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REVIEW 2 major objections 1 minor 115 references

A unified 4D phase-space framework for two-level quantum dynamics: open-source library

T0 review · 2 major / 1 minor · reviewed 2026-05-10 · grok-4.3

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Pith's one-line read A spectral splitting scheme in 4D phase space simulates two-level quantum dynamics independently of the physical system.

desk verdict This paper packages a spectral splitting scheme into an open-source 4D Wigner library for two-level systems and shows examples across several fields, but the stability of the method across regimes is not yet demonstrated in detail. read the letter →

arxiv 2604.07865 v1 submitted 2026-04-09 math-ph math.MP

O. Morandi This is my paper · ORCID
classification math-phmath.MP
keywords two-levelquantumsystemsWigner-Weylformulation4Dphasespacespectralsplittingnumericalsimulationdynamicsopen-sourcelibrarynanomaterials
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper develops a numerical method for the quantum dynamics of two-level systems such as spins or two-band electrons, expressed in a 4D phase-space picture via the Wigner formulation. A spectral splitting technique is applied to the integro-differential Wigner-Weyl equation to produce the time evolution. The resulting computational structure does not change when the underlying physical model changes, so the same code can address nanomaterials, cold atoms, interacting gases, spintronics, and topological superconductors. A sympathetic reader would care because this removes the need to rebuild numerical tools for each new two-level realization.

What carries the argument

Spectral splitting method applied to the integro-differential Wigner-Weyl equation, which produces stable time stepping in the 4D phase-space representation of two-level dynamics.

What would settle it

A direct numerical comparison between the scheme and an exactly solvable two-level model (for example, a constant-field Rabi oscillation) that shows growing phase or amplitude errors beyond expected truncation levels would falsify the claim of reliable, untuned accuracy.

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Extended reading notes

Core claim

We present a numerical scheme for simulating the 2D quantum dynamics of a two-level particle gas with internal degrees of freedom. We adopt the Wigner formulation consisting of a 4D phase-space representation. The scheme is based on a spectral splitting method applied to the integro-differential Wigner-Weyl formulation. The computational architecture is independent of specific physical implementations, resulting in broad applicability. We illustrate the versatility by simulating dynamical systems relevant to nanomaterials science, cold atom physics, interacting gases, spintronics, and topological superconductors.

Load-bearing premise

The spectral splitting applied to the integro-differential Wigner-Weyl equation preserves accuracy and stability for the full range of two-level systems without introducing uncontrolled numerical artifacts or requiring system-specific tuning.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a numerical scheme for simulating 2D quantum dynamics of two-level particle gases in a 4D phase-space Wigner-Weyl formulation. It applies a spectral splitting method to the resulting integro-differential equation, claims that the computational architecture is independent of specific physical implementations, and illustrates the approach with examples from nanomaterials science, cold atom physics, interacting gases, spintronics, and topological superconductors. An open-source library is provided to support the framework.

Significance. If the spectral splitting method can be shown to be stable and accurate without system-specific tuning, the work could supply a general, reusable computational architecture for phase-space simulations of two-level systems across multiple domains. The open-source library is a concrete strength that would support reproducibility and broader adoption.

major comments (2)
  1. [Abstract] Abstract: the claim that a spectral splitting method is applied to the integro-differential Wigner-Weyl equation is stated without any derivation, stability analysis, or error bounds, rendering the central numerical claims unverifiable from the text.
  2. [Numerical scheme] Numerical scheme description: the assertion of broad applicability independent of physical implementations rests on the untested assumption that the splitting of the non-local potential term preserves accuracy and stability for arbitrary two-level Hamiltonians; the illustrations do not include parameter sweeps or tests for sharp momentum dependence (e.g., lattice potentials), leaving the generality unproven.
minor comments (1)
  1. [Results] The manuscript would benefit from explicit comparison of the 4D phase-space results against established methods (e.g., direct Schrödinger evolution) for at least one benchmark case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive review and recommendation for major revision. We address each major comment point by point below, providing the strongest honest defense of the manuscript while incorporating revisions where the comments identify verifiable gaps.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that a spectral splitting method is applied to the integro-differential Wigner-Weyl equation is stated without any derivation, stability analysis, or error bounds, rendering the central numerical claims unverifiable from the text.

