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Journal of Physics A Mathematical and Theoretical

Beyond the geometric phase: intrinsic fluctuation-induced ‘pollutions’ for adiabatic evolution of a nonlinear quantum state

Qi Zhang

Published Jul 4, 2012• 1 citation
Publisher

Abstract

The adiabatic geometric phase of a wavefunction whose dynamics is governed by the two-mode Gross–Pitaevskii equation is investigated. We show that, for a general state beyond eigenstate, the intrinsic dynamical fluctuations neglected by the conventional adiabatic theorem will inevitably give rise to phase factors which greatly depend on the details of the adiabatic process (adiabatic parameter evolution). Thus, contrary to the general belief that the total phase is only composed of the dynamical phase and geometric phase, intriguing intrinsic ‘pollution’ terms are discovered. An experimental scheme for detecting the ‘pollution’ terms based on optical waveguides is proposed.

DOI
10.1088/1751-8113/45/29/295302

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The adiabatic geometric phase of a wavefunction whose dynamics is governed by the two-mode Gross–Pitaevskii equation is investigated. We show that, for a general state beyond eigenstate, the intrinsic dynamical fluctuations neglected by the conventional adiabatic theorem will inevitably give rise to phase factors which greatly depend on the details of the adiabatic process (adiabatic parameter evolution). Thus, contrary to the general belief that the total phase is only composed of the dynamical phase and geometric phase, intriguing intrinsic ‘pollution’ terms are discovered. An experimental scheme for detecting the ‘pollution’ terms based on optical waveguides is proposed.
We investigate the geometric phase or Berry phase of adiabatic quantum evolution in the Bose-Einstein condensate (BEC) systems governed by nonlinear Gross-Pitaevskii(GP) equations. We study how this phase is modified by the nonlinearity and find that the Bogoliubov fluctuations around the eigenstates are accumulated during the nonlinear adiabatic evolution and contribute a finite phase of geometric nature. A two-mode BEC model is used to illustrate our theory. Our theory is applicable to other nonlinear systems such as paraxial wave equation for nonlinear optics and Ginzburg-Landau equations for complex order parameters in condensed-matter physics.
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.
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 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.
Abstract A generalization of the adiabatic theorem for quantum systems governed by pseudo-Hermitian Hamiltonians and details of its demonstration are given. Introducing a modified time-dependent metric giving a precise description of the quantum unitary evolution where the obtained effective Hamiltonian is observable because it is mean value is real. We show that an eigenstate of a pseudo-Hermitian Hamiltonian slowly transported will acquire a real generalized geometrical phase factor which contains two contributions: the first one corresponds to the conventional Berry’s phase as expected and a new geometrical term that we call the metric geometrical phase. We will apply our results in the cases of the famous time dependent brachistochrone problem and a non-Hermitian displaced harmonic oscillator.
The geometric phase is a common phenomenon in a variety of physical systems, with significant applications in optics. We report the first experimental demonstration of the adiabatic geometric phase in nonlinear frequency conversion, wherein the coupling between the signal and idler frequencies constitutes the intrinsic two-level dynamics of the system. We observe a variety of effects associated with the geometric phase, including the adiabatic broadening of bandwidth, asymmetric transmission for opposite propagation directions, conjugation of the phase for orthogonal eigenstates, and the nonreciprocity associated with the pump field bias. Our work paves the way towards all-optically controlled geometric phase elements for wavefront shaping, isolation, guiding, and quantum optical applications, harnessing spectral and spatial correlations.