Collision of two interacting electrons on a mesoscopic beamsplitter: exact solution in the classical limit

TL;DR

This paper models the classical dynamics of two interacting electrons in a mesoscopic beamsplitter, revealing how Coulomb repulsion influences collision outcomes and providing a phase diagram for bunching and antibunching behaviors.

Contribution

It presents an exact classical solution for electron collisions in a mesoscopic beamsplitter, separating center-of-mass and relative motion, and predicts a universal phase diagram of scattering outcomes.

Findings

Classical equations of motion describe electron guiding center dynamics.

A universal phase diagram of bunching and antibunching outcomes.

Interaction increases the effective barrier height, affecting collision results.

Abstract

Experiments on collisions of isolated electrons guided along the edges in quantum Hall setups can mimic mixing of photons with the important distinction that electrons are charged fermions. In the so-called electronic Hong-Ou-Mandel (HOM) setup uncorrelated pairs of electrons are injected towards a beamsplitter. If the two electron wave packets were identical, Fermi statistics would force the electrons to scatter to different detectors, yet this quantum antibunching may be confounded by Coulomb repulsion. Here we model an electronic HOM experiment using a quadratic 2D saddle point potential for the beamsplitter and unscreened Coulomb interaction between the two injected electrons subjected to a strong out-of-plane magnetic field. We show that classical equations of motion for the drift dynamics of electrons' guiding centers take on the form of Hamilton equations for canonically conjugated variables subject to the saddle point potential and the Coulomb potential where the dynamics of the center-of-mass coordinate and the relative coordinate separate. We use these equations to determine collision outcomes in terms of a few experimentally tuneable parameters: the initial energies of the uncorrelated electrons, relative time delay of injection and the shape of the saddle point potential. A universal phase diagram of deterministic bunching and antibunching scattering outcomes is presented with a single energy scale characterizing the increase of the effective barrier height due to interaction of coincident electrons. We suggest clear-cut experimental strategies to detect the predicted effects and give analytical estimates of conditions when the classical dynamics is expected to dominate over quantum effects.