Universal Analytic Solution for the Quantum Transport of Structured Matter-Waves in Magnetic Optics

Abstract

We present a closed-form analytic solution for the propagation of an arbitrary charged scalar state in a non-uniform magnetic field. The dynamics are governed by classical beam optics parameters (Courant-Snyder parameters), the Twiss functions, and phase advance, revealing a direct map between quantum evolution and its classical counterpart. The solution decomposes into three components, exhibiting a complex rotation dependent on the sign of the orbital angular momentum (OAM) projection, alongside an intrinsic distortion from interference governed by a generalized Gouy phase. For a relevant Glaser-type magnetic field and a half-blocked twisted electron, we demonstrate that asymmetry reveals interference-driven dynamics beyond rigid rotation. Our fully relativistic framework provides a practical tool for predicting beam behavior in particle accelerators and electron microscopes.

Figures (2)

• Figure 1: Absolute value for the expansion coefficients $c_{n,l}^{arc}$ for the half-blocked twisted state for two initial values of the OAM projection $l_a=1$ - left panel and $l_a=-3$ - right panel. Radial number is zero in both cases $n_a=0$. The plots are on a logarithmic scale.
• Figure 2: Rotation angles $\varphi_{\pm}(z)$ calculated with the help of the Eq.\ref{['eq:ang']} for the case of the Glaser field Eq.\ref{['eq:GZF']} with the parameters $a=4$ and $c=15$ (upper left and upper right panels). Parameter $b(z)$ (upper middle panel) calculated based on the analytical expressions LewisNUF21 of the solution to the Eq.\ref{['eq:EPeq']} with the Glaser field Eq.\ref{['eq:GZF']} and $b_0=1$, $b_0'=0$. Intensity distributions (lower panel) $|\psi_{arc}|^2$ of the half-blocked twisted state Eq.\ref{['eq:psiarc']} for different positions $z$ inside the Glaser field calculated with the help of the Eq.\ref{['eq:psifull']}, Eq.\ref{['eq:psiparts']} and Eq.\ref{['eq:cnl']}.