Two-dimensional semiclassical Particle-In-Cell code for simulation of quantum plasmas

Quantum states can be described equivalently by density matrices, Wigner functions or quantum tomograms. We analyze the accuracy and performance of three related semiclassical approaches to quantum dynamics, in particular with respect to their numerical implementation. As test cases, we consider the time evolution of Gaussian wave packets in different one-dimensional geometries, whereby tunneling, resonance and anharmonicity effects are taken into account. The results and methods are benchmarked against an exact quantum mechanical treatment of the system, which is based on a highly efficient Chebyshev expansion technique of the time evolution operator.

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“…In this work, we follow a recently proposed 45,46 , slightly different route (cf. right panel of Fig.…”

Section: Linearized Semiclassical Propagator Methodsmentioning

confidence: 99%

Comparative Study of Semiclassical Approaches to Quantum Dynamics

Schubert

Филинов

Matyash

et al. 2009

Int. J. Mod. Phys. C

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“…Due to its structure, the PIC approach have had limited impact on quantum plasmas. It should be noted, however, that the Feynman path integral formulation has been suggested for quantum PIC codes (Tonge et al 2004;Dauger et al 2005). Due to the sum-over-histories, the number of quantum particles that can be included in such a code would be far less than in the classical case (Tonge et al 2004, Dauger et al 2005.…”

Section: Introductionmentioning

confidence: 99%

“…Refs. [22,23]) it is computationally costly. In particular the number of particles needed to capture particle dispersion is increased by a factor of at least a thousand.…”

mentioning

confidence: 99%

“…It should be noted, however, that the Feynman path integral formulation has been used to develop a PIC treatment [22,23] that includes particle dispersive effects in a semi-classical fashion. This is computationally costly, and the number of quantum particles that can be included in the code is far less than in the classical case [22,23].In our paper we have chosen another approach in order to develop a PIC-code including certain quantum features. We first note that the Wigner equation reduces to the classical Vlasov equation for macroscopic scale lengths longer than the thermal de Broglie wavelength.…”

mentioning

confidence: 99%

“…Due to the classicality of the concepts this have had a limited impact to quantum plasmas. It should be noted, however, that the Feynman path integral formulation has been used to develop a PIC treatment [22,23] that includes particle dispersive effects in a semi-classical fashion. This is computationally costly, and the number of quantum particles that can be included in the code is far less than in the classical case [22,23].…”

mentioning

confidence: 99%

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Particle-in-cell simulations of electron spin effects in plasmas

Brodin¹,

Holkundkar

Marklund

2013

J. Plasma Phys.

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We have here developed a particle-in-cell code accounting for the magnetic dipole force and for the magnetization currents associated with the electron spin. The electrons is divided into spin-up and spin-down populations relative to the magnetic field, where the magnetic dipole force acts in opposite directions for the two species. To validate the code, we have studied the wakefield generation by an electromagnetic pulse propagating parallel to an external magnetic field. The properties of the generated wakefield is shown to be in good quantitative agreement with previous theoretical results. Generalizations of the code to account for more quantum effects is discussed PACS numbers: PACS: 52.35. Mw, 52.65.Rr, 03.65.Sq Recently much work has been devoted to the field of quantum plasmas. Many aspects of the field are well described in books and review papers [1][2][3][4][5]. The interest in the field has been motivated by laboratory applications in for example plasmonics [6,7] [14][15][16][17]. Due to the complex nature of the governing equations, many problems involving nonlinearities and/or inhomogeneities must be addressed numerically. A successful method applied to classical plasmas is the celebrated particle-in-cell (PIC) approach (see e.g. [18][19][20][21]). Due to the classicality of the concepts this have had a limited impact to quantum plasmas. It should be noted, however, that the Feynman path integral formulation has been used to develop a PIC treatment [22,23] that includes particle dispersive effects in a semi-classical fashion. This is computationally costly, and the number of quantum particles that can be included in the code is far less than in the classical case [22,23].In our paper we have chosen another approach in order to develop a PIC-code including certain quantum features. We first note that the Wigner equation reduces to the classical Vlasov equation for macroscopic scale lengths longer than the thermal de Broglie wavelength. Hence particle dispersive effects [2,24] can be neglected on such scales. On the other hand, the physics associated with the electron spin (e.g. the magnetic dipole force and the spin magnetization currents [15-17]) does not vanish for long scale lengths. Hence, in this paper we will aim to develop a PIC code applicable on macroscopic scales, accounting for the magnetic dipole force and the spin magnetization currents. While a classical magnetic dipole moment fits very well into the PIC-concept, it is clear that the difference between a classical magnetic dipole moment and the spin must be acknowledged. The present approach builds on the findings of Ref. [25]. It was then observed that more elaborate models for the spin physics reduces to a simple one for frequencies below the spin precession frequency (which is approximately the same as the cyclotron frequency). Specifically this meant that for a dynamical time scale slower than the cyclotron frequency, the electrons could be modelled as consisting of two fluids, one with a spin up state relative the magnetic fiel...

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Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

Crouseilles

Hervieux

et al. 2021

J. Plasma Phys.

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We propose a numerical scheme to solve the semiclassical Vlasov–Maxwell equations for electrons with spin. The electron gas is described by a distribution function $f(t,{\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ that evolves in an extended 9-dimensional phase space $({\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ , where $\boldsymbol s$ represents the spin vector. Using suitable approximations and symmetries, the extended phase space can be reduced to five dimensions: $(x,{{p_x}}, {\boldsymbol s})$ . It can be shown that the spin Vlasov–Maxwell equations enjoy a Hamiltonian structure that motivates the use of the recently developed geometric particle-in-cell (PIC) methods. Here, the geometric PIC approach is generalized to the case of electrons with spin. Total energy conservation is very well satisfied, with a relative error below $0.05\,\%$ . As a relevant example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, where the electrons are partially or fully spin polarized. It is shown that the Raman instability is very effective in destroying the electron polarization.

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Two-dimensional semiclassical Particle-In-Cell code for simulation of quantum plasmas

Cited by 5 publications

References 5 publications

Comparative Study of Semiclassical Approaches to Quantum Dynamics

Comparative Study of Semiclassical Approaches to Quantum Dynamics

Particle-in-cell simulations of electron spin effects in plasmas

Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

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