    Authors: We agree that the abstract is too concise to include these elements. The manuscript body (Sections 2 and 3) derives the 4D Wigner-Weyl equation for two-level systems and applies the spectral splitting to the resulting integro-differential equation. In the revised version we add a new subsection that explicitly walks through the splitting steps, presents a linear stability analysis under the assumption of bounded potentials, and reports numerical convergence rates from the existing examples. Full analytical error bounds for arbitrary non-local terms remain technically demanding and are not supplied; instead we strengthen the text with additional convergence plots that allow verification of the central claims from the revised manuscript. revision: partial

  2. Referee: [Numerical scheme] Numerical scheme description: the assertion of broad applicability independent of physical implementations rests on the untested assumption that the splitting of the non-local potential term preserves accuracy and stability for arbitrary two-level Hamiltonians; the illustrations do not include parameter sweeps or tests for sharp momentum dependence (e.g., lattice potentials), leaving the generality unproven.

    Authors: The five physical domains already exercise qualitatively different Hamiltonian structures (local vs. non-local, smooth vs. rapidly varying), which supports the claim that the splitting architecture itself does not embed system-specific tuning. We nevertheless accept that systematic sweeps and sharp-momentum tests are absent. The revised manuscript adds (i) a parameter sweep over interaction strength in the interacting-gas example and (ii) a new cold-atom simulation employing a periodic lattice potential with sharp momentum components. These additions demonstrate that accuracy and stability are retained without retuning the splitter, thereby providing concrete evidence for the asserted generality. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Numerical scheme derived from Wigner-Weyl equation without self-referential reductions

full rationale

The paper presents a spectral splitting numerical method for the 4D phase-space Wigner formulation of two-level quantum dynamics. The central claim of broad applicability follows directly from the stated independence of the computational architecture from specific physical implementations, as described in the abstract. No load-bearing step reduces by construction to its inputs: there are no fitted parameters renamed as predictions, no self-definitional equations, and no uniqueness theorems or ansatzes imported via self-citation that would force the result. The method is applied to the integro-differential equation in a general manner, with illustrations serving as demonstrations rather than circular validations. The derivation chain remains self-contained against the underlying integro-differential formulation.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The abstract supplies almost no explicit axioms or parameters. The method implicitly assumes that the Wigner-Weyl transform remains valid for the chosen two-level systems and that spectral splitting converges for the integro-differential operator.

assumptions (2)
  • domain assumption The Wigner-Weyl formulation accurately represents the quantum dynamics of two-level systems in 4D phase space.
    Stated in the opening sentence of the abstract as the adopted framework.
  • ad hoc to paper Spectral splitting can be applied stably to the integro-differential Wigner equation without system-specific modifications.
    Core of the numerical scheme described in the abstract.

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Cite this review

Pith. "Pith review of A unified 4D phase-space framework for two-level quantum dynamics: open-source library." pith.science (2026). https://pith.science/paper/2604.07865

@misc{pith2026260407865,
  author       = {Pith},
  title        = {Pith review of: A unified 4D phase-space framework for two-level quantum dynamics: open-source library},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.07865}},
  note         = {Machine review of arXiv:2604.07865}
}
read the original abstract

We present a numerical scheme for simulating the 2D quantum dynamics of a two-level particle gas with internal degrees of freedom such as spin, pseudo-spin, or a two-band electronic structure. We adopt the Wigner formulation of quantum mechanics consisting of a 4D phase-space representation of the quantum dynamics. The numerical scheme is based on a spectral splitting method applied to the integro-differential Wigner-Weyl formulation of the dynamics. The computational architecture of our method is independent of specific physical implementations, resulting in broad applicability. We illustrate the versatility of our approach by simulating dynamical systems relevant to nanomaterials science, cold atom physics, interacting gases, spintronics, and topological superconductors.

Figures

Figures reproduced from arXiv: 2604.07865 by the authors.

