Wigner Potential Research Articles

On the momentum resolution limit in solving the discrete Wigner transport equation

The Wigner transport equation can take quantum-mechanical effects into account via the Wigner potential performing the non-local potential correlation. The discretization of the Wigner potential in the phase space makes the maximum correlation length finite and inversely proportional to the grid size (or resolution) in the momentum space (Δk). It has been known that the use of too small Δk leads to unphysical simulation results and there has been a lower bound called the momentum resolution limit for getting physically correct results. Here, we show that these numerical artifacts result from the abrupt regime change (quantum to classical or vice versa) across the contact between the simulated device and its boundary. We also demonstrate that its effect can be mitigated by introducing non-rectangular window functions to the Wigner potential.

Solving the Wigner equation for chemically relevant scenarios: Dynamics in 2D.

The signed particle Monte Carlo (SPMC) approach has been used in the past to model steady-state and transient dynamics of the Wigner quasi-distribution for electrons in low-dimensional semiconductors. Here, we make a step toward high-dimensional quantum phase-space simulation in chemically relevant scenarios by improving the stability and memory demands of SPMC in 2D. We do so by using an unbiased propagator for SPMC to improve trajectory stability and applying machine learning to reduce memory demands for storage and manipulation of the Wigner potential. We perform computational experiments on a 2D double-well toy model of proton transfer and demonstrate stable pico-second-long trajectories that require only a modest computational effort.

Absence of Eigenvalues in the Continuous Spectrum for Klein-Gordon Operators

We construct the one-dimensional analogous of von-Neumann Wigner potential to the relativistic Klein-Gordon operator, in which is defined taking asymptotic mathematical rules in order to obtain existence conditions of eigenvalues embedded in the continuous spectrum. Using our constructed potential, we provide an explicit and analytical example of the Klein-Gordon operator with positive eigenvalues embedded in the so called relativistic region. Even so in this not standard example, we present the region of the continuum where those eigenvalues cannot occur. Besides, the absence of eigenvalues in the continuous spectrum for Klein-Gordon operators is proven to a broad general potential classes, including the minimally coupled electric Coulomb potential. Considering known techniques available in literature for Schrodinger operators, we demonstrate an expression for Klein-Gordon operator written in Schrodinger's form, whereby is determined the mathematical spectrum region of absence of eigenvalues.

A computational approach for investigating Coulomb interaction using Wigner\u2013Poisson coupling

Entangled quantum particles, in which operating on one particle instantaneously influences the state of the entangled particle, are attractive options for carrying quantum information at the nanoscale. However, fully-describing entanglement in traditional time-dependent quantum transport simulation approaches requires significant computational effort, bordering on being prohibitive. Considering electrons, one approach to analyzing their entanglement is through modeling the Coulomb interaction via the Wigner formalism. In this work, we reduce the computational complexity of the time evolution of two interacting electrons by resorting to reasonable approximations. In particular, we replace the Wigner potential of the electron–electron interaction by a local electrostatic field, which is introduced through the spectral decomposition of the potential. It is demonstrated that for some particular configurations of an electron–electron system, the introduced approximations are feasible. Purity, identified as the maximal coherence for a quantum state, is also analyzed and its corresponding analysis demonstrates that the entanglement due to the Coulomb interaction is well accounted for by the introduced local approximation.

Wigner equation for general electromagnetic fields: The Weyl-Stratonovich transform

Gauge-invariant Wigner theories are formulated in terms of the kinetic momentum, which---being a physical quantity---is conserved after a change of the gauge. These theories rely on a transform of the density matrix, originally introduced by Stratonovich, which generalizes the Weyl transform by involving the vector potential. We thus present an alternative derivation of the Weyl-Stratonovich transform, which bridges the concepts and notions used by the different, available gauge-invariant approaches and thus links physically intuitive with formal mathematical viewpoints. Furthermore, an explicit form of the Wigner equation, suitable for numerical analysis and corresponding to general, inhomogeneous, and time-dependent electromagnetic conditions, is obtained. For a constant magnetic field, the equation reduces to two models: in the case of a constant electric field, this is the ballistic Boltzmann equation, where classical particles are driven by local forces. The second model, derived for general electrostatic conditions, involves novel physics, where the magnetic field acts locally via the Liouville operator, while the electrostatics is determined by the manifestly nonlocal Wigner potential. A significant consequence of our work is the fact that now the constant magnetic field case can be treated with existing numerical approaches developed for the standard, scalar potential Wigner theory. Therefore, in order to demonstrate the feasibility of the approach, a stochastic method is applied to simulate a physically intuitive evolution problem.

