Correlated Tunnelling of Two Electrons through a Barrier in Quantum Wires

TL;DR

This paper investigates the quantum tunnelling of two correlated electrons through a potential barrier in one-dimensional wires, providing exact solutions for antisymmetric cases and methods for symmetric cases.

Contribution

It introduces a novel approach using Sommerfeld-Maluzhinets transformation and Bethe ansatz to solve correlated two-electron tunnelling problems.

Findings

Exact solution for antisymmetric subproblem.

Method for solving symmetric subproblem.

Reduction to independent planar scattering problems.

Abstract

We study the tunnelling trough a potential barrier of the system of two quantum correlated particles. The system is considered in one dimension. The interaction with the barrier and between particles is approximated by $\delta$-potentials. Assuming that the particles have the same masses, we reduce the problem to the set of independent planar scattering problems corresponding to two-body flux symmetry with respect to the barrier. In order to solve the problems we apply the method of Sommerfeld-Maluzhinets integral transformation, which requires the solution of a set of functional equations. We show that the sub-problem which is antisymmetric with respect to the center of mass allows the solution in frames of the Bethe ansatz. So we give the exact solution of this problem. For the subproblem which is symmetric we describe the class of solutions for the functional equations and give an approach for obtaining the solution.