Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Workshop on Computational Electronics

Wigner paths method in quantum transport with dissipation

Paolo Bordone, Andrea Bertoni, ROSSELLA BRUNETTI, Carlo Jacoboni

Published Nov 7, 2002• 0 citations
Publisher

Abstract

The Wigner formulation of quantum mechanics based on the concept of Wigner function (WF), is particularly suitable for the study of quantum transport in mesoscopic systems since it allows to describe quantum mechanical effects in an (r,p) phase space, in analogy to what is done for classical systems. Recently we have introduced the concept of Wigner paths (WPs) in phase space which both provide a pictorial representation of the quantum evolution of the system of interest and constitute a useful tool for providing numerical solutions of the quantum equation describing the time evolution of the system. The theory has been applied to the following cases: (a) The study of the transport properties of the transient regime for homogeneous systems. (b) The comparison between quantum and classical treatment of the scattering mechanism. (c) The study of the reflection of the WF from an infinite potential barrier. Furthermore, the Wigner approach, originally formulated in terms of single-time WF has been extended by introducing the two-time WF. The Fourier transform of this function contains independent information about both momentum and energy of the electron. The Monte Carlo procedure based on the generation of WPs can be used to evaluate the two-time WF of an electron interacting with the phonon gas.

DOI
10.1109/iwce.2000.869932

Related papers

Wigner paths for electrons interacting with phonons

M. Pascoli, Paolo Bordone, ROSSELLA BRUNETTI +1 more

Wigner trajectories in phase space provide a pictorial representation of the quantum evolution of a system of interest in the Wigner-function formulation of quantum mechanics with many analogies with the description of classical evolutions. The existence of Wigner trajectories has been discussed so far only for ballistic coherent electrons moving in an external potential profile. We prove in this paper the existence of Wigner paths that are defined also in the presence of electron-phonon coupling, and go over some interpretative problems related to Wigner trajectories. Wigner paths can be used to determine the evolution of the Wigner function in time during scattering processes. This general result opens new fields of application of the concept of Wigner function, mainly in connection with the numerical study of quantum transport in mesoscopic systems.

Physical review. B, Condensed matter•Aug 15, 1998•39 citations

Momentum distribution functions of strongly correlated systems of particles: Wigner approach and path integrals

Alexander S. Larkin, Vladimir S. Filinov, В. Е. Фортов

The new numerical version of the Wigner approach to quantum mechanics for treatment thermodynamic properties of the strongly interacting systems of particles has been developed for extreme conditions, when there are no small physical parameters and analytical approximations used in different kind of perturbation theories can not be applied. The new path integral representation of the quantum Wigner function in the phase space has been developed for canonical ensemble. Explicit analytical expression of the Wigner function has been obtained in linear and harmonic approximations. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. Preliminary calculations of the momentum distribution function of the Coulomb systems of particles have been carried out. Comparison with classical Maxwell-Boltzmann distribution shows the significant influence of quantum effects on the high energy asymptotics ("tails') of the calculated momentum distribution functions, which resulted in appearance of sharp oscillations.

Journal of Physics Conference Series•Nov 1, 2016•1 citation

Quantum dynamics in Wigner and tomography representations

Vladimir S. Filinov, G. Schubert, Pavel R. Levashov +3 more

In the standard formulation of quantum mechanics states of a system are described by wave functions or density operators . However quantum description of the system can be given in many other ways , for example, in Wigner-Moyal or Feynman formulations . All these representation are equivalent in the the sense of physical results, but the form of presentation of quantum mechanics is different as wave functions, Wigner-Liouville functions or other representations of density operators differ in essential way from each other. In this work we are dealing with Wigner and probability representation of quantum mechanics. The probability representations or tomography representation was recently proposed in terms of marginal distribution functions (MDF). MDF describing the quantum states are positive distribution functions connected with wave functions or density matrices by known integral transformation. Sign conservation of the of the MDF can be valuable in computer simulations to overcome the "sign problem". Important advantage of the probability representation is that the quantum transitions between quantum states are descried by non-negative probabilities. This work is devoted to the development of a new stochastic approaches to numerical solution of the evolution equations for Wigner-Liouviile (WL) and marginal distribution functions. To solve the WL equation we combine both molecular dynamics and Monte Carlo methods and compute traces of the dynamical operators. The results obtained for a system of electrons and random scatterers clearly demonstrate that the many-particle interaction between the electrons leads to an enhancement of the conductivity compared to the case of noninteracting electrons. To obtain evolution of MDF we developed new stochastic approach to solution of the generalized Langevin equation (GLE). GLE is derived from Kolmogorov equations for Green function of evolution equation for MDF. We discuss the basic relations and main ideas of this approach, compare obtained numerical results with results of independent finite-difference calculations for quantum oscillator and quantum particles crossing the finite well and tunneling through the Gaussian barrier.

IEEE International Conference on Plasma Science (ICOPS)•Jun 1, 2008•0 citations

Wigner transport equation with finite coherence length

Carlo Jacoboni, Paolo Bordone

The use of the Wigner function for the study of quantum transport in open systems present severe criticisms. Some of the problems arise from the assumption of infinite coherence length of the electron dynamics outside the system of interest. In the present work the theory of the Wigner function is revised assuming a finite coherence length. A new dynamical equation is found, corresponding to move the Wigner momentum off the real axis, and a numerical analysis is performed for the case of study of the onedimensional potential barrier.

arXiv:1305.6527•May 28, 2013•0 citations

Wigner function approach to the quantum Brownian motion of a particle in a potential

W. T. Coffey, Yuri P. Kalmykov, С. В. Титов +1 more

Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.

