Disentangling theorem and scattering functions for long-range coherent tunneling

Abstract

A closed form of the disentangling theorem is used to derive an exact expression for the quantum mechanical intermediate scattering function describing long-range coherent quantum tunneling. The result applies to a single particle in a large periodic nearest-neighbor tight-binding system in one spatial dimension, with one localized site per unit cell. The result is exact up to the assumption of orthogonal localized states and the substitution of the coordinate operator with a discrete representation diagonal in the on-site basis. The intermediate scattering function is expressed in terms of modified Bessel functions, and consists of a symmetric real part and antisymmetric imaginary part. The real and imaginary parts both exhibit decaying oscillations reflecting the oscillatory dynamics among neighboring sites combined with the long-term spreading of the wave function from any initial site. The imaginary part is significant only when the thermal energy is comparable to or smaller than the width of the tight-binding energy band, and represents quantum recoil or the asymmetry of energy exchange probability in quasielastic scattering from the coherent system. The one-dimensional result is extended, in the form of $\mathbf{k}$-space integrals, to describe the coherent tunneling dynamics in hexagonal and honeycomb systems. The prospects for observing the phenomenology of the analytical line shapes are discussed with respect to the practical implementation of helium-3 surface spin echo spectroscopy.