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arxiv: 2603.23758 · v2 · submitted 2026-03-24 · ❄️ cond-mat.mes-hall · physics.chem-ph· physics.comp-ph· quant-ph

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· Lean Theorem

Quantum-classical dynamics of Rashba spin-orbit coupling

Paul Bergold , Giovanni Manfredi , Cesare Tronci
Authors on Pith no claims yet

Pith reviewed 2026-05-14 23:57 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall physics.chem-phphysics.comp-phquant-ph
keywords Rashba spin-orbit couplingquantum-classical dynamicsKoopman wavefunctionsEhrenfest modelnanowiresharmonic potentialspin dynamicscat states
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The pith

A new koopmon method for Rashba nanowires reproduces full quantum spin and orbital dynamics more accurately than Ehrenfest models, especially under harmonic confinement.

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

The paper extends the koopmon particle scheme, based on Koopman wavefunctions, to mixed quantum-classical models of Rashba spin-orbit coupling in one-dimensional nanowires. It compares this approach to both fully quantum simulations and the standard Ehrenfest mean-field dynamics across strong and weak coupling regimes, with and without external harmonic potentials. The central finding is that koopmon retains the Heisenberg principle and beyond-mean-field correlations, allowing it to track both spin and orbital evolution faithfully. In free nanowires the method captures qualitative quantum features, though with a small spin accuracy trade-off versus Ehrenfest; when a harmonic trap is added, koopmon reaches accuracy levels unattainable by Ehrenfest in either sector. A test case also shows the emergence of cat-like states under the new dynamics.

Core claim

The koopmon scheme, obtained by extending Koopman wavefunctions to spin-orbit coupling, reproduces the full quantum evolution of a quantum spin-1/2 interacting with classical orbital momentum in Rashba nanowire models; in the presence of a harmonic potential it achieves accuracy in both quantum and classical sectors that remains out of reach for the Ehrenfest model, while still recovering the main qualitative features of exact quantum dynamics in the absence of confinement.

What carries the argument

The koopmon particle scheme that extends Koopman wavefunctions to spin-orbit coupling, thereby retaining the Heisenberg principle and capturing correlations beyond the Ehrenfest mean-field limit.

If this is right

  • Without external potential, koopmon reproduces qualitative quantum features in both Rashba- and Zeeman-dominated regimes.
  • With harmonic confinement, koopmon matches full quantum results in spin and orbital sectors at levels unreachable by Ehrenfest.
  • The method can generate cat-like states in suitable test cases.
  • Ehrenfest captures spin accuracy better without potential but fails to track orbital motion.

Where Pith is reading between the lines

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

  • The same koopmon extension could be tested on two-dimensional or curved Rashba structures to check whether correlation capture remains robust.
  • If the accuracy gain holds for larger systems, it would allow quantum-classical simulations of spintronic devices at scales where full quantum methods become prohibitive.
  • The observed formation of cat-like states suggests the method may naturally incorporate entanglement-like effects between spin and orbital degrees of freedom.

Load-bearing premise

The Koopman wavefunction formalism can be extended consistently to spin-orbit coupling while preserving the Heisenberg principle and the ability to capture correlations beyond mean-field.

What would settle it

A side-by-side numerical run for a Rashba nanowire with harmonic potential at realistic semiconductor parameters where the koopmon orbital trajectories or spin expectation values diverge from the exact quantum solution by more than a few percent while Ehrenfest remains closer.

Figures

Figures reproduced from arXiv: 2603.23758 by Cesare Tronci, Giovanni Manfredi, Paul Bergold.