Quantum hydrodynamic model by moment closure of Wigner equation

In this paper, we derive the quantum hydrodynamics models based on the moment closure of the Wigner equation. The moment expansion adopted is of the Grad type first proposed by Grad [“On the kinetic theory of rarefied gases,” Commun. Pure Appl. Math. 2(4), 331–407 (1949)10.1002/cpa.3160020403]. The Grad's moment method was originally developed for the Boltzmann equation. Recently, a regularization method for the Grad's moment system of the Boltzmann equation was proposed by Cai et al. [Commun. Pure Appl. Math. “Globally hyperbolic regularization of Grad's moment system” (in press)] to achieve the global hyperbolicity so that the local well-posedness of the moment system is attained. With the moment expansion of the Wigner function, the drift term in the Wigner equation has exactly the same moment representation as in the Boltzmann equation, thus the regularization applies. The moment expansion of the nonlocal Wigner potential term in the Wigner equation turns out to be a linear source term, which can only induce very mild growth of the solution. As a result, the local well-posedness of the regularized moment system for the Wigner equation remains as for the Boltzmann equation.

Device modeling in the Wigner picture

Basic computational aspects of the Wigner function approach for modeling and simulation of electronic transport are discussed, beginning with the coherent problem, followed by the dissipative problem including scattering events, which has been recently restated in terms of a scattering-induced Wigner function correction. These alternative formulations of the computational task are discussed along with a method for separation of the Wigner potential into classical (force) and quantum components.

A quasi-particle model of the electron–Wigner potential interaction

The Wigner equation was used for the calculation of carrier transport in mesoscopic devices. The carrier transport has a coherent part determined by the Wigner potential Vw and a dissipative part accounting for the interaction with phonons. Models have been developed which solve the equation by using quasi-particles evolving over pieces of classical Newton trajectories. In a backward evolution approach [1], the interaction with the Wigner potential is interpreted as a scattering process. The quantum information is carried by the particle weight. In the ensemble model proposed in [2], the quantum information from Vw is included by the particle affinity. Both weight and affinity are artificial numerical quantities. We propose a model which interprets the Wigner equation with a Boltzmann scattering term as a Boltzmann equation with a generation term. The quantum information is carried by the sign of the quasi-particles. In all other aspects quasi-particles behave as classical particles. The sign has a physical meaning since particles of opposite sign can annihilate. The model ensures a seamless transition between classical and quantum regions. Negative values of the Wigner function are explained in a direct way.

Stochastic interpretation of the Wigner transport in nanostructures

A stochastic interpretation of the quantum transport in nanoscale electronic devices is proposed. The Wigner potential is treated as a scattering source which determines the electron–potential interaction. A particle picture where each scattering event generates positive and negative particles is associated to the transport process.

A wigner function-based quantum ensemble monte carlo study of a resonant tunneling diode

We present results of resonant tunneling diode operation achieved from a particle-based quantum ensemble Monte Carlo (EMC) simulation that is based on the Wigner distribution function (WDF). Methods of including the Wigner potential into the EMC, to incorporate natural quantum phenomena, via a particle property we call the affinity are discussed. Dissipation is included via normal Monte Carlo procedures and the solution is coupled to a Poisson solver to achieve fully selfconsistent results.

A Wigner Function Based Ensemble Monte Carlo Approach for Accurate Incorporation of Quantum Effects in Device Simulation

We present results of both Gaussian wave-packet tunneling though a single barrier structure and RTD operation achieved from a particle-based Ensemble Monte Carlo (EMC) simulation that is based on the Wigner distribution function (WDF). Methods of including the Wigner potential into the EMC, to incorporate naturally quantum phenomena, via a particle property we call the affinity are discussed. Results showing tunneling and correlation build-up in both cases are presented.

Scattering from a truncated von Neumann–Wigner potential

In 1929, von Neumann and Wigner [Phys. Z. 30, 465 (1929)] demonstrated the existence of local potentials which support a bound state with energy embedded in the continuous spectrum. In this paper, we study the $s$-wave scattering produced by a class of von Neumann--Wigner potentials which are perturbed by truncating them at some large radius $r=a.$ For the class of potentials considered, we obtain an analytic expression for the Jost function. The original continuum bound state is replaced by a well isolated complex zero of the Jost function (equivalent to a pole of the $S$ matrix), but this does not produce a conventional Breit-Wigner resonance. Instead, we find twin peaks in the cross section, separated by a very narrow gap associated with the Jost function zero. Our study continues with an account of bound states and virtual states supported by the potential, and with an investigation of the form of asymptotically large zeros of the Jost function. We compare the results of this paper with the significantly different results we obtained using a different perturbation [Phys. Rev. A 52, 3932 (1995)].

A Limiting Absorption Principle for Schrödinger Operators with Generalized Von Neumann–Wigner Potentials II. The Proof

In this series of papers we prove the limiting absorption principle over a given interval for a class of Hamiltonians which contains the original one of von Neumann and Wigner. More specifically, the Hamiltonians are of the form −Δ+csinb|x|/|x|β+V(x), where 2/3<β≤1,V(x) is a short range potential and I is a given compact subinterval of the open positive axis which does not contain the pointb2/4.