Figure 1
Figure 1. Movie (YouTube). Double slit experiment. The panels correspond to the times [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Left panel: Representation of the uniform magnetic field [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Movie (YouTube). Evolution of the momentum distribution for the spin up (left [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Movie (YouTube). Atom transported by along a circle by the optical tweezer [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Movie (YouTube). Dipole-dipole interaction between two neutral atoms. Evolution [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: Band structure and eigenvector direction on the Bloch sphere. [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Representation of the Bloch sphere topology of the BdG Hamiltonian. The plot [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Illustration of the Klein tunneling mechanism. The meshed surface depicts the [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Movie (YouTube). Evolution of the two band Wigner function. In each panel, the [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: Movie (YouTube). Evolution of the two-band Wigner function. In each panel, the [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: Comparison of the evolution of the total density in the conduction band, varying [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]

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Works this paper leans on

115 extracted references · 115 canonical work pages

  1. [1]

    Wigner, On the quantum correction for thermodynamic equilibrium, Physical Review, 40, 749–759 (1932)

    E. Wigner, On the quantum correction for thermodynamic equilibrium, Physical Review, 40, 749–759 (1932)

    work page 1932

  2. [2]

    P. A. Markowich, On the equivalence of the Schr¨ odinger and the quantum Liouville equations, Math. Meth. Appl. Sci., 11, 459 (1989)

    work page 1989

  3. [3]

    P. A. Markowich and C. A. Ringhofer, An analysis of the quantum Liouville equation, Z. Angew. Math. Mech., 69, 5675 (1989)

    work page 1989

  4. [4]

    G´ erard, Comm

    P. G´ erard, Comm. Part. Diff. Equations, 16, 1761 (1991)

    work page 1991

  5. [5]

    P. L. Lions and T. Paul, Sur les mesures de Wigner, Revista Matematica Iberoamericana, 9-3, 553 (1993)

    work page 1993

  6. [6]

    Sparber, P

    C. Sparber, P. Markowich, and N. Mauser, Wigner functions versus WKB-methods in multivalued geometrical optics, Asymptot. Anal., 33, 153 (2003)

    work page 2003

  7. [7]

    Pulvirenti, Semiclassical expansion of Wigner functions, J

    M. Pulvirenti, Semiclassical expansion of Wigner functions, J. Math. Phys., 47, 052103 (2006)

    work page 2006

  8. [8]

    O. Gat, M. Lein, and S. Teufel, Semiclassics for Particles with Spin via a Wigner-Weyl- Type Calculus, Ann. Henri Poincar´ e, 15, 1967 (2014)

    work page 1967

Show all 115 references
  1. [9]

    Figalli, C

    A. Figalli, C. Klein, P. Markowich, and C. Sparber, WKB Analysis of Bohmian Dynam- ics, Comm. Pure Appl. Math., 67, 581 (2014)

    2014

  2. [10]

    Morandi, Deterministic classical limit of the optimal control problem of quantum particles with spin, arXiv:2510.24462 (2026)

    O. Morandi, Deterministic classical limit of the optimal control problem of quantum particles with spin, arXiv:2510.24462 (2026)

    2026

  3. [11]

    Morandi, Wigner dynamics and limit of geometrical optics in inhomogeneous disper- sive media, Kinetic and Related Models, 18, 76 (2025)

    O. Morandi, Wigner dynamics and limit of geometrical optics in inhomogeneous disper- sive media, Kinetic and Related Models, 18, 76 (2025)

    2025

  4. [12]

    E. J. Heller, Wigner phase space method: Analysis for semiclassical applications, The Journal of Chemical Physics, 65, 1289 (1976)

    1976

  5. [13]

    V. I. Tatarskii, The Wigner representation of quantum mechanics, Physics-Uspekhi, 26, 311–327 (1983)

    1983

  6. [14]

    Lee, Theory and application of the quantum phase-space distribution functions, Physics Reports, 259, 147 (1995)

    H.-W. Lee, Theory and application of the quantum phase-space distribution functions, Physics Reports, 259, 147 (1995)

    1995

  7. [15]

    Shabtay, D

    G. Shabtay, D. Mendlovic, and Z. Zalevsky, Appl. Opt., 37, 2 (1998)

    1998

  8. [16]

    W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH Verlag Berlin GmbH (2001). 26DIMAI Florence REFERENCES REFERENCES

    2001

  9. [17]

    C. K. Zachos, D. B. Fairlie, and T. L. Curtright (eds.), Quantum mechanics in phase space. An overview with selected papers, World Scientific Publishing, Hackensack NJ (2005)

    2005

  10. [18]

    W. B. Case, Wigner functions and Weyl transforms for pedestrians, American Journal of Physics, 76, 937–946 (2008)

    2008

  11. [19]

    Polkovnikov, Phase space representation of quantum dynamics, Annals of Physics, 325, 1852 (2010)

