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Chinese Physics B

New method for calculating the Berry connection in atom—molecule systems

Fu-Cheng Cui, Bao-Jun Wu

Published Nov 1, 2012• 0 citations
Publisher

Abstract

In the mean-field theory of atom—molecule systems, where the bosonic atoms combine to form molecules, there is no usual U(1) symmetry, which presents an apparent hurdle for calculating the Berry connection in these systems. We develop a perturbation expansion method of Hannay's angle suitable for calculating the Berry curvature in the atom—molecule systems. With this Berry curvature, the Berry connection can be computed naturally. We use a three-level atom—molecule system to illustrate our results. In particular, with this method, we compute the curvature for Hannay's angle analytically, and compare it to the Berry curvature obtained with the second-quantized model of the same system. An excellent agreement is found, indicating the validity of our method.

DOI
10.1088/1674-1056/21/11/110303

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In the mean-field theory of atom—molecule systems, where the bosonic atoms combine to form molecules, there is no usual U(1) symmetry, which presents an apparent hurdle for calculating the Berry connection in these systems. We develop a perturbation expansion method of Hannay's angle suitable for calculating the Berry curvature in the atom—molecule systems. With this Berry curvature, the Berry connection can be computed naturally. We use a three-level atom—molecule system to illustrate our results. In particular, with this method, we compute the curvature for Hannay's angle analytically, and compare it to the Berry curvature obtained with the second-quantized model of the same system. An excellent agreement is found, indicating the validity of our method.
In the mean-field theory of atom-molecule systems,where the bosonic atoms combine to form molecules,there is no usual U(1) symmetry,which presents an apparent hurdle for calculating the Berry connection in these systems.We develop a perturbation expansion method of Hannay’s angle suitable for calculating the Berry curvature in the atom-molecule systems.With this Berry curvature,the Berry connection can be computed naturally.We use a three-level atom-molecule system to illustrate our results.In particular,with this method,we compute the curvature for Hannay’s angle analytically,and compare it to the Berry curvature obtained with the second-quantized model of the same system.An excellent agreement is found,indicating the validity of our method.
In the mean-field theory of atom-molecule systems, where bosonic atoms combine to form molecules, there is no usual U(1)\text{U}(1)U(1) symmetry, presenting an apparent hurdle for defining the Berry phase and Berry curvature for these systems. We define a Berry connection for this system, with which the Berry phase and Berry curvature can be naturally computed. We use a three-level atom-molecule system to illustrate our results. In particular, we have computed the mean-field Berry curvature of this system analytically, and compared it to the Berry curvature computed with the second-quantized model of the same system. An excellent agreement is found, indicating the validity of our definition.
In the present paper, we investigate the Berry phase and the Hannay angle of an interacting two-mode boson system and obtain their analytic expressions in explicit forms. The relation between the Berry phase and the Hannay angle is discussed. We find that, in the large-particle-number limit, the classical Hannay angle equals the particle number derivative of the quantum Berry phase except for a sign. This relationship is applicable to other many-body boson systems where the coherent-state description is available and the total particle number is conserved. The measurement of the classical Hannay angle in the many-body systems is briefly discussed as well.
The Berry phase, which was discovered more than two decades ago, provides very deep insight into the geometric structure of quantum mechanics. Its classical counterpart, Hannay's angle, is defined if closed curves of action variables return to the same curves in phase space after a time evolution. In this paper we study the Berry phase and Hannay's angle in a quantum-classical hybrid system under the Born-Oppenheimer approximation. By the term quantum-classical hybrid system, we mean a composite system consists of a quantum subsystem and a classical subsystem. The effects of subsystem-subsystem couplings on the Berry phase and Hannay's angle are explored. The results show that the Berry phase has been changed sharply by the couplings, whereas the couplings have a small effect on the Hannay's angle.