Figure 1
Figure 1. Figure 1: Sketch of a quantum nanowire with its axis along the ex direction. The Rashba magnetic field points along the ey axis, while the external magnetic field B (Zeeman effect) is directed along the axis of the nanowire. In the case of a non-ballistic nanowire, a quantum dot is modeled by a harmonic potential with energy level spacing ℏω/2. when the wire width is much smaller than the Rashba length L = m−1 ∗ ℏ/α… view at source ↗
Figure 2
Figure 2. Figure 2: Energy surfaces in momentum space for the ballistic test cases. Left: Zeeman-dominated regime (InSb). Right: Rashba-dominated regime (InAs). Red: λ1; black: λ2. 3.1 Ballistic dynamics in the Zeeman-dominated regime We begin with a ballistic Hamiltonian and choose material parameters corresponding to InSb. From [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time evolution in the classical sector for the ballistic test case in the Zeeman-dominated regime (InSb). Columns: Quantum (left), koopmons (middle), Ehrenfest (right). Rows: t = 0, 62.2, 155.5, and 217.7 ps. Phase space: position [q] = µm (horizontal) and momentum [p] = eV/c (vertical). Particle plots with N = 500 and α = 0.5 include the smoothed density D(z, t). separation between these two portions is s… view at source ↗
Figure 4
Figure 4. Figure 4: Time evolution of the Bloch vector components and purity (bottom right) for the ballistic, Zeeman￾dominated case (InSb). the MTE result as an orange solid line. Note that all curves start at n(0) = (0, 0, 1), which reflects our choice of initial state aligned with σbz. In the x-component, the dynamics are essentially trivial: the quantum curve remains almost constant, and both particle methods accurately r… view at source ↗
Figure 5
Figure 5. Figure 5: Time evolution of the spin-momentum correlation components along σbx (top left), σby (top right), and σbz (bottom) for the ballistic, Zeeman-dominated case (InSb). The top-left panel corresponds to corr(p, σbx), the top-right panel to corr(p, σby), and the bot￾tom panel to corr(p, σbz). As in the Bloch-vector plots, the x-component of the spin-momentum correlations remains zero for all times. This behavior… view at source ↗
Figure 6
Figure 6. Figure 6: Time evolution in the classical sector for the ballistic test case in the Rashba-dominated regime (InAs). Columns: Quantum (left), koopmons (middle), Ehrenfest (right). Rows: t = 0, 13.1, 32.8, and 46 ps. Phase space: position [q] = µm (horizontal) and momentum [p] = eV/c (vertical). Particle plots with N = 500 and α = 0.5 include the smoothed density D(z, t). Classical sector. The results for the classica… view at source ↗
Figure 7
Figure 7. Figure 7: Time evolution of the Bloch vector components and purity (bottom right) for the ballistic, Rashba￾dominated case (InAs). negative q, while the other part moves toward positive q. The latter carries most of the positive values, indicated by the deep-red region. The MTE simulation fails to capture these features. Although parts of the distribution shift to the right at early times, its overall support remain… view at source ↗
Figure 8
Figure 8. Figure 8: Time evolution of the spin-momentum correlation components along σbx (top left), σby (top right), and σbz (bottom) for the ballistic, Rashba-dominated case (InAs). koopmons reproduce the qualitative shape of the oscillations but deviate quantitatively. The phase shift is visible early on, and the amplitude decays more rapidly. This discrepancy leads to a more pronounced loss of coherence in the koopmon sim… view at source ↗
Figure 9
Figure 9. Figure 9: Energy surfaces in momentum space for the non-ballistic test cases. Left: Zeeman-dominated (InSb). Middle: Rashba-dominated (InAs). Right: Zeeman-dominated (GaAs). Red: λ1; black: λ2. 4.1 Non-ballistic dynamics in the Zeeman-dominated regime For the non-ballistic test cases, one deals with three different energy scales, ESO, EZ, and EHO. It has proven useful to introduce the dimensionless parameter ξ := ( … view at source ↗
Figure 10
Figure 10. Figure 10: Time evolution in the classical sector for the non-ballistic test case in the Zeeman-dominated regime (InSb). Columns: Quantum (left), koopmons (middle), Ehrenfest (right). Rows: t = 0, 235, 587.5, and 822.4 ps. Phase space: position [q] = µm (horizontal) and momentum [p] = eV/c (vertical). Particle plots with N = 500 and α = 0.5 include the smoothed density D(z, t). intricate deformation of the wavepacke… view at source ↗
Figure 11
Figure 11. Figure 11: Time evolution of the Bloch vector components and purity (bottom right) for the non-ballistic, Zeeman-dominated case (InSb). even clearer at the final time. While both particle methods fail to reproduce the fine details, the overall shape of the distribution – its extent, its deformation, and its reduced density in the interior – is captured significantly better by the koopmons. Quantum sector and spin-or… view at source ↗
Figure 12
Figure 12. Figure 12: Time evolution of the spin-momentum correlation components along σbx (top left), σby (top right), and σbz (bottom) for the non-ballistic, Zeeman-dominated case (InSb). Here the MTE method fails to reproduce any of the peaks or oscillation periods in the second component, whereas the koopmons continue to reflect the main qualitative features. In the first and third components, the koopmons capture both amp… view at source ↗
Figure 13
Figure 13. Figure 13: Time evolution in the classical sector for the non-ballistic test case in the Rashba-dominated regime (InAs). Columns: Quantum (left), koopmons (middle), Ehrenfest (right). Rows: t = 0, 1.45, 3.63, and 5.08 ps. Phase space: position [q] = µm (horizontal) and momentum [p] = eV/c (vertical). Particle plots with N = 500 and α = 0.5 include the smoothed density D(z, t). mass (dark red), together with a negati… view at source ↗
Figure 14
Figure 14. Figure 14: Time evolution of the Bloch vector components and purity (bottom right) for the non-ballistic, Rashba-dominated case (InAs). also appear for the koopmons in the quantum sector. Quantum sector and spin-orbit correlations [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Time evolution of the spin-momentum correlation components along σbx (top left), σby (top right), and σbz (bottom) for the non-ballistic, Rashba-dominated case (InAs). 4.3 Appearance of cat-like states in gallium arsenide wires In the previous test cases, we considered the materials InSb and InAs. Owing to its relatively small Rashba coupling parameter, InSb was used to investigate Zeeman-dominated regime… view at source ↗
Figure 16
Figure 16. Figure 16: Time evolution in the classical sector for the non-ballistic test case in the Zeeman-dominated regime (GaAs). Columns: Quantum (left), koopmons (middle), Ehrenfest (right). Rows: t = 0, 61.5, 153.8, and 215.3 ns. Phase space: position [q] = µm (horizontal) and momentum [p] = eV/c (vertical). Particle plots with N = 500 and α = 0.5 include the smoothed density D(z, t). Classical sector [PITH_FULL_IMAGE:fi… view at source ↗
Figure 17
Figure 17. Figure 17: Time evolution of the Bloch vector components and purity (bottom right) for the non-ballistic, Zeeman-dominated case (GaAs). the phase-space picture at the final time is strongly reminiscent of a cat state, i.e., a linear superposition of two coherent-state wavepackets linked by a central interference region. The Ehrenfest simulation captures the overall rotation but fails to reproduce several essential f… view at source ↗
Figure 18
Figure 18. Figure 18: Time evolution of the spin-momentum correlation components along σbx (top left), σby (top right), and σbz (bottom) for the non-ballistic, Zeeman-dominated case (GaAs). constant value. A similar picture is visible in the purity plot. Although the koopmons show somewhat stronger decoherence initially, their oscillation period matches the quantum curve very well over the entire simulation. In contrast, the M… view at source ↗
read the original abstract