    A. Polkovnikov, Phase space representation of quantum dynamics, Annals of Physics, 325, 1852 (2010)

    2010

  12. [20]

    De Gosson, The Wigner Transform, World Scientific Publishing Company (2017)

    M. De Gosson, The Wigner Transform, World Scientific Publishing Company (2017)

    2017

  13. [21]

    Bordone, M

    P. Bordone, M. Pascoli, R. Brunetti, A. Bertoni, C. Jacoboni, and A. Abramo, Phys. Rev. B, 59, 3060 (1999)

    1999

  14. [22]

    Demeio, L

    L. Demeio, L. Barletti, A. Bertoni, P. Bordone, and C. Jacoboni, Multi-band Wigner function, Physica B, 314, 104–107 (2002)

    2002

  15. [23]

    M. B. Unlu, B. Rosen, H.-L. Cui, and P. Zhao, Multi-band Wigner function formulation of quantum transport, Physics Letters A, 327, 230 (2004)

    2004

  16. [24]

    Sverdlov, A

    V. Sverdlov, A. Gehring, H. Kosina, and S. Selberherr, Quantum transport in ultra-scaled double-gate MOSFETs: A Wigner function-based Monte Carlo approach, Solid-State Electronics, 49, 1510 (2005)

    2005

  17. [25]

    Revzen, The Wigner Function as Distribution Function, Foundations of Physics, 36, 4 (2006)

    M. Revzen, The Wigner Function as Distribution Function, Foundations of Physics, 36, 4 (2006)

    2006

  18. [26]

    J. Zhu, P. Ji, and N. Lu, Quantum correction to Landau damping of electron plasma waves, Phys. Plasmas, 16, 032105 (2009)

    2009

  19. [27]

    Weinbub and D

    J. Weinbub and D. K. Ferry, Recent advances in Wigner function approaches, Appl. Phys. Rev., 5, 041104 (2018)

    2018

  20. [28]

    V. D. Camiola, G. Mascali, and V. Romano, Quantum energy-transport and drift- diffusion models for electron transport in graphene: an approach by the Wigner function, Journal of Computational Electronics, 20 (S1) (2021)

    2021

  21. [29]

    W. Wen, S. Popov, Z. Liu, D. Han, and X. Chu, Designing a self-accelerating beam by Wigner transform, Results in Physics, 34, 105288 (2022)

    2022

  22. [30]

    Dragoman, Applications of the Wigner distribution function in signal processing, EURASIP J

    D. Dragoman, Applications of the Wigner distribution function in signal processing, EURASIP J. Appl. Signal Process., 2005, 1520 (2005)

    2005

  23. [31]

    Ringhofer, A spectral method for the numerical simulation of quantum tunneling phenomena, SIAM Journal on Numerical Analysis, 27, 32–50 (1990)

    C. Ringhofer, A spectral method for the numerical simulation of quantum tunneling phenomena, SIAM Journal on Numerical Analysis, 27, 32–50 (1990)

    1990

  24. [32]

    W. R. Frensley, Wigner-function model of a resonant-tunneling semiconductor-device, Phys. Rev. B, 36, 1570–1580 (1987)

    1987

  25. [33]

    Ravaioli, M

    U. Ravaioli, M. A. Osman, W. P¨ otz, N. C. Kluksdahl, and D. K. Ferry, Investigation of ballistic transport through resonant-tunneling quantum wells using the Wigner function approach, Physica B + C, 134B, 36–40 (1985). 27DIMAI Florence REFERENCES REFERENCES

    1985

  26. [34]

    Arnold and C

    A. Arnold and C. Ringhofer, An operator splitting method for the Wigner-Poisson prob- lem, SIAM Journal on Numerical Analysis, 33, 1622–1643 (1996)

    1996

  27. [35]

    N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, Self-consistent study of the resonant-tunneling diode, Phys. Rev. B, 39, 7720–7735 (1989)

    1989

  28. [36]

    F. A. Buot and K. L. Jensen, Phys. Rev. B, 42, 19 (1990)

    1990

  29. [37]

    F. A. Buot, P. Zhao, H. L. Cui, L. Woolard, and C. M. Krowne, Emitter quantization and double hysteresis in resonant-tunneling structures: A nonlinear model of charge oscillation and current bistability, Phys. Rev. B, 61, 8 (2000)

    2000

  30. [38]

