Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-27T13:07:07.468Z Has data issue: false hasContentIssue false
  • English
  • Français

Vlasov–Maxwell equations with spin effects

Published online by Cambridge University Press:  27 April 2023

Nicolas Crouseilles ,
Paul-Antoine Hervieux ,
Xue Hong  and
Giovanni Manfredi Open the ORCID record for Giovanni Manfredi [Opens in a new window]
Nicolas Crouseilles*
Affiliation:
Univ Rennes and Inria centre de l'université de Rennes and IRMAR UMR 6625, 35042 Rennes, France
Paul-Antoine Hervieux
Affiliation:
Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France
Xue Hong
Affiliation:
Univ Rennes and Inria centre de l'université de Rennes and IRMAR UMR 6625, 35042 Rennes, France
Giovanni Manfredi*
Affiliation:
Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France
*
†Email addresses for correspondence: nicolas.crouseilles@inria.fr, giovanni.manfredi@ipcms.unistra.fr
†Email addresses for correspondence: nicolas.crouseilles@inria.fr, giovanni.manfredi@ipcms.unistra.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a numerical method to solve the Vlasov–Maxwell equations for spin-1/2 particles, in a semiclassical approximation where the orbital motion is treated classically while the spin variable is fully quantum. Unlike the spinless case, the phase-space distribution function is a $2\times 2$ matrix, which can also be represented, in the Pauli basis, as one scalar function $f_0$ and one three-component vector function $\boldsymbol f$. The relationship between this ‘vectorial’ representation and the fully scalar representation on an extended phase space first proposed by Brodin et al. (Phys. Rev. Lett., vol. 101, 2008, p. 245002) is analysed in detail. By means of suitable approximations and symmetries, the vectorial spin-Vlasov–Maxwell model can be reduced to two-dimensions in the phase space, which is amenable to numerical solutions using a high-order grid-based Eulerian method. The vectorial model enjoys a Poisson structure that paves the way to accurate Hamiltonian split-time integrators. As an example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, and compare the results with those obtained earlier with the scalar spin-Vlasov–Maxwell model and a particle-in-cell code.

