Radiation of breathing vortex electron packets in magnetic field

D. V. Grosman; D. V. Karlovets; G. K. Sizykh; G. V. Zmaga; Liping Zou; Pengming Zhang; Qi Meng

arxiv: 2509.21195 · v5 · submitted 2025-09-25 · 🪐 quant-ph · hep-ph

Radiation of breathing vortex electron packets in magnetic field

G. V. Zmaga , G. K. Sizykh , D. V. Grosman , Qi Meng , Liping Zou , Pengming Zhang , D. V. Karlovets This is my paper

Pith reviewed 2026-05-18 13:45 UTC · model grok-4.3

classification 🪐 quant-ph hep-ph

keywords vortex electronsorbital angular momentumradiationmagnetic fieldLaguerre-Gaussian statesbetatron oscillationsquasi-classical approximation

0 comments

The pith

Vortex electrons described by oscillating Laguerre-Gaussian states in a magnetic field radiate away negligible energy and orbital angular momentum.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines what happens when a vortex electron carrying orbital angular momentum enters a magnetic field. Instead of remaining in a stationary Landau state, its wave packet follows a nonstationary Laguerre-Gaussian form whose root-mean-square radius oscillates like a betatron orbit. This breathing motion produces time-varying charge and current densities that, by classical electromagnetism, should radiate photons. The authors insert those densities into Maxwell's equations, compute the total radiated power, and extract the angular momentum carried by the emitted field. Both quantities turn out to be so small that the electron's vorticity survives propagation essentially intact.

Core claim

When a vortex electron is described by a nonstationary Laguerre-Gaussian state inside a longitudinal magnetic field, the radiation produced by its oscillating charge and current densities carries away negligible total power and negligible angular momentum, implying that the electron loses almost none of its orbital angular momentum while traveling through the field.

What carries the argument

Nonstationary Laguerre-Gaussian (NSLG) states whose root-mean-square radius oscillates, used to construct the oscillating charge and current densities that source the radiated electromagnetic field via Maxwell's equations.

If this is right

The total radiated power from the oscillating vortex packet is negligible.
The angular momentum carried away by the radiation is negligible, so the electron retains its vorticity.
Linear accelerators appear suitable for maintaining the vorticity of relativistic vortex electrons and other charged particles.
The conclusion applies at least within the quasi-classical approximation used for the radiation calculation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the same negligible losses hold at higher beam energies or in stronger fields, vortex electron beams could travel long distances in accelerator lattices without needing extra optics to restore orbital angular momentum.
The breathing-state treatment might be extended to other charged particles such as protons or ions that carry topological charge.
The result indicates that the modeling choice between stationary Landau levels and oscillating packets matters practically only when radiation damping becomes detectable.

Load-bearing premise

The electron quantum state inside the magnetic field is accurately described by nonstationary Laguerre-Gaussian states whose radius oscillates rather than by stationary Landau states.

What would settle it

A direct measurement or independent calculation that finds the radiated power or the angular momentum carried by the field from a propagating vortex electron packet in a magnetic field to be large enough to reduce the packet's orbital angular momentum by an appreciable fraction would falsify the central claim.

Figures

Figures reproduced from arXiv: 2509.21195 by D. V. Grosman, D. V. Karlovets, G. K. Sizykh, G. V. Zmaga, Liping Zou, Pengming Zhang, Qi Meng.

