We study spin effects in twisted-electron propagation through the entrance or exit region of an axially symmetric magnetic coil. Starting from the Foldy-Wouthuysen reduction of the Dirac equation, we derive a paraxial spinor equation in which the longitudinally varying solenoidal field produces, in addition to the usual diagonal Zeeman term, a transverse Pauli coupling proportional to the fringe-field gradient. The scalar transverse dynamics is treated exactly by the Ermakov mapping, which absorbs the longitudinally dependent focusing into a metaplectic scaling transformation and reduces the orbital evolution to that of a stationary two-dimensional oscillator. On this background, the transverse Pauli term is treated perturbatively and yields an explicit first-order correction for arbitrary realistic solenoidal profiles. Axial symmetry implies conservation of the total projection of angular momentum, so each spin flip is accompanied by a compensating one-quantum change of orbital angular momentum. In addition, the linear coordinate structure of the perturbation restricts the first-order dynamics to at most two neighboring radial sidebands for each incoming oscillator component. We derive the corresponding transition amplitudes and show how their phases are governed jointly by the Ermakov accumulation and the diagonal spin-orbital rotation. The resulting framework provides a direct way to quantify spin-orbit mixing of twisted electrons in realistic magnetic lenses and solenoidal beam-line elements, and it identifies a route toward controlled spin-OAM conversion in engineered sequences of magnetic-field edges.
We study spin effects in twisted-electron propagation through the entrance or exit region of an axially symmetric magnetic coil. Starting from the Foldy-Wouthuysen reduction of the Dirac equation, we derive a paraxial spinor equation in which the longitudinally varying solenoidal field produces, in addition to the usual diagonal Zeeman term, a transverse Pauli coupling proportional to the fringe-field gradient. The scalar transverse dynamics is treated exactly by the Ermakov mapping, which absorbs the longitudinally dependent focusing into a metaplectic scaling transformation and reduces the orbital evolution to that of a stationary two-dimensional oscillator. On this background, the transverse Pauli term is treated perturbatively and yields an explicit first-order correction for arbitrary realistic solenoidal profiles. Axial symmetry implies conservation of the total projection of angular momentum, so each spin flip is accompanied by a compensating one-quantum change of orbital angular momentum. In addition, the linear coordinate structure of the perturbation restricts the first-order dynamics to at most two neighboring radial sidebands for each incoming oscillator component. We derive the corresponding transition amplitudes and show how their phases are governed jointly by the Ermakov accumulation and the diagonal spin-orbital rotation. The resulting framework provides a direct way to quantify spin-orbit mixing of twisted electrons in realistic magnetic lenses and solenoidal beam-line elements, and it identifies a route toward controlled spin-OAM conversion in engineered sequences of magnetic-field edges.
We study relativistic twisted (orbital-angular-momentum) states of a massive charged particle propagating through an axially symmetric, longitudinally inhomogeneous solenoid field and a co-directed accelerating or decelerating electric field. Starting from the Dirac equation and using controlled spinless and paraxial approximations, we show that the transverse envelope obeys an effective nonstationary Schrödinger equation governed by a Caldirola--Kanai Hamiltonian. The longitudinal energy gain or loss encoded in f(z)=[E0−V(z)]2−m2 generates an effective gain or damping rate γ(z)=∂zf(z)/[2f(z)] and a z-dependent oscillator frequency ω(z)=p0Ω(z)/f(z). Exploiting the Ermakov mapping (unitary equivalence of Caldirola--Kanai systems), we obtain a closed-form propagated twisted wave function by transforming the stationary Landau basis. The transverse evolution is controlled by a single scaling function b(z) that satisfies a generalized Ermakov--Pinney equation with coefficients determined by Ez(z) and Bz(z). In the limiting cases of uniform acceleration with Bz=0 and of solenoid focusing with negligible acceleration, our solution reduces to previously known analytic results, providing a direct bridge to established models.
We investigate the Dirac oscillator in a helically twisted spacetime endowed with a uniform axial torsion. Starting from an orthonormal coframe, we compute the Levi--Civita spin connection explicitly and separate the geometric contribution from the axial contortion. Retaining the matrix β in the radial Moshinsky coupling, we show that the second-order problem is the ordered product Π^+Π^− rather than the square of a single operator. The resulting radial dynamics is a coupled, self-adjoint two-component system in which the spin connection supplies the correct cylindrical radial operator, while the off-diagonal metric generates the helical combination m/r−ωk and a Coulomb-like geometric term. A finite-element solution reproduces the planar Dirac-oscillator spectrum in the flat limit and reveals asymmetric dependence on the longitudinal momentum, avoided level crossings, and a supersymmetric zero mode at E=Mc2. The axial torsion and longitudinal momentum preserve this zero mode, whereas the helical twist lifts it quadratically. Sector-resolved thermodynamic functions are obtained from the relativistic bound-state spectrum. The explicit spinors further determine longitudinal vector and axial currents, and a Witten-index analysis identifies the helical twist as the deformation that removes the protected zero mode.
