Stability and accuracy of a pseudospectral scheme for the Wigner function equation

Thomann, Andrea; Borzì, Alfio

doi:10.1002/num.22072

Cited by 8 publications

(3 citation statements)

References 31 publications

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“…Therefore, in many physically interesting situations a numerical approach is needed to solve the equation. Among the existing numerical schemes developed for this type of equation 54 – 59 , the spectral split-operator method 60 seems to be highly efficient 61 – 65 . This method allows us to look at the Moyal equation ( 14 ) as an example of a continuous dynamical system in phase space for which there exists a unitary time evolution operator such that where is the WDF at an arbitrary time instant , and corresponds to the WDF defined at the initial time .…”

Section: Theoretical Frameworkmentioning

confidence: 99%

Phase-space studies of backscattering diffraction of defective Schrödinger cat states

Kołaczek

Spisak

Wołoszyn

2021

Sci Rep

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The coherent superposition of two well separated Gaussian wavepackets, with defects caused by their imperfect preparation, is considered within the phase-space approach based on the Wigner distribution function. This generic state is called the defective Schrödinger cat state due to this imperfection which significantly modifies the interference term. Propagation of this state in the phase space is described by the Moyal equation which is solved for the case of a dispersive medium with a Gaussian barrier in the above-barrier reflection regime. Formally, this regime constitutes conditions for backscattering diffraction phenomena. Dynamical quantumness and the degree of localization in the phase space of the considered state as a function of its imperfection are the subject of the performed analysis. The obtained results allow concluding that backscattering communication based on the defective Schrödinger cat states appears to be feasible with existing experimental capabilities.

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Section: Theoretical Frameworkmentioning

confidence: 99%

Phase-space studies of backscattering diffraction of defective Schrödinger cat states

Kołaczek

Spisak

Wołoszyn

2021

Sci Rep

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“…Efficient numerical schemes based on non-uniform meshes are developed for deterministic discretization method (Kim, 2007;Costolanski and Kelley, 2010;Kim and Kim, 2015;Schulz and Mahmood, 2016) or techniques based on the fast Fourier transform via the split-operator method. (Gómez et al, 2014;Spisak et al, 2015;Cabrera et al, 2015;Thomann and Borz, 2017). Besides, stochastic methods based on different variants of the Monte Carlo techniques are available (Jacoboni et al, 2001;Nedjalkov et al, 2004;Querlioz and Dollfus, 2010;Muscato and Wagner, 2016).…”

Section: Introductionmentioning

confidence: 99%

The Effect of Elastic and Inelastic Scattering on Electronic Transport in Open Systems

Kulinowski

Wołoszyn

Radecka

et al. 2019

International Journal of Applied Mathematics and Computer Science

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The purpose of this study is to apply the distribution function formalism to the problem of electronic transport in open systems, and to numerically solve the kinetic equation with a dissipation term. This term is modeled within the relaxation time approximation and contains two parts, corresponding to elastic or inelastic processes. The collision operator is approximated as a sum of the semi-classical energy dissipation term and the momentum relaxation term, which randomizes the momentum but does not change the energy. As a result, the distribution of charge carriers changes due to the dissipation processes, which has a profound impact on the electronic transport through the simulated region discussed in terms of the current–voltage characteristics and their modification caused by the scattering. Measurements of the current–voltage characteristics for titanium dioxide thin layers are also presented, and compared with the results of numerical calculations.

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“…The existed deterministic solvers, including the finite difference schemes [9,10] and the spectral collocation methods [11][12][13], always require the potentials to decay fast and vanish at infinities, i.e., the localized potentials, since the Wigner kernel is evaluated by the Poisson summation formula, and thus is not applicable for the unbounded potentials. For the potentials of the polynomial type, on the other hand, as an equivalent series form of the pseudo-differential operator, the Moyal expansion reduces to a finite series [14,15] and the resulting equation can be solved by either the spectral method [16] or the Hermite expansion [17]. In this work, we attempt to combine the advantages of the above two to evolve the Wigner quantum dynamics in the presence of unbounded potentials.…”

Section: Introductionmentioning

confidence: 99%

Numerical Methods for the Wigner Equation with Unbounded Potential

2018

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Unbounded potentials are always utilized to strictly confine quantum dynamics and generate bound or stationary states due to the existence of quantum tunneling. However, the existed accurate Wigner solvers are often designed for either localized potentials or those of the polynomial type. This paper attempts to solve the time-dependent Wigner equation in the presence of a general class of unbounded potentials by exploiting two equivalent forms of the pseudodifferential operator: integral form and series form (i.e., the Moyal expansion). The unbounded parts at infinities are approximated or modeled by polynomials and then a remaining localized potential dominates the central area. The fact that the Moyal expansion reduces to a finite series for polynomial potentials is fully utilized. Using a spectral collocation discretization which conserves both mass and energy, several typical quantum systems are simulated with a high accuracy and reliable estimation of macroscopically measurable quantities is thus obtained.

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Stability and accuracy of a pseudospectral scheme for the Wigner function equation

Cited by 8 publications

References 31 publications

Phase-space studies of backscattering diffraction of defective Schrödinger cat states

Phase-space studies of backscattering diffraction of defective Schrödinger cat states

The Effect of Elastic and Inelastic Scattering on Electronic Transport in Open Systems

Numerical Methods for the Wigner Equation with Unbounded Potential

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