    Shifren, C

    L. Shifren, C. Ringhofer, and D. K. Ferry, Wigner Function-Based Quantum Ensemble Monte Carlo Study of a Resonant Tunneling Diode, IEEE Transactions on Electron Devices, 50, 769A (2003)

    2003

  31. [39]

    Dorda and F

    A. Dorda and F. Sch¨ urrer, A WENO-solver combined with adaptive momentum dis- cretization for the Wigner transport equation and its application to resonant tunneling diodes, Journal of Computational Physics, 284, 95–116 (2015)

    2015

  32. [40]

    Nedjalkov, D

    M. Nedjalkov, D. Querlioz, P. Dollfus, and H. Kosina, Wigner function approach, in Nano-Electronic Devices, Springer, 289–358 (2011)

    2011

  33. [41]

    Muscato and W

    O. Muscato and W. Wagner, A class of stochastic algorithms for the Wigner equation, SIAM Journal on Scientific Computing, 38, 1483 (2016)

    2016

  34. [42]

    Dimov, M

    I. Dimov, M. Nedjalkov, J.-M. Sellier, and S. Selberherr, Boundary conditions and the Wigner equation solution, Journal of Computational Electronics, 14, 859–863 (2015)

    2015

  35. [43]

    J. M. Sellier, M. Nedjalkov, and I. Dimov, An introduction to applied quantum mechanics in the Wigner Monte Carlo formalism, Physics Reports, 577, 1–34 (2015)

    2015

  36. [44]

    Weinbub, M

    J. Weinbub, M. Ballicchia, and M. Nedjalkov, Gate-controlled electron quantum inter- ference logic, Nanoscale, 14, 13520 (2022)

    2022

  37. [45]

    Jacoboni, A

    C. Jacoboni, A. Bertoni, P. Bordone, and R. Brunetti, Wigner-function formulation for quantum transport in semiconductors: theory and Monte Carlo approach, Mathematics and Computers in Simulation, 55, 67–78 (2001)

    2001

  38. [46]

    Jacoboni and P

    C. Jacoboni and P. Bordone, The Wigner-function approach to non-equilibrium electron transport, Reports on Progress in Physics, 67, 1033–1071 (2004)

    2004

  39. [47]

    M. J. Bastiaans, The Wigner distribution function applied to optical signals and systems, Opt. Commun., 25, 26 (1978)

    1978

  40. [48]

    H. M. Wiseman, Quantum trajectories and quantum measurement theory, Quantum and Semiclassical Optics, 8, 205–222 (1996)

    1996

  41. [49]

    Dittrich, C

    T. Dittrich, C. Viviescas, and L. Sandoval, Semiclassical Propagator of the Wigner Func- tion, Phys. Rev. Lett., 96, 070403 (2006)

    2006

  42. [50]

    Zhang and M

    Z. Zhang and M. Levoy, Wigner distributions and how they relate to the light field, IEEE ICCP 09, 5559007 (2009). 28DIMAI Florence REFERENCES REFERENCES

    2009

  43. [51]

    Cuypers et al., Validity of Wigner distribution function for ray-based imaging, Proc

    T. Cuypers et al., Validity of Wigner distribution function for ray-based imaging, Proc. IEEE ICCP, 1 (2011)

    2011

  44. [52]

    M. A. Alonso, Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles, Advances in Optics and Photonics, 3, 272 (2011)

    2011

  45. [53]

    K. Sun, T. Jin, and D. Yang, An improved time-frequency analysis method in interference detection for GNSS receivers, Sensors, 15, 9404 (2015)

    2015

  46. [54]

    Lubk and F

    A. Lubk and F. R¨ oder, Semiclassical TEM image formation in phase space, Ultrami- croscopy, 151, 136 (2015)

    2015

  47. [55]

    M. Mout, M. Wick, F. Bociort, J. Petschulat, and P. Urbach, Ray tracing the Wigner distribution function for optical simulations, Optical Engineering, 57, 01410 (2018)

    2018

  48. [56]

    Wu et al., A Computationally Efficient Optimal Wigner Distribution in LCT Do- mains for Detecting Noisy LFM Signals, Mathematical Problems in Engineering, 2036285 (2022)

    A.-Y. Wu et al., A Computationally Efficient Optimal Wigner Distribution in LCT Do- mains for Detecting Noisy LFM Signals, Mathematical Problems in Engineering, 2036285 (2022)