We present a microscopic calculation of spontaneous photon emission by twisted (paraxial) electrons propagating through inhomogeneous, axisymmetric magnetic fields. We construct exact electron states that incorporate transverse mode structure and wavefront curvature by combining the Foldy-Wouthuysen transformation with a geometric framework based on Lewis-Ermakov invariants and metaplectic transformations. We show that the evolution of such structured states corresponds to an open path in the space of quadratic forms, giving rise to a geometric contribution to the emission amplitude that cannot be eliminated by gauge choice or adiabatic arguments. The inverse radius of curvature of the electron wavefront controls a finite-window boundary contribution to the emission amplitude in regions where the external magnetic field vanishes locally, this contribution vanishes when the observation window is extended to infinity. Presented mechanism generalizes Landau-level radiation to nonasymptotic, structured electron states and establishes a direct connection between noncyclic geometric evolution and photon emission.
We demonstrate that a weak nonlinear magnetic entrance edge induces a tennis-racket (Dzhanibekov) instability in the shell-resolved orbital pseudospin dynamics of twisted electrons propagating in a nominally uniform solenoidal field. Starting from a Maxwell-consistent thin-edge extension of the entrance field, we derive an effective fixed-shell Hamiltonian in which linear Schwinger pseudospin precession acquires an anisotropic quadratic correction. In the symmetric aligned limit, an exact linear eigenstate (a Laguerre-Gaussian vortex state) becomes a hyperbolic fixed point of the large-shell dynamics, producing recurrent reversals of the mean pseudospin projection. These reversals appear in real space as repeated conversions of the transverse profile between Laguerre-Gaussian vortex and Hermite-Gaussian multi-lobed states. The unavoidable Lewis-Ermakov breathing of realistic wave packets does not generate a separate mechanism; it naturally modulates the nonlinear strength and sets the growth time scale. Microscope-scale estimates show that the required regime is accessible with standard octupole correctors in a transmission electron microscope.
Electron diffraction from nanogratings provides a platform for free-electron interferometry, yet controlled manipulation of electron spin in such geometries remains largely unexplored. In particular, the role of the self-generated magnetic field arising from electron motion and the feasibility of coherent spin control without disrupting diffraction coherence have not been quantitatively investigated. In this article, a self-consistent Maxwell-Pauli framework is developed to study spin-resolved electron diffraction from nanogratings in the presence of magnetic fields. The model incorporates geometric confinement, image-charge interactions, self-generated magnetostatic fields, and externally applied magnetic fields. Numerical simulations show that the intrinsic magnetic self-field produced by the electron probability current is several orders of magnitude too weak to induce measurable spin mixing, demonstrating that nanogratings act as spin-conserving beam splitters under field-free conditions. When a uniform magnetic field is applied upstream of the nanograting, coherent Larmor precession enables controlled spin rotation without modifying the diffraction geometry or degrading coherence. The magnetic field required for a π spin rotation scales inversely with the interaction length and electron de Broglie wavelength λdB. Furthermore, a downstream nonuniform magnetic field applied after the nanograting imparts a spatially varying Zeeman phase, producing opposite transverse momentum shifts for the two spin components. The spin-dependent transverse dynamics is analyzed using Husimi Q-function phase-space maps, which visualize spin-dependent population redistribution and momentum separation. The proposed approach enables tunable spatial separation of spin-resolved free electron beams and establishes an all-magnetic route for coherent spin rotation, control, and interferometry.
Twisted light has received a lot of attention in recent years. And there are justified reasons for this. The properties of twisted light are unusual and differ fundamentally from the plane-wave properties. The phase fronts of these light beams have a helical structure, giving them the name twisted. This helical phase structure originates from the well-defined orbital angular momentum (OAM) that they carry along their propagation direction and in addition to the spin angular momentum (SAM). Moreover, they have a characteristic intensity profile, which consists of concentric rings with a central minimum, and a phase singularity in the beam center. This thesis is based on two main parts, each dealing with a different aspect of twisted light. These investigations are carried out using Bessel beams, one type of twisted light. Twisted light beams are cylindrically symmetrical. Therefore, they are defined on the basis of circularly polarized light. This fact leads to the first question: Is it possible to generate twisted light in other polarization states? In this work, we derive how this can be achieved by using the superposition principle and how this can be used to construct linearly, radially, or azimuthally polarized Bessel beams. The focus in the second part is on excitations of atoms by twisted Bessel beams. In particular, we analyze how the use of twisted light can modify individual atomic multipole transitions. The obtained results show that for an efficient modification a precise localization of the target atom is required. In the last chapter, both aspects are combined. We analyze transitions between magnetic hyperfine states in a single trapped atom (or ion), which is exposed to an additional external magnetic field. The presented calculations for these multipole excitations are based on a formalism that accounts for the alignment of the applied external magnetic field with respect to the light propagation direction, and additionally different polarization states of the radiation field are considered. The obtained results indicate that twisted light could be of particular interest for precision experiments. One possible application is the use in single-ion clocks. The work shows that high multipole orders of the vortex beams allow excitations in the intensity minimum in the center of these beams without causing large disturbances, and thus may enable a significant reduction of the light shift in atomic transitions.