Physical Chemistry Chemical Physics•Jan 1, 2007•59 citations

Wigner approach to quantum statistical mechanics and quantum generalization molecular dynamics method. Part 2

Vladimir S. Filinov

A new numerical approach for the calculation of quantum time correlation functions in the canonical and micro-canonical ensembles has been developed within the Wigner formulation of quantum statistical mechanics. For electrons in disordered systems of scatterers the numerical results for the tensor of static electrical conductivity have been obtained according to the quantum Kubo formula in the one-electron approximation in the canonical ensemble. The second quantity considered is the flux position time correlation function characterizing the momentum flow for tunnelling particles at the barrier crossing from one potential well to another. The difficulties with studies of exponentially small effects in the temperature averaging for the canonical ensemble has been overcome by a transition to the micro-canonical ensemble with a fixed total energy. The related integral Wigner-Liouville type equations have been obtained and a numerical approach combining both molecular dynamics and Monte Carlo methods for solving this equation has also been developed. Numerical results have been obtained for the momentum flow of a particle tunnelling through the barrier dividing two potential wells in two- and three-dimensional cases.

Molecular Physics•Aug 1, 1996•38 citations
The Wigner formulation of quantum mechanics based on the concept of Wigner function (WF), is particularly suitable for the study of quantum transport in mesoscopic systems since it allows to describe quantum mechanical effects in an (r,p) phase space, in analogy to what is done for classical systems. Recently we have introduced the concept of Wigner paths (WPs) in phase space which both provide a pictorial representation of the quantum evolution of the system of interest and constitute a useful tool for providing numerical solutions of the quantum equation describing the time evolution of the system. The theory has been applied to the following cases: (a) The study of the transport properties of the transient regime for homogeneous systems. (b) The comparison between quantum and classical treatment of the scattering mechanism. (c) The study of the reflection of the WF from an infinite potential barrier. Furthermore, the Wigner approach, originally formulated in terms of single-time WF has been extended by introducing the two-time WF. The Fourier transform of this function contains independent information about both momentum and energy of the electron. The Monte Carlo procedure based on the generation of WPs can be used to evaluate the two-time WF of an electron interacting with the phonon gas.
Wigner trajectories in phase space provide a pictorial representation of the quantum evolution of a system of interest in the Wigner-function formulation of quantum mechanics with many analogies with the description of classical evolutions. The existence of Wigner trajectories has been discussed so far only for ballistic coherent electrons moving in an external potential profile. We prove in this paper the existence of Wigner paths that are defined also in the presence of electron-phonon coupling, and go over some interpretative problems related to Wigner trajectories. Wigner paths can be used to determine the evolution of the Wigner function in time during scattering processes. This general result opens new fields of application of the concept of Wigner function, mainly in connection with the numerical study of quantum transport in mesoscopic systems.
The new numerical version of the Wigner approach to quantum mechanics for treatment thermodynamic properties of the strongly interacting systems of particles has been developed for extreme conditions, when there are no small physical parameters and analytical approximations used in different kind of perturbation theories can not be applied. The new path integral representation of the quantum Wigner function in the phase space has been developed for canonical ensemble. Explicit analytical expression of the Wigner function has been obtained in linear and harmonic approximations. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. Preliminary calculations of the momentum distribution function of the Coulomb systems of particles have been carried out. Comparison with classical Maxwell-Boltzmann distribution shows the significant influence of quantum effects on the high energy asymptotics ("tails') of the calculated momentum distribution functions, which resulted in appearance of sharp oscillations.
In the standard formulation of quantum mechanics states of a system are described by wave functions or density operators . However quantum description of the system can be given in many other ways , for example, in Wigner-Moyal or Feynman formulations . All these representation are equivalent in the the sense of physical results, but the form of presentation of quantum mechanics is different as wave functions, Wigner-Liouville functions or other representations of density operators differ in essential way from each other. In this work we are dealing with Wigner and probability representation of quantum mechanics. The probability representations or tomography representation was recently proposed in terms of marginal distribution functions (MDF). MDF describing the quantum states are positive distribution functions connected with wave functions or density matrices by known integral transformation. Sign conservation of the of the MDF can be valuable in computer simulations to overcome the "sign problem". Important advantage of the probability representation is that the quantum transitions between quantum states are descried by non-negative probabilities. This work is devoted to the development of a new stochastic approaches to numerical solution of the evolution equations for Wigner-Liouviile (WL) and marginal distribution functions. To solve the WL equation we combine both molecular dynamics and Monte Carlo methods and compute traces of the dynamical operators. The results obtained for a system of electrons and random scatterers clearly demonstrate that the many-particle interaction between the electrons leads to an enhancement of the conductivity compared to the case of noninteracting electrons. To obtain evolution of MDF we developed new stochastic approach to solution of the generalized Langevin equation (GLE). GLE is derived from Kolmogorov equations for Green function of evolution equation for MDF. We discuss the basic relations and main ideas of this approach, compare obtained numerical results with results of independent finite-difference calculations for quantum oscillator and quantum particles crossing the finite well and tunneling through the Gaussian barrier.
The use of the Wigner function for the study of quantum transport in open systems present severe criticisms. Some of the problems arise from the assumption of infinite coherence length of the electron dynamics outside the system of interest. In the present work the theory of the Wigner function is revised assuming a finite