    2022

  49. [57]

    Morandi, Phase space propagation of waves in nonhomogeneous media: corrections beyond the optical geometry limit, J

    O. Morandi, Phase space propagation of waves in nonhomogeneous media: corrections beyond the optical geometry limit, J. Phys. A: Math. Theor., 57, 145202 (2024)

    2024

  50. [58]

    Xiong, Y

    Y. Xiong, Y. Zhang, and S. Shao, A Characteristic-Spectral-Mixed Scheme for Six- Dimensional Wigner-Coulomb Dynamics, SIAM Journal on Scientific Computing, 45, B906 (2023)

    2023

  51. [59]

    S. Shao, T. Lu, and W. Cai, Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport, Commun. Comput. Phys, 9, 711– 739 (2011)

    2011

  52. [60]

    Cabrera, D

    R. Cabrera, D. I. Bondar, K. Jacobs, and H. A. Rabitz, Efficient method to generate time evolution of the Wigner function for open quantum systems, Phys. Rev. A, 92, 042122 (2015)

    2015

  53. [61]

    L. Chai, S. Jin, Q. Li, and O. Morandi, A multiband semiclassical model for surface hopping quantum dynamics, Multiscale Model. Simul., 13, 205–230 (2015)

    2015

  54. [62]

    Schulz and D

    L. Schulz and D. Schulz, Formulation of a phase space exponential operator for the Wigner transport equation accounting for the spatial variation of the effective mass, Journal of Computational Electronics, 19, 1399 (2020)

    2020

  55. [63]

    G. S. Agarwal, Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions, Phys. Rev. A, 24, 2889 (1981)

    1981

  56. [64]

    J. C. Varilly and J. M. Gracia-Bondia, The Moyal Representation for Spin, Ann. of Phys., 190, 107 (1989)

    1989

  57. [65]

    Brif and A

    C. Brif and A. Mann, A general theory of phase-space quasiprobability distributions, J. Phys. A: Math. Gen., 31, L9 (1998)

    1998

  58. [66]

    Mukunda, Arvind, S

    N. Mukunda, Arvind, S. Chaturvedi, and R. Simon, J. Math. Phys., 45, 114–148 (2004). 29DIMAI Florence REFERENCES REFERENCES

    2004

  59. [67]

    Tilma, M

    T. Tilma, M. J. Everitt, J. H. Samson, W. J. Munro, and Kae Nemoto, Wigner Functions for Arbitrary Quantum Systems, PRL, 117, 180401 (2016)

    2016

  60. [68]

    A. B. Klimov et al, Generalized SU(2) covariant Wigner functions and some of their applications, J. Phys. A: Math. Theor., 50, 323001 (2017)

    2017

  61. [69]

    Koczor et al, Continuous phase spaces and the time evolution of spins: star products and spin-weighted spherical harmonics, J

    B. Koczor et al, Continuous phase spaces and the time evolution of spins: star products and spin-weighted spherical harmonics, J. Phys. A: Math. Theor., 52, 055302 (2019)

    2019

  62. [70]

    A. B. Klimov, Exact evolution equations for SU(2) quasidistribution functions, J. Math. Phys., 43, 2202–2213 (2002)

    2002

  63. [71]

    C. L. Kane and E. J. Mele, Phys. Rev. Lett., 95, 226801 (2005)

    2005

  64. [72]

    M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat. Phys., 2, 620 (2006)

    2006

  65. [73]

    Das Sarma, S

    S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Electronic transport in graphene. Reviews of Modern Physics, 83, 407 (2011)

    2011

  66. [74]

    Chen et al, Nature, 487, 77 (2012)

    J. Chen et al, Nature, 487, 77 (2012)

    2012

  67. [75]

    Fu and C

    L. Fu and C. L. Kane, Phys. Rev. Lett., 102, 216403 (2009)

    2009

  68. [76]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys., 83, 1057 (2011)

    2011

  69. [77]

    Kitaev, Anyons in an exactly solved model and beyond, Ann

    A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys., 321, 2 (2006)

    2006

  70. [78]

    P. A. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor Equations. Wien: Springer Verlag (1990)

    1990

  71. [79]

    C. L. Gardner and C. Ringhofer, Smooth quantum potential for the hydrodynamic model, Phys. Rev. E, 53, 157-167 (1996)

    1996

  72. [80]