Keywords

plasma simulationquantum plasmaplasma dynamics
Type
Research Article
Information
Journal of Plasma Physics , Volume 89 , Issue 2 , April 2023 , 905890215
Check for updates
Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beaurepaire, E., Merle, J.-C., Daunois, A. & Bigot, J.-Y. 1996 Ultrafast spin dynamics in ferromagnetic nickel. Phys. Rev. Lett. 76 (22), 4250.CrossRefGoogle ScholarPubMed
Bigot, J.-Y. & Vomir, M. 2013 Ultrafast magnetization dynamics of nanostructures. Ann. Phys. 525 (1–2), 2–30.CrossRefGoogle Scholar
Bigot, J.-Y., Vomir, M. & Beaurepaire, E. 2009 Coherent ultrafast magnetism induced by femtosecond laser pulses. Nat. Phys. 5 (7), 515–520.CrossRefGoogle Scholar
Brodin, G., Holkundkar, A. & Marklund, M. 2013 Particle-in-cell simulations of electron spin effects in plasmas. J. Plasma Phys. 79 (4), 377–382.CrossRefGoogle Scholar
Brodin, G., Marklund, M., Zamanian, J., Ericsson, Å. & Mana, P.L. 2008 Effects of the g factor in semiclassical kinetic plasma theory. Phys. Rev. Lett. 101 (24), 245002.CrossRefGoogle Scholar
Brodin, G., Marklund, M., Zamanian, J. & Stefan, M. 2011 Spin and magnetization effects in plasmas. Plasma Phys. Control. Fusion 53 (7), 074013.CrossRefGoogle Scholar
Choi, G.-M., Min, B.-C., Lee, K.-J. & Cahill, D.G. 2014 Spin current generated by thermally driven ultrafast demagnetization. Nat. Commun. 5 (1), 1–8.CrossRefGoogle ScholarPubMed
Crestetto, A., Crouseilles, N., Li, Y. & Massot, J. 2022 Comparison of high-order Eulerian methods for electron hybrid model. J. Comput. Phys. 451, 110857.CrossRefGoogle Scholar
Crouseilles, N., Einkemmer, L. & Faou, E. 2015 Hamiltonian splitting for the Vlasov–Maxwell equations. J. Comput. Phys. 283, 224–240.CrossRefGoogle Scholar
Crouseilles, N., Hervieux, P.-A., Li, Y., Manfredi, G. & Sun, Y. 2021 Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects. J. Plasma Phys. 87 (3).CrossRefGoogle Scholar
Crouseilles, N., Mehrenberger, M. & Sonnendrücker, E. 2010 Conservative semi-lagrangian schemes for Vlasov equations. J. Comput. Phys. 229 (6), 1927–1953.CrossRefGoogle Scholar
Dauger, D., Decyk, V. & Dawson, J. 2005 Using semiclassical trajectories for the time-evolution of interacting quantum-mechanical systems. J. Comput. Phys. 209 (2), 559–581.CrossRefGoogle Scholar
Forslund, D., Kindel, J. & Lindman, E. 1975 Theory of stimulated scattering processes in laser-irradiated plasmas. Phys. Fluids 18, 1002.CrossRefGoogle Scholar
Ghizzo, A., Bertrand, P., Shoucri, M., Johnston, T., Fualkow, E. & Feix, M. 1990 A Vlasov code for the numerical simulation of stimulated Raman scattering. J. Comput. Phys. 90 (2), 431–457.CrossRefGoogle Scholar
Hinschberger, Y. & Hervieux, P.-A. 2012 Foldy-Wouthuysen transformation applied to the interaction of an electron with ultrafast electromagnetic fields. Phys. Lett. A 376 (6), 813–819.CrossRefGoogle Scholar
Huot, F., Ghizzo, A., Bertrand, P., Sonnendrücker, E. & Coulaud, O. 2003 Instability of the time splitting scheme for the one-dimensional and relativistic Vlasov–Maxwell system. J. Comput. Phys. 185 (2), 512–531.CrossRefGoogle Scholar
Hurst, J., Hervieux, P.-A. & Manfredi, G. 2018 Spin current generation by ultrafast laser pulses in ferromagnetic nickel films. Phys. Rev. B 97, 014424.CrossRefGoogle Scholar
Hurst, J., Morandi, O., Manfredi, G. & Hervieux, P.-A. 2014 Semiclassical Vlasov and fluid models for an electron gas with spin effects. Eur. Phys. J. D 68 (6), 1–11.CrossRefGoogle Scholar
Hurst, J., Morandi, O., Manfredi, G. & Hervieux, P.-A. 2017 Phase-space methods for the spin dynamics in condensed matter systems. Phil. Trans. R. Soc. Lond. A 375 (20160199).Google ScholarPubMed
Kraus, M., Kormann, K., Morrison, P.J. & Sonnendrücker, E. 2017 Gempic: geometric electromagnetic particle-in-cell methods. J. Plasma Phys. 83 (4).CrossRefGoogle Scholar
Krieger, K., Dewhurst, J., Elliott, P., Sharma, S. & Gross, E. 2015 Laser-induced demagnetization at ultrashort time scales: predictions of TDDFT. J. Chem. Theory Comput. 11 (10), 4870–4874.CrossRefGoogle ScholarPubMed
Krieger, K., Elliott, P., Müller, T., Singh, N., Dewhurst, J., Gross, E. & Sharma, S. 2017 Ultrafast demagnetization in bulk versus thin films: an ab initio study. J. Phys.: Condens. Matter 29 (22), 224001.Google ScholarPubMed
Li, F., Decyk, V.K., Miller, K.G., Tableman, A., Tsung, F.S., Vranic, M., Fonseca, R.A. & Mori, W.B. 2021 Accurately simulating nine-dimensional phase space of relativistic particles in strong fields. J. Comput. Phys. 438, 110367.CrossRefGoogle Scholar
Li, Y., Sun, Y. & Crouseilles, N. 2020 Numerical simulations of one laser-plasma model based on Poisson structure. J. Comput. Phys. 405, 109172.CrossRefGoogle Scholar
Manfredi, G., Hervieux, P.-A. & Crouseilles, N. 2022 Spin effects in ultrafast laser-plasma interactions. Eur. Phys. J. Spec. Top. 1–7. https://doi.org/10.1140/epjs/s11734-022-00669-5.Google Scholar
Manfredi, G., Hervieux, P.-A. & Hurst, J. 2019 Phase-space modeling of solid-state plasmas. Rev. Mod. Plasma Phys. 3 (1), 1–55.CrossRefGoogle Scholar
Marklund, M. & Morrison, P. 2011 Gauge-free hamiltonian structure of the spin Maxwell–Vlasov equations. Phys. Lett. A 375 (24), 2362–2365.CrossRefGoogle Scholar
Marklund, M., Zamanian, J. & Brodin, G. 2010 Spin kinetic theory–quantum kinetic theory in extended phase space. Transp. Theory Stat. Phys. 39 (5–7), 502–523.CrossRefGoogle Scholar
Moldabekov, Z.A., Bonitz, M. & Ramazanov, T. 2018 Theoretical foundations of quantum hydrodynamics for plasmas. Phys. Plasmas 25 (3), 031903.CrossRefGoogle Scholar
Qin, H., Liu, J., Xiao, J., Zhang, R., He, Y., Wang, Y., Sun, Y., Burby, J., Ellison, L. & Zhou, Y. 2015 Canonical symplectic Particle-In-Cell method for long-term large-scale simulations of the Vlasov-Maxwell equations. Nucl. Fusion 56 (1), 014001.CrossRefGoogle Scholar
Schellekens, A., Kuiper, K., De Wit, R. & Koopmans, B. 2014 Ultrafast spin-transfer torque driven by femtosecond pulsed-laser excitation. Nat. Commun. 5 (1), 1–7.CrossRefGoogle ScholarPubMed
Tonge, J., Dauger, D.E. & Decyk, V.K. 2004 Two-dimensional semiclassical particle-in-cell code for simulation of quantum plasmas. Comput. Phys. Commun. 164 (1–3), 279–285.CrossRefGoogle Scholar
Xiao, J., Qin, H., Liu, J., He, Y., Zhang, R. & Sun, Y. 2015 Explicit high-order non-canonical symplectic Particle-In-Cell algorithms for Vlasov-Maxwell systems. Phys. Plasmas 22 (11), 112504.CrossRefGoogle Scholar
Zamanian, J., Marklund, M. & Brodin, G. 2010 Scalar quantum kinetic theory for spin-1/2 particles: mean field theory. New J. Phys. 12 (4), 043019.CrossRefGoogle Scholar
Zerroukat, M., Wood, N. & Staniforth, A. 2006 The parabolic spline method for conservative transport problems. Intl J. Numer. Meth. Fluids 51 (11), 1297–1318.CrossRefGoogle Scholar