A Limiting Absorption Principle for Schrödinger Operators with Generalized Von Neumann–Wigner Potentials I. Construction of Approximate Phase

In this series of papers we prove the limiting absorption principle over a given interval for a class of Hamiltonians which contains the original one of von Neumann and Wigner. More specifically, the Hamiltonians are of the form −Δ+csinb|x|/|x|β+V(x), where 2/3<β≤1,V(x) is a short range potential and I is a given compact subinterval of the open positive axis which does not contain the pointb2/4.

Scattering from a shifted von Neumann-Wigner potential.

We study the s-wave scattering produced by perturbing a family of potentials that support a single bound s state with energy embedded in the continuous spectrum. Such bound states were first studied by von Neumann and Wigner [Z. Phys. 20, 465 (1929)], and we refer to the class of potentials that we perturb as von Neumann--Wigner potentials. We solve the radial Schr\odinger equation exactly for von Neumann--Wigner potentials which have been perturbed by a radial displacement. We analyze the Jost function and its zeros, including how the latter moves as the perturbation is changed. Depending on the amount of the displacement, the perturbation may produce a sharp resonance at an energy shifted from that of the unperturbed continuum bound state, a virtual state, or a conventional bound state with negative energy. In all cases, no physically acceptable solution of the perturbed Schr\odinger equation has the energy of the unperturbed continuum bound state, and the scattering phase shift that is defined from the Jost function has a discontinuity of magnitude \ensuremath{\pi} at that energy. The continuum bound state is recovered in the limit as the perturbation goes to zero, but the limit is nonuniform, and we demonstrate a number of anomalies resulting from this.

Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling.

We solve a form of the Bloch equation for the density matrix in a manner that identifies a «quantum» correction to the classical potential. This correction is given by the difference between the Bohm self-potential and a form of the Wigner potential that have both been used in previous simulations of semiconductor devices. The net potential appearing in the density is then a nonlocally smoothed average of the total semiclassical potential, which itself is composed of the classical potential and the quantum correction. However, the effective potential, which appears in hydrodynamic equations for device modeling, is shown to be the difference between this nonlocally smoothed potential and the local value of the potential

Generalized Fourier transforms for nonlinear systems

A simple generalization is presented of the usual Fourier transforms using generalized exponential and Fourier series developed previously for nonlinear systems. The expressions are given in terms of Jacobi elliptic functions in a form as close as possible to the Fourier transform. There is a pair of transforms for each value of the Jacobi parameter m. Simple applications are presented, one of which gives the generalized Yukawa potential and another that gives the generalized Breit–Wigner potential: Both correspond to the generalized Klein–Gordon equation.

Distorted-field approximation to the static pion potential.

The interaction potentials for two static sources of the ..pi..-meson field are calculated in the distorted-field approximation. The spin- and isospin-dependent parts of the potential are shown to be close to the corresponding one-pion-exchange potentials. In addition, there is a very strong Wigner potential that is not present in the one-pion-exchange potential. All of the components of the potential depend significantly on the shape of the assumed pion-nucleon Yukawa interaction.

Mean excitation energies in the nuclear muon capture

The parametrization of the total muon capture rates in terms of a mean nuclear excitation energy ${\overline{E}}^{\ensuremath{'}}$, proposed in a recent letter of Christillin, Dellafiore, and Rosa-Clot (CDR), is examined. We show that this parametrization ignores the importance of the allowed transitions in light nuclei and fails to reproduce the experimentally observed trend of the ${T}_{g}$ giant resonance energy as a function of nuclear mass number. The removal of contributions due to higher multipoles in heavier nuclei aggravates this discrepancy. The failure of ${\overline{E}}^{\ensuremath{'}}$, extracted by CDR, to correspond to physically meaningful nuclear excitation energies in heavier nuclei is attributable to the intrinsic arbitrariness in its definition, and to the oversimplifying assumption of the nuclear Hamiltonian to be a sum of kinetic energy and Wigner potential alone. A first-order improvement of the Primakoff approximation is discussed.

Application of Sum Rules to Electron-Deuteron Scattering

A measurement of elastic and inelastic electron-deuteron scattering has been made at a momentum transfer of 206 Mev/c and an electron energy of 175 Mev for comparison with the Drell-Schwartz sum rules. The measured value of the non-energy-weighted sum rule is in good agreement with theory. The experimental result for the energy weighted sum rule is 30% larger than the value given for a pure Wigner potential and is consistent with a Rosenfeld two-body interaction if the analysis is restricted to central forces. The analysis is extended to include tensor forces for comparison with the prediction of the Gartenhaus potential. It is found that the gauge terms introduce major ambiguities when tensor forces are included.