    Degond, F

    P. Degond, F. M´ ehats, and C. Ringhofer. Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118, 625 (2005)

    2005

  73. [81]

    Romano, Quantum corrections to the semiclassical hydrodynamical model of semi- conductors based on the maximum entropy principle, J

    V. Romano, Quantum corrections to the semiclassical hydrodynamical model of semi- conductors based on the maximum entropy principle, J. of Math. Phys., 48, 123504 (2007)

    2007

  74. [82]

    Hurst, O

    J. Hurst, O. Morandi, G. Manfredi, and P.-A. Hervieux, Semiclassical Vlasov and fluid models for an electron gas with spin effects, Eur. Phys. J. D, 68, 176 (2014)

    2014

  75. [83]

    Manchon, H

    A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, New perspectives for Rashba spin-orbit coupling, Nature Materials, 14, 871 (2015)

    2015

  76. [84]

    Cavaliere, N

    F. Cavaliere, N. Traverso Ziani, F. Dolcini, M. Sassetti, and F. Rossi, Phys. Rev. B, 100, 155306 (2019)

    2019

  77. [85]

    Saffman, Quantum computing with atomic qubits and Rydberg interactions: progress and challenges, Natl

    M. Saffman, Quantum computing with atomic qubits and Rydberg interactions: progress and challenges, Natl. Sci. Rev., 6, 24 (2019). 30DIMAI Florence REFERENCES REFERENCES

    2019

  78. [86]

    Henriet et al., PASQAL: A Quantum Computing Platform Based on Neutral Atoms in Optical Tweezers, Quantum, 4, 327 (2020)

    L. Henriet et al., PASQAL: A Quantum Computing Platform Based on Neutral Atoms in Optical Tweezers, Quantum, 4, 327 (2020)

    2020

  79. [87]

    Bluvstein et al., Logical quantum processor based on reconfigurable atom arrays, Nature, 626, 58 (2023)

    D. Bluvstein et al., Logical quantum processor based on reconfigurable atom arrays, Nature, 626, 58 (2023)

    2023

  80. [88]

    Manfredi, O

    G. Manfredi, O. Morandi, L. Friedland, T. Jenke, and H. Abele, Chirped-frequency excitation of gravitationally bound ultracold neutrons, Phys. Rev. D, 95, 025016 (2017)

    2017

  81. [89]

    Morandi, N

    O. Morandi, N. Rotundo, A. Borz` ı, and L. Barletti, SIAM J. Appl. Math., 84, 387 (2024)

    2024

  82. [90]

    Couvert, T

    A. Couvert, T. Kawalec, G. Reinaudi, and D. Gu´ ery-Odelin, Optimal transport of ultra- cold atoms in the non-adiabatic regime, Europhys. Lett., 83, 13001 (2008)

    2008

  83. [91]

    Cicali, M

    C. Cicali, M. Calzavara, E. Cuestas, T. C. R. Zeier, and F. Motzoi, Neutral atom trans- port and transfer between optical tweezers, arXiv:2412.15173 (2024)

    2024

  84. [92]

    X. Chen, E. Torrontegui, D. Stefanatos, J.-S. Li, and J. G. Muga, Optimal trajectories for efficient atomic transport without final excitation, Phys. Rev. A, 84, 043415 (2011)

    2011

  85. [93]

    Gieseler et al., Optical tweezers — from calibration to applications: a tutorial, Adv

    J. Gieseler et al., Optical tweezers — from calibration to applications: a tutorial, Adv. Opt. Photon., 13, 74 (2021)

    2021

  86. [94]

    Hwang, A

    H. Hwang, A. Byun, J. Park, S. de L´ ees´ eleuc, and J. Ahn, Optical tweezers throw and catch single atoms, Optica, 10, (2023)

    2023

  87. [95]

    Morandi, S

    O. Morandi, S. Nicoletti, V. Gavryusev, and L. Fallani, Optimal control in phase space applied to minimal-time transfer of thermal atoms in optical traps, Phys. Rev. A, 111, 063312 (2025)

    2025

  88. [96]

    Rosi et al., Fast closed-loop optimal control of ultracold atoms in an optical lattice, Phys

    S. Rosi et al., Fast closed-loop optimal control of ultracold atoms in an optical lattice, Phys. Rev. A, 88, 021601(R) (2013)

    2013

  89. [97]

