Evolution of a twisted electron wave packet perturbed by an inhomogeneous electric field

TL;DR

The paper investigates the evolution of a twisted electron Laguerre-Gaussian wave packet carrying orbital angular momentum $l$ under a static, inhomogeneous electric field that breaks axial symmetry. A perturbative Green's-function framework is developed to compute the first-order correction $\\Psi^{(1)}$ for two model fields: a fully localized perturbation $\\phi(\\mathbf{r})=\\lambda\\delta(\\mathbf{r}-\\mathbf{r}_0)$ and a transverse field with $\\phi(x)=-E_0 d\\, f((x-a)/d)$. The results show that the external field induces strong azimuthal density modulation, mixes OAM channels, and splits the initial $l$-fold zero into $l$ singly charged vortices arranged on a circle in the delta-like case, with less regular but still vortex-rich patterns for the transverse field and parity-dependent effects. After the perturbation ends, the vortex cores expand self-similarly with a linear-in-time growth of their radii, implying potential for beam diagnostics and controlled manipulation of twisted electron beams in electrostatic environments.

Abstract

Laguerre-Gaussian (LG) wave packets, known for their vortex structure and nonzero orbital angular momentum (OAM), are of great interest in various scientific fields. Here we study the nonrelativistic dynamics of a spatially-localized electron LG wave packet interacting with an inhomogeneous external electric field that violates the axial symmetry of the initial wave function. We focus on the analysis of the electron density and demonstrate how it is affected by the external field. Within the first order of perturbation theory, we calculate the electron wave function and reveal that the electric field may significantly alter the wave packet's structure and distort its qualitative form. We demonstrate that due to the interaction with the external field, the degenerate zeros of the initial wave function located on the $z$ axis split into multiple nondegenerate nodes in the transverse plane representing separate single-charge vortices. This mechanism resembles the analogous effects known in topological optics. These findings provide new insights into controlling and manipulating twisted matter beams and into their possible instabilities.

Figures (9)

• Figure 1: Simplified scheme of the main setup under consideration. A twisted electron wave packet with a given $z$ projection $l$ of orbital angular momentum evolves under the influence of an external electric field directed along the $x$ axis. The electric field is present in a finite spatial layer $2d$ and has a smooth $x$-dependent profile. The external field breaks the initial rotational symmetry of the vortex state. By means of first-order perturbation theory, we investigate the dynamics of the wave packet and examine its topological properties.
• Figure 2: Electron probability density for a twisted wave packet interacting with a $\delta$-like perturbation \ref{['eq:app_pot']}. Rows (a)-(d) correspond to odd OAM projections $l=1$, $3$, $5$, and $7$, respectively. Left column displays the unperturbed density $|\Psi^{(0)}|^{2}$; right column contains the plots for $|\Psi^{(0)}+\Psi^{(1)}|^{2}$. When the off-axis $\delta$-like potential at $\rho_{0}=10$ a.u. is included [panels (a2)-(d2)], the ring structure acquires an $l$-fold azimuthal modulation consisting of $l$ maxima (red). The coupling strengths are $\lambda=30$ a.u., $0.2$ a.u., $0.007$ a.u., and $0.0002$ a.u. for the four $l$ values, respectively. The other parameters are the following: the electron energy is $\bar{p}^2/(2m) = 2$ keV, $\sigma = 0.02$ a.u., and $t = 3500$ a.u. The average speed of the electron in units of the spped of light amounts to $\bar{v}/c \approx 0.09$, which justifies our nonrelativistic treatment of the problem.
• Figure 3: The same as in Fig. \ref{['fig:app_1']} for even values of the OAM projection: $l=2$, $4$, $6$, and $8$ [rows (a)--(d)]. The coupling strengths are $\lambda=3$ a.u., $0.045$ a.u., $0.002$ a.u., and $4\times10^{-5}$ a.u., respectively. The rest parameters are the same as in Fig. \ref{['fig:app_1']}.
• Figure 4: Nodal structure of the twisted electron packet perturbed by the localized potential \ref{['eq:app_pot']}. Blue (red) curves trace the zero level of $\mathrm{Re}\,\Psi$ ($\mathrm{Im}\,\Psi$); their intersections (black dots) mark density zeros. Rows (a)--(d) correspond to odd orbital charges $l=1$, $3$, $5$, and $7$, respectively. Left column shows the unperturbed state $\Psi^{(0)}$, where a single $l$-fold node resides at the origin. Right column depicts the full wave function $\Psi^{(0)}+\Psi^{(1)}$. The $\delta$-like potential located at $\rho_{0}=10$ a.u. splits the central node into $l$ first-order vortices whose positions form a regular polygon: one off-axis zero for $l=1$, three for $l=3$ and so on. The coupling strength $\lambda$ and other parameters are the same as in Fig. \ref{['fig:app_1']}.
• Figure 5: The same as in Fig. \ref{['fig:app_3']} for even values of the OAM projection: $l=2$, $4$, $6$, and $8$. The coupling strength $\lambda$ and other parameters are the same as in Fig. \ref{['fig:app_2']}.