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Physical Review A

Quantal phase factors accompanying adiabatic changes in the case of continuous spectra

Mustapha Maamache, Y. Saadi

Published Nov 12, 2008• 21 citations
Publisher

Abstract

By defining ``a virtual gap'' for the continuous spectrum through the notion of eigendifferential (Weyl's packet) and using the differential projector operator, we present a rigorous demonstration and discussion of the quantum adiabatic theorem and the validity condition of the adiabatic approximation for systems having a nondegenerate continuous spectrum. We show that a quantal system in an eigenstate, of operators with a continuous nondegenerate eigenvalue spectrum, slowly transported around a closed curve $C$ by varying parameters in its Hamiltonian, will acquire a generalized geometrical phase factor where an explicit formula is derived in terms of the eigenstates of the Hamiltonian. Examples are given for illustration.

DOI
10.1103/physreva.78.052109

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By defining ``a virtual gap'' for the continuous spectrum through the notion of eigendifferential (Weyl's packet) and using the differential projector operator, we present a rigorous demonstration and discussion of the quantum adiabatic theorem and the validity condition of the adiabatic approximation for systems having a nondegenerate continuous spectrum. We show that a quantal system in an eigenstate, of operators with a continuous nondegenerate eigenvalue spectrum, slowly transported around a closed curve CCC by varying parameters in its Hamiltonian, will acquire a generalized geometrical phase factor where an explicit formula is derived in terms of the eigenstates of the Hamiltonian. Examples are given for illustration.
Abstract A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp{iγ(C)} in addition to the familiar dynamical phase factor. An explicit general formula for γ(C) is derived in terms of the spectrum and eigenstates of Ĥ(R) over a surface spanning C. If C lies near a degeneracy of Ĥ, γ(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration γ(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.
The dynamics of a nonstationary quantum system whose Hamiltonian explicitly depends on time is called adiabatic if a system state that is an eigenstate of the Hamiltonian at the initial instant of time remains close to this eigenstate throughout the evolution. The degree of such closeness depends on the smallness of the parameter that determines the rate of change of the Hamiltonian. It is usually believed that one of the factors playing a decisive role for the stability of the adiabatic dynamics is the structure of the spectrum of the Hamiltonian. As the quantum adiabatic theorem states in its usual formulation, deviations from the adiabatic evolution can be estimated from above by the ratio of the rate of change of the Hamiltonian to the minimum distance between the energy of the state that approximates the adiabatic dynamics and the rest of the spectrum of the Hamiltonian. We analyze this dependence and prove theorems showing that the efficiency of the adiabatic approximation is more influenced by the rate of change of the Hamiltonian eigenvectors than by the dynamics of the spectrum. In a vast majority of physically meaningful cases, it turns out that controlling the dynamics of eigenvectors is sufficient for ensuring the adiabaticity, regardless of the dynamics of the spectrum as such.
We prove the adiabatic theorem for quantum evolution without the traditional gap condition. We show that the theorem holds essentially in all cases where it can be formulated. In particular, our result implies that the adiabatic theorem holds also for eigenvalues embedded in the continuous spectrum. If there is information on the Hölder continuity of the spectral measure, then one can also estimate the rate at which the adiabatic limit is approached. The adiabatic theorem of Quantum Mechanics describes the long time behavior of the solutions of an initial value problem where the Hamiltonian generating the evolution depends slowly in time. Traditionally, the theorem is stated for Hamiltonians that have an eigenvalue which is separated by a gap from the rest of the spectrum. Folk wisdom is that a gap condition is a sine qua non for the adiabatic theorem to hold. In particular, according to this folk wisdom, one does not expect a general adiabatic theorem for Hamiltonians that have an eigenvalue embedded in, say, the continuous spectrum. Our purpose is to show that this folk wisdom is wrong, and there is a general adiabatic theorem without a gap condition. All one really needs for the adiabatic theorem is a spectral projection for the Hamiltonian that depends smoothly on time.
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.