Hannay angles coincide with the quantum Berry phases.
The ``geometrical phase factor,'' found by Berry in his study of the quantum adiabatic theorem, has a classical analogue discovered recently by Hannay and Berry himself. This classical analogue consists of an extra angle swept by the system in its classical evolution during an adiabatic excursion. As in Berry's phase, Hannay's angle reflects a nontrivial holonomy of the system and it has a purely geometric character. In this paper we calculate Hannay's angles for the classical Grassmannian system associated to a quantum spin in a magnetic field. We also show their close relation to the corresponding quantum-mechanical phase of Berry for the same system.