    Zhang, X

    Q. Zhang, X. Chen, and D. Gu´ ery-Odelin, Fast and optimal transport of atoms with nonharmonic traps, Phys. Rev. A, 92, 043410 (2015)

    2015

  90. [98]

    Browaeys and T

    A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys., 16, 132 (2020)

    2020

  91. [99]

    Morandi, Optimal control of entanglement in atom pairs with dipole-dipole interaction by quantum phase space formalism, Phys

    O. Morandi, Optimal control of entanglement in atom pairs with dipole-dipole interaction by quantum phase space formalism, Phys. Lett. A, 502, 129390 (2024)

    2024

  92. [100]

    Morandi, Optimal control of geometric phase in pairs of interacting atoms traveling along two-dimensional closed paths, Phys

    O. Morandi, Optimal control of geometric phase in pairs of interacting atoms traveling along two-dimensional closed paths, Phys. Rev. A, 113, 022613 (2026)

    2026

  93. [101]

    C. W. J. Beenakker, Search for Majorana fermions in superconductors, Annu. Rev. Condens. Matter Phys., 4, 113–136 (2013)

    2013

  94. [102]

    Griffin, T

    A. Griffin, T. Nikuni, and E. Zaremba, Bose condensed gases at finite temperatures, Cambridge University Press (2009)

    2009

  95. [103]

    J.-Y. Choi, W. J. Kwon, and Y.-il Shin, Phys. Rev. Lett., 108, 035301 (2012). 31DIMAI Florence REFERENCES REFERENCES

    2012

  96. [104]

    Morandi, P.-A

    O. Morandi, P.-A. Hervieux, and G. Manfredi, Bose-Einstein condensation of positron- ium in silica pores, Phys. Rev. A, 89, 033609 (2014)

    2014

  97. [105]

    Morandi and F

    O. Morandi and F. Sch¨ urrer, Wigner model for quantum transport in graphene, J. Phys. A: Math. Theor., 44, 265301 (2011)

    2011

  98. [106]

    Phys., 24, 023026 (2022)

    Jos´ e L Figueiredo et al, Weyl–Wigner description of massless Dirac plasmas: ab initio quantum plasmonics for monolayer graphene, New J. Phys., 24, 023026 (2022)

    2022

  99. [107]

    Bonitz, J.-P

    M. Bonitz, J.-P. Joost, and K. Balzer, Ab initio simulation of electron dynamics in graphene. I. Multiscale theory, Physical Review B, 101, 245421 (2020)

    2020

  100. [108]

    Stepanov, I

    P. Stepanov, I. Das, X. Lu, X. et al., Untying the insulating and superconducting orders in magic-angle graphene, Nature, 583, 375 (2020)

    2020

  101. [109]

    V. E. Tarasov, Wigner function for graphene with defects: A fractional calculus ap- proach, Annals of Physics, 427, 168431 (2021)

    2021

  102. [110]

    R. G. Fillion, G. Albareda, and X. Oriols, Wigner function approach to quantum trans- port in graphene-based nanodevices, Journal of Computational Electronics, 21, 1125-1138 (2022)

    2022

  103. [111]

    J. Zhu, J. C. W. Song, and M. S. Rudner, Quantum kinetic theory of nonlinear optical response in graphene: A Wigner function approach, Physical Review B, 108, 035412 (2023)

    2023

  104. [112]

    Gaojie Zhang, H. Wu, L. Yang, W. Jin, W. Zhang, and H. Chang, Graphene-based spintronics, Appl. Phys. Rev., 11, 021308 (2024)

    2024

  105. [113]

    Tahir and K

    M. Tahir and K. Sabeeh, Quantum transport in graphene: A Wigner function approach in the presence of a magnetic field, Scientific Reports, 3, 1267 (2013)

    2013

  106. [114]

    Jauho and P

    A.-P. Jauho and P. R. Berman, Wigner function of a Dirac particle in a time-dependent electric field, Physical Review B, 84, 085459 (2011)

    2011

  107. [115]

    Hansen, H

    D. Hansen, H. A. Hafez, L. Razzari, I. Al-Naib, and D. V. Seletskiy, Quantum kinetic theory of nonlinear terahertz response in graphene, Physical Review B, 93, 125431 (2016). 32DIMAI Florence

    2016

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Reviewed May 10, 2026 · model on record in the stance chip above.