Entanglement in Elastic Electron Scattering: Perturbation theory misses fundamental aspects of Bragg scattering

TL;DR

This work shows that elastic electron scattering cannot always be described by a fixed perturbative potential, because the probe and sample can become entangled, making the electron's reduced state mixed with purity $\mathrm{tr}(\hat{\rho}^2) < 1$ when the scatterer has finite mass. A density-operator formalism is developed, with $\hat{\rho}_{in} = |k_0\rangle|I\rangle\langle I|\langle k_0|$ and $\hat{\rho}$ obtained after projecting onto elastic channels and tracing over the sample; entanglement thus modifies the observable electron state unless $\langle F_{\boldsymbol{k}'}|F_{\boldsymbol{k}}\rangle = 1$. The paper demonstrates two concrete examples—Bragg scattering on nanoparticles and a symmetric two-beam HRTEM case—where the entanglement factor $e^{-(q^2 + {q'}^2 - \boldsymbol{q}'\cdot\boldsymbol{q})\sigma_0^2}$ and related terms alter coherence and image contrast, with decoherence times depending on the scatterer mass $M$ and the CM localization $\sigma_0$. In the heavy-mass limit the conventional theory is recovered, but for finite, especially nanoscale scatterers, entanglement can produce measurable deviations, offering a route to probe quantum aspects of scattering and potentially inform longstanding issues such as the Stobbs factor.

Abstract

Elastic electron scattering is one of the primary means of investigating materials on the atomic scale. It is usually described by modeling the sample as a fixed, static, perturbative potential, thereby completely neglecting the quantum nature of the atoms inside. In this work, we present a quantum treatment of elastic electron scattering. We show that the interaction of the probe beam and the sample results in entanglement between the two systems, which can have far-reaching consequences, particularly on coherence and image contrast. As a timely example, we discuss decoherence in Bragg scattering on nanoparticles. We also investigate under which conditions the conventional scattering theory is recovered.