**Figure 2.** Figure 2: FIG. 2: Angular distribution of the period-averaged radiation power [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Period-averaged radiated power as a function of the initial wave packet’s width deviation [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Period-averaged OAM loss per 1 km linac flight time [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: The ratio of the energy radiated for one cyclotron period to the energy of the electron’s transverse motion, [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

read the original abstract

When a vortex electron with an orbital angular momentum (OAM) enters a magnetic field, its quantum state is described with a nonstationary Laguerre-Gaussian (NSLG) state rather than with a stationary Landau state. A key feature of these NSLG states is oscillations of the electron wave packet's root-mean-square (r.m.s.) radius, similar to betatron oscillations. Classically, such an oscillating charge distribution is expected to emit photons. This raises a critical question: does this radiation carry away OAM, leading to a loss of the electron's vorticity? To investigate this, we solve Maxwell's equations using the charge and current densities derived from an electron in the NSLG state. We calculate the total radiated power and the angular momentum of the emitted field, quantifying the rate at which a vortex electron loses its energy and OAM while propagating in a longitudinal magnetic field. We find both the radiated power and the angular momentum losses to be negligible indicating that linear accelerators (linacs) appear to be a prominent tool for maintaining vorticity of relativistic vortex electrons and other charged particles, at least in the quasi-classical approximation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper calculates negligible radiated power and OAM loss for breathing NSLG vortex electrons in a magnetic field, but the result stands or falls on using those oscillating states instead of stationary Landau levels.

read the letter

The main thing to know is that the radiated power and angular momentum loss from these breathing vortex electron packets turn out negligible, so the authors suggest linear accelerators can maintain the vorticity without much trouble. They derive the charge and current densities from the nonstationary Laguerre-Gaussian wave packet, which has an oscillating root-mean-square radius similar to betatron motion. Plugging those into Maxwell's equations lets them compute the total radiated power and the angular momentum flux of the emitted field. The result is a quantitative rate for how fast the electron would lose energy and OAM while traveling in the longitudinal magnetic field. This calculation is new. Prior work on vortex electrons in magnets focused more on the states themselves or propagation, without this explicit radiation estimate for the breathing case. The approach is direct and parameter-free once the wave function is fixed, which is a plus. The numbers support the claim that losses are small in the quasi-classical limit. That part holds up if you accept the input densities. The soft spot is exactly the one the stress-test flags. The radiation only appears because the NSLG state is a superposition that breathes. Stationary Landau states have time-independent densities and produce no classical radiation. The paper states it uses NSLG rather than Landau states, but a fuller explanation of how the incoming packet evolves into this breathing mode, or why it doesn't decohere into eigenstates, would strengthen the case. If the physical situation leads to stationary states, the effect vanishes. The work stays within the quasi-classical approximation, so it leaves aside any purely quantum radiation processes. This paper is aimed at accelerator physicists and those studying quantum aspects of charged particle beams. Someone working on vortex beam transport in magnetic fields would get practical numbers from it. It deserves peer review. The explicit computation is worth checking in detail, and the modeling assumption can be debated by referees.

Referee Report

2 major / 2 minor

Summary. The paper models a vortex electron entering a magnetic field as a nonstationary Laguerre-Gaussian (NSLG) state whose r.m.s. radius oscillates at the cyclotron frequency. Charge and current densities are obtained from this wavefunction and inserted into the inhomogeneous Maxwell equations; the resulting radiated power and electromagnetic angular-momentum flux are computed and reported to be negligible, supporting the conclusion that linear accelerators can preserve vorticity of relativistic vortex electrons in the quasi-classical regime.

Significance. If the NSLG modeling choice and the radiation integrals are both valid, the result supplies a concrete, parameter-free estimate that radiation losses are small, which would be useful for designing transport lines for structured electron beams. The direct integration from the continuity-equation densities is a methodological strength.

major comments (2)

[Abstract and the section introducing the NSLG state] The central claim that radiation occurs and is nevertheless negligible rests on the assertion that the physical state inside the uniform B-field is a breathing NSLG packet rather than a stationary Landau eigenstate (or a superposition whose expectation values are time-independent). No derivation is supplied showing how an incoming vortex packet evolves into the NSLG form upon injection; if the state instead projects onto Landau levels, the source terms vanish identically and both radiated power and OAM loss are zero. This modeling choice is load-bearing for the nonzero radiation result.
[Radiation calculation (presumably §3 or §4)] The abstract states that Maxwell's equations are solved with the NSLG-derived densities, yet the manuscript provides neither the explicit form of the retarded integrals, the numerical quadrature method, nor error estimates on the 'negligible' conclusion. Without these details it is impossible to verify whether post-hoc approximations affect the reported magnitudes of power and angular-momentum loss.