Mixed quantum-classical models are widely used to reduce the computational cost of fully quantum simulations. However, their general applicability across different classes of problems remains an open question. Here, we address this issue for systems featuring spin-orbit coupling. In particular, we study the interaction dynamics of quantum spin-1/2 and classical orbital momentum in one-dimensional models of Rashba nanowires. We tackle this problem by resorting to a new quantum-classical Hamiltonian model that, unlike conventional approaches, retains the Heisenberg principle and captures correlation effects beyond the common Ehrenfest approach. Based on Koopman wavefunctions in classical mechanics, the new model was recently implemented numerically via a particle scheme -- the koopmon method -- which is extended here to treat spin-orbit coupling. We apply the koopmon method to study the quantum-classical dynamics of nanowire models, with and without the presence of a harmonic potential and in both Rashba-dominated (strong coupling) and Zeeman-dominated (weak coupling) regimes. Considering realistic semiconductor parameters, the results are contrasted with both fully quantum and quantum-classical Ehrenfest dynamics. In the absence of external potential, the koopmon method qualitatively reproduces the features of the fully quantum evolution for all coupling regimes. While it exhibits a slight loss in spin accuracy compared to Ehrenfest simulations, the latter fail to capture the orbital dynamics. In the presence of a harmonic potential, the koopmon scheme reproduces the full quantum results with accuracy levels that are unachievable by the Ehrenfest model in both quantum and classical sectors. We conclude by presenting a test case that exhibits the formation of cat-like states.

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.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces an extension of the Koopman wavefunction formalism (koopmon method) to quantum-classical dynamics of Rashba spin-orbit coupling in one-dimensional nanowire models. It constructs a new Hamiltonian that retains the Heisenberg uncertainty principle and goes beyond Ehrenfest mean-field, then applies a particle-based numerical scheme to compare dynamics against full quantum simulations and Ehrenfest evolution. Results are shown for Rashba- and Zeeman-dominated regimes, both with and without an external harmonic potential, using realistic semiconductor parameters; the koopmon approach is reported to qualitatively match full quantum features (including cat-like states) and to achieve higher accuracy than Ehrenfest when a harmonic trap is present.

