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Геометрическая фаза для когерентного состояния группы su(3)

Х. Х. Муминов, Ю. Я. Юсефи

Published Jan 1, 2014• 0 citations
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Abstract

In the frameworks of SU(3) coherent states approach the geometrical phase for quantum-mechanical S=1 spin systems is calculated, which generalized the concept of Berry phase due to taking into account excitations of quadrupole degrees of freedom of spin dynamics.

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We set the general formalism for calculating Berry's phase in quantum systems with Hamiltonian belonging to the algebra of a semisimple Lie group of any rank in the framework of generalized coherent states. Within this approach the geometric properties of Berry's connection are also studied, both in the Abelian and non-Abelian cases. In particular we call attention to the non-Abelian case where we make use of a vectorial generalization of coherent states. In this respect a thorough and self-contained exposition of the formalism of vector coherent states is given. The specific examples of the groups SU(3) and Sp(2) are worked out in detail.
Spin coherent states, proposed by several workers in the early 70s, are shown to contain the natural generalization to higher spins of the phase discovered earlier by Pancharatnam for polarized light. They also provide a simple setting in which the geometric phase of Berry and of Aharonov and Anandan can be exhibited and some of its further properties, noted by Samuel and Bhandari, can be illustrated. It is pointed out how many of the foregoing results might have been deduced in the 70s by proceeding in strict analogy with the earlier work of Pancharatnam. A similar point is made in connection with the geometric phase of SU(1,1) coherent states. In addition to its possible historical interest, this approach shows how the concept of the geometric phase can be introduced and developed in a largely kinematic fashion as an addendum to the standard quantum theory of angular momentum.
The quantum dynamics of time-dependent Hamiltonians of the kind H(t)=U(t)H(0)U(t)° is studied in the framework of generalized coherent states. It is shown that the quantum and the classical dynamics are isomorphic and that the phase of ‖ψ(t)〉 is simply the classical action. In adiabatic approximation it is easy to extract Berry’s phase from this. A discussion of generalized coherent states (CS’s) necessary to deal with excited states is given; in particular for SU(2), SU(1,1) and E(2) their structure is completely characterized, showing that the deep connection with geometric quantization extends to these generalized CS’s. This technique may be employed to estimate higher-order correction to the adiabatic approximation.
The exchange phase for two spins is studied here from the point of view of the quantization of a fermion in the framework of Nelson’s stochastic mechanics. This introduces a direction vector attached to a space–time point depicting the spin degrees of freedom. In this formalism, a fermion appears as a scalar particle attached with a magnetic-flux quantum, and a quantum spin can be described in terms of an SU(2) gauge bundle. This helps us to recast the Berry–Robbins formalism where the exchange phase appears as an unfamiliar geometric phase arising out of the ‘exchange rotation’ in a transported spin basis in terms of gauge currents. However, for polarized fermions, the exchange phase is found to be given by the Berry phase.