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Journal of Physics A Mathematical and General

The Berry phase for a threefold degenerate state

Arnout Jozef Ceulemans, M. Szopa

Published Oct 7, 1991• 6 citations
Publisher

Abstract

Different expressions for calculating the Berry phase of adiabatic processes are reviewed and their limitations are discussed. These expressions are then applied to the case of a circuit surrounding a triple degeneracy. It is shown that the most general formula for the Berry phase requires the use of the full SU(3) invariance group of a T state coupled to in + tau 1 + tau 2 modes. The T*( in + tau 2 ) Jahn-Teller problem appears as a special case with SO(3) symmetry.

DOI
10.1088/0305-4470/24/19/014

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Different expressions for calculating the Berry phase of adiabatic processes are reviewed and their limitations are discussed. These expressions are then applied to the case of a circuit surrounding a triple degeneracy. It is shown that the most general formula for the Berry phase requires the use of the full SU(3) invariance group of a T state coupled to in + tau 1 + tau 2 modes. The T*( in + tau 2 ) Jahn-Teller problem appears as a special case with SO(3) symmetry.
A formalism is developed for calculating Berry phases for nonadiabatic time-periodic quantum systems when a dynamical invariant is known. It is found that, when the invariant is periodic and has a nondegenerate spectrum, this method allows a convenient way to obtain generalized Berry phases and the proper cyclic initial states. The method is applied to the generalized harmonic oscillator and the two-level system, where the invariant operators are explicitly constructed. Formulae for the conventional Berry's phases are readily obtained by taking the adiabatic limit of the exact results.
The Hamiltonians of two- and three-level atomic systems interacting with nearly resonant electro-magnetic fields are described. The geometric phase related to the σ z operator of a two-level system is analysed and it is shown that this phase is well defined also for an open circuit. For a closed circuit this phase becomes equivalent to twice the Berry phase. For three-level atomic system the eigenenergies of the Hamiltonian are obtained by solving a cubic equation. Simplified forms for these eigenenergies and a condition for degeneracy are obtained. For non-degenerate three-level system geometric phases related to Cartan subalgebra are analysed under the adiabatic approximation and their relations with the three-level Berry phases are clarified. We show how to use the general theory of non-Abelian connections to the Hamiltonian of a three-level atomic system which includes a degeneracy.