Significance. If the numerical comparisons hold under scrutiny, the work supplies a practical mixed quantum-classical route for spin-orbit systems that preserves key quantum features at lower cost than full quantum propagation. The explicit Hamiltonian construction and direct side-by-side tests against independent quantum benchmarks constitute a concrete advance over standard Ehrenfest treatments for this class of problems.

major comments (2)
  1. [§4] §4 (harmonic-potential results): the claim that koopmon accuracy is 'unachievable by the Ehrenfest model in both quantum and classical sectors' rests on visual or qualitative agreement; quantitative error metrics (e.g., L2 norms on spin and orbital observables, or fidelity to the full-quantum wavefunction) and their dependence on particle number or time-step are not reported, making it impossible to judge whether the improvement is systematic or parameter-specific.
  2. [§3.1] §3.1 (Koopman extension): the retention of the Heisenberg principle under Rashba coupling is asserted via the wavefunction representation, yet the explicit form of the extended Koopman Hamiltonian (including the spin-orbit term) and the proof that the commutator structure is preserved are not derived; without this step the central advantage over Ehrenfest remains formal rather than demonstrated.
minor comments (3)
  1. [Abstract] The abstract states a 'slight loss in spin accuracy' relative to Ehrenfest in the free-nanowire case; this should be quantified (e.g., by reporting the time-averaged spin expectation error) so readers can weigh the trade-off against the improved orbital dynamics.
  2. [Figures] Figure captions and axis labels in the comparison plots should explicitly state the number of koopmon particles, the time-step, and the full-quantum basis size used, to allow reproducibility.
  3. [§5] The final cat-state test case is presented without a quantitative measure of coherence or visibility; adding a simple fidelity or Wigner-function overlap metric would strengthen the claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments. We address the two major points below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [§4] §4 (harmonic-potential results): the claim that koopmon accuracy is 'unachievable by the Ehrenfest model in both quantum and classical sectors' rests on visual or qualitative agreement; quantitative error metrics (e.g., L2 norms on spin and orbital observables, or fidelity to the full-quantum wavefunction) and their dependence on particle number or time-step are not reported, making it impossible to judge whether the improvement is systematic or parameter-specific.

    Authors: We agree that quantitative error metrics would make the comparison more rigorous. In the revised manuscript we will add L2-norm errors for the spin and orbital observables (relative to the full-quantum benchmark) as functions of time, together with a brief discussion of their dependence on particle number and time step. This will allow readers to assess the systematic nature of the improvement over Ehrenfest dynamics. revision: yes

  2. Referee: [§3.1] §3.1 (Koopman extension): the retention of the Heisenberg principle under Rashba coupling is asserted via the wavefunction representation, yet the explicit form of the extended Koopman Hamiltonian (including the spin-orbit term) and the proof that the commutator structure is preserved are not derived; without this step the central advantage over Ehrenfest remains formal rather than demonstrated.

    Authors: The extended Koopman Hamiltonian that incorporates the Rashba term is written explicitly in Eq. (3) of Section 3.1, and the preservation of the commutator structure follows from the underlying Koopman-wavefunction representation. To make this step fully explicit we will insert a short derivation of the Hamiltonian and a one-paragraph proof that the relevant commutators remain unchanged under the Rashba coupling. This addition will be placed in the revised version of Section 3.1. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior Koopman framework; central results independent of fitted inputs

full rationale

The paper extends the koopmon particle scheme (previously implemented for classical mechanics via Koopman wavefunctions) to Rashba spin-orbit coupling in nanowire models. The derivation chain relies on explicit Hamiltonian construction and numerical integration that are self-contained within the manuscript. Central claims are validated by direct comparison to independent fully quantum simulations and Ehrenfest dynamics, which serve as external benchmarks rather than internal fits. No self-definitional reduction, fitted parameter renamed as prediction, or load-bearing uniqueness theorem from overlapping authors appears in the reported regimes. The retention of Heisenberg uncertainty follows from the Koopman representation as adopted from prior work, but this does not force the numerical outcomes. Overall, the results remain falsifiable against external quantum benchmarks, justifying a low circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the validity of extending Koopman wavefunctions to include Rashba spin-orbit terms while preserving quantum-classical consistency.

axioms (2)
  • standard math Standard quantum mechanics for spin-1/2 particles
    Invoked for the quantum spin sector in the nanowire models.
  • domain assumption Classical mechanics for orbital momentum
    Orbital degrees of freedom are treated classically in the mixed model.
invented entities (1)
  • Koopman wavefunctions extended to spin-orbit coupling no independent evidence
    purpose: To retain Heisenberg principle and correlation effects in the quantum-classical Hamiltonian
    Introduced via prior Koopman work and extended here to Rashba systems

pith-pipeline@v0.9.0 · 5591 in / 1309 out tokens · 50378 ms · 2026-05-14T23:57:17.771259+00:00 · methodology

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Forward citations

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