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Adiabatic geometric phase of a general quantum state

Biao Wu, Jie Liu

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
The Berry phase acquired by an eigenstate that experienced a nonlinear adiabatic evolution is investigated thoroughly. The circuit integral of the Berry connection of the instantaneous eigenstate cannot account for the adiabatic geometric phase, while the Bogoliubov excitations around the eigenstates are found to be accumulated during the nonlinear adiabatic evolution and contribute a finite phase of geometric nature. A two-mode model is used to illustrate our theory. Our theory is applicable to Bose-Einstein condensate, nonlinear light propagation, and Ginzburg-Landau equations for complex order parameters in condensed-matter physics.
In a quantum system initially in the n-th eigenstate, an adiabatic evolution of the Hamiltonian ensures that the system remains in the corresponding instantaneous eigenstate while acquiring a phase factor. This phase has two components: one resulting from standard time evolution and another associated with the dependence of the eigenstate on the varying Hamiltonian, known as the Berry phase. In this work, we explore the concept of geometric amplitudes in the context of a Hermitian Hamiltonian with imaginary eigenvalues. We introduce the notion of geometric amplitude and provide a novel derivation of this concept. Our study reveals that a system undergoing cyclic evolution under adiabatic conditions acquires an additional amplitude factor of purely geometric origin. To illustrate this idea, we apply it to a concrete case: a generalized harmonic oscillator.