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

Abstract

By defining "a virtual gap" for the continuous spectrum through the notion of eigendifferential ͑Weyl's packet͒ 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.

Non-adiabatic quantum pumping by a randomly moving quantum dot

“…The effective Berry phase here is φ k 0 (t) = k 0 γ(t). The same result one gets by applying the formula for the BP for a continuous eigenstate [29] which in our notations reads…”

“…(21) and Fig.4 that the survival amplitude indeed decays in time as t −1/2 as predicted by Eq. (29). What the latter formula fails to predict though are the oscillations of the corresponding Bessel function.…”

“…An important limiting case is that of the adiabatically slow motion of the barrier. When the change of position is adiabatic then at time t the scattering state is given by the corresponding moving eigenstate |k 0 up to a BP factor [23,29] |χ k 0 (t) = e i φ k 0 (t) |k 0 .…”

“…). (15) and correspond to unnormalized Weyl differentials [29,33] in the moving basis representation of the initial state and its mirror counterpart.…”

“…‡ Strictly speaking applying the corresponding formula from [29] to the kernel (12) gives rise also to an imaginary part of the phase. The reson for this is that in their derivation the authors of [29] assumed that the kernel A(k, k ′ ; 0) is always diagonal which is clearly not the case for (12). =γ(t) e i(E(k)−E(k 0 ))t/ F (k, k 0 , e ikγ(t) , e ik 0 γ(t) ) +G(k, k 0 , e ikγ(t) , e ik 0 γ(t) ) with the functions F (k, k 0 , e ikγ(t) , e ik 0 γ(t) ), G(k, k 0 , e ikγ(t) , e ik 0 γ(t) ) andF (k, k 0 , e ikγ(t) , e ik 0 γ(t) ), G(k, k 0 , e ikγ(t) , e ik 0 γ(t) ) obtained by integrating the right hand of Eq.…”