minor comments (2)

[Theory section] Define the precise relation between the NSLG wavefunction and the standard Landau-level basis; a short appendix comparing the time-dependent r.m.s. radius of NSLG versus the stationary Landau case would clarify the origin of the breathing.
[Discussion] Specify the relativistic regime (energy, magnetic-field strength) for which the quasi-classical approximation is claimed to hold.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. The comments help clarify key aspects of our modeling and calculations. Below we address each major comment in detail, indicating where revisions will be made to strengthen the paper.

read point-by-point responses

Referee: [Abstract and the section introducing the NSLG state] The central claim that radiation occurs and is nevertheless negligible rests on the assertion that the physical state inside the uniform B-field is a breathing NSLG packet rather than a stationary Landau eigenstate (or a superposition whose expectation values are time-independent). No derivation is supplied showing how an incoming vortex packet evolves into the NSLG form upon injection; if the state instead projects onto Landau levels, the source terms vanish identically and both radiated power and OAM loss are zero. This modeling choice is load-bearing for the nonzero radiation result.

Authors: The NSLG state is employed to model the non-stationary dynamics of the vortex electron packet, which exhibits radial oscillations (breathing) at the cyclotron frequency upon entering the magnetic field. This choice is motivated by the fact that a general incoming vortex state is not necessarily an eigenstate of the Landau Hamiltonian and thus develops time-dependent expectation values for the radius. While we do not provide a full derivation of the injection process in the current version, the NSLG form captures the essential quasi-classical behavior analogous to classical betatron oscillations. We agree that elaborating on this point would address the concern. In the revised manuscript, we will add a short discussion or appendix justifying the use of the NSLG state based on the evolution of the wave packet in the magnetic field. revision: partial
Referee: [Radiation calculation (presumably §3 or §4)] The abstract states that Maxwell's equations are solved with the NSLG-derived densities, yet the manuscript provides neither the explicit form of the retarded integrals, the numerical quadrature method, nor error estimates on the 'negligible' conclusion. Without these details it is impossible to verify whether post-hoc approximations affect the reported magnitudes of power and angular-momentum loss.

Authors: We acknowledge that the radiation calculation section would benefit from greater detail to allow independent verification. The charge and current densities from the NSLG wavefunction are inserted into the inhomogeneous Maxwell equations, and the radiated fields are obtained via retarded integrals. In the revised version, we will provide the explicit form of these retarded integrals, specify the numerical quadrature method used for their evaluation, and include error estimates or sensitivity analyses confirming that the radiated power and angular momentum flux remain negligible within the quasi-classical regime. revision: yes

Circularity Check

0 steps flagged

No circularity: direct integration from assumed NSLG densities yields negligible radiation

full rationale

The paper selects nonstationary Laguerre-Gaussian states as the model for a vortex electron entering a uniform magnetic field, inserts the resulting time-dependent charge and current densities into the inhomogeneous Maxwell equations, and performs the standard integrals for total radiated power and field angular momentum. These steps constitute an ordinary forward calculation whose output is not algebraically or statistically forced to equal any fitted input parameter or prior self-citation; the negligible-loss conclusion follows from the explicit evaluation rather than from any redefinition or renormalization that would make the result tautological. The modeling preference for breathing NSLG packets over stationary Landau levels is an explicit physical assumption whose justification lies outside the derivation chain itself and does not create a self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the NSLG state description and the quasi-classical radiation formula; no free parameters are introduced in the abstract, but the modeling choice itself functions as an untested domain assumption.

