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arXiv:quant-ph/9911083 · Physical Review Letters

Off-Diagonal Geometric Phases

Nicola Manini, F. Pistolesi

Published Oct 9, 2000• 135 citations
arXivHTMLPDF

Abstract

We investigate the adiabatic evolution of a set of nondegenerate eigenstates of a parametrized Hamiltonian. Their relative phase change can be related to geometric measurable quantities that extend the familiar concept of Berry phase to the evolution of more than one state. We present several physical systems where these concepts can be applied, including an experiment on microwave cavities for which off-diagonal phases can be determined from published data.

DOI
10.1103/physrevlett.85.3067

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We investigate the adiabatic evolution of a set of nondegenerate eigenstates of a parametrized Hamiltonian. Their relative phase change can be related to geometric measurable quantities that extend the familiar concept of Berry phase to the evolution of more than one state. We present several physical systems where these concepts can be applied, including an experiment on microwave cavities for which off-diagonal phases can be determined from published data.
A geometric phase is found for the adiabatic evolution of a general quantum state;it reduces to Berry phase when applied to eigenstates. This new phase applies to both linear quantum systems and nonlinear quantum systems,and should be particularly useful for the latter for which the adiabatic evolution of a general quantum state cannot be described in terms of a superposition of eigenstates.For linear quantum systems,our new phase becomes a statistical average of the Berry phases for all eigenstates involved.
We unveil that even an infinitesimal deviation of the initial state from the eigenstates of the initial Hamiltonian can yield a significant change in the geometric phase accompanying adiabatic evolution without degeneracies of the energy levels. This leads to the counterintuitive result that the values of two geometric phases computed from the general definition of the Berry phase [Samuel and Bhandari, Phys. Rev. Lett. 60, 2339 (1988)] can dramatically differ from each other in the adiabatic limit even if their evolution paths are sufficiently close in a pointwise manner. The result of this work indicates that the widely accepted theory of geometric phases is not very consistent in the context of adiabatic evolution involving superposed eigenstates.
One milestone in quantum physics is Berry's seminal work [Proc.~R.~Soc.~Lond.~A \textbf{392}, 45 (1984)], in which a quantal phase factor known as geometric phase was discovered to solely depend on the evolution path in state space. Here, we unveil that even an infinitesimal deviation of the initial state from the eigenstate of the initial Hamiltonian can yield a significant change of the geometric phase accompanying an adiabatic evolution. This leads to the surprising observation that two geometric phases can dramatically differ from each other even if their evolution paths are sufficiently close in a pointwise manner.
In this paper, we have investigated the off-diagonal Berry phase of nonlinear systems and presented its explicit expression. The results show that, for nonlinear systems, the off-diagonal berry phase contains a new term in addition to the dynamical phase, the geometric phase and the nonlinear phase. This new term can describe a cross effect between the Bogoliubov excitation around one eigenstate and another instantaneous eigenstate, while the Bogoliubov excitations are found to be accumulated during the adiabatic evolution and contribute a finite phase of geometric nature. As an application, the off-diagonal Berry phase of a two-well trapped Bose-Einstein condensate system is calculated.
The concept of a geometric phase (Berry's phase) is generalized to the case of noneigenstates, which is applicable to both linear and nonlinear quantum systems. This is particularly important to nonlinear quantum systems, where, due to the lack of the superposition principle, the adiabatic evolution of a general state cannot be described in terms of eigenstates. For linear quantum systems, our new geometric phase reduces to a statistical average of Berry's phases. Our results are demonstrated with a nonlinear two-level model.
The background and evolvement of phase problem in quantum mechanics is overviewed.Berry phase which was brought up originally by Berry M.V.for adiabatic evolution in linear systems is limited in practical applications.Berry phase is generalized to nonadiabatic evolution(AA phase),noncyclic quantum evolution(off-diagonal geometric phase)and nonlinear systems(geometric phase of general quantum states).It is pointed out that the off-diagonal geometric phase in nonlinear system is a new area for research.The prospect of this subject is viewed as well.