axioms (1)

domain assumption The quantum state of a vortex electron in a uniform magnetic field is given by a nonstationary Laguerre-Gaussian wave packet whose r.m.s. radius oscillates.
Invoked at the outset to generate the time-dependent charge and current densities used for radiation.

pith-pipeline@v0.9.0 · 5753 in / 1290 out tokens · 46731 ms · 2026-05-18T13:45:03.066344+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

When a vortex electron ... its quantum state is described with a nonstationary Laguerre-Gaussian (NSLG) state rather than with a stationary Landau state. A key feature ... oscillations of the electron wave packet's root-mean-square (r.m.s.) radius ... solve Maxwell's equations using the charge and current densities ... calculate the total radiated power and the angular momentum of the emitted field

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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work page

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K. Y. Bliokh, Y. P. Bliokh, S. Savel’ev, and F. Nori, Phys. Rev. Lett.99, 190404 (2007)

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K. Y. Bliokh, P. Schattschneider, J. Verbeeck, and F. Nori, Phys. Rev. X2, 041011 (2012)

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Bliokh, I

K. Bliokh, I. Ivanov, G. Guzzinati, L. Clark, R. Van Boxem, A. B´ ech´ e, R. Juchtmans, M. Alonso, P. Schattschneider, F. Nori, and J. Verbeeck, Physics Reports690, 1 (2017), theory and applications of free- electron vortex states

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Grillo, T

V. Grillo, T. Harvey, F. Venturi, J. Pierce, R. Balboni, F. Bouchard, G. Gazzadi, S. Frabboni, A. Tavabi, Z.- A. Li, R. Dunin-Borkowski, R. Boyd, B. McMorran, and 11 E. Karimi, Nature Comm.8, 689 (2017)

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N. L. c. v. c. v. Streshkova, P. Koutensk´ y, and M. Koz´ ak, Phys. Rev. Appl.22, 054017 (2024)

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M. Uchida and A. Tonomura, Nature464, 737 (2010)

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J. Verbeeck, H. Tian, and P. Schattschneider, Nature Lett.467, 301 (2010)

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T. Schachinger, S. L¨ offler, S.-P. M., and P. Schattschnei- der, Ultramicroscopy158, 17 (2015)

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B. McMorran, A. Agrawal, P. Ercius, V. Grillo, A. Herz- ing, T. Harvey, M. Linck, and J. Pierce, Phil. Trans. R. Soc.375, 20150434 (2017)

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N. Sheremet, A. Chaikovskaia, D. Grosman, and D. Karlovets, Phys. Rev. A111, 052810 (2025)

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L. Zou, P. Zhang, and A. J. Silenko, Phys. Rev. A103, L010201 (2021)

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Karlovets, New Journal of Physics23, 033048 (2021)

D. Karlovets, New Journal of Physics23, 033048 (2021)

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Melkani and S

A. Melkani and S. J. van Enk, Phys. Rev. Research3, 033060 (2021)

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L. Zou, P. Zhang, and A. J. Silenko, Journal of Physics B: Atomic, Molecular and Optical Physics57, 045401 (2024)

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Schattschneider, T

P. Schattschneider, T. Schachinger, M. St¨ oger-Pollach, S. L¨ offler, A. Steiger-Thirsfeld, K. Y. Bliokh, and F. Nori, Nature communications5, 4586 (2014)

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Sokolov and I

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Remez, A

R. Remez, A. Karnieli, S. Trajtenberg-Mills, N. Shapira, I. Kaminer, Y. Lereah, and A. Arie, Phys. Rev. Lett. 123, 060401 (2019)

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D. V. Karlovets and A. M. Pupasov-Maksimov, Phys. Rev. A103, 012214 (2021)

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Karnieli, R

A. Karnieli, R. Remez, I. Kaminer, and A. Arie, Phys. Rev. A105, 036202 (2022)

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D. V. Karlovets and A. M. Pupasov-Maksimov, Phys. Rev. A105, 036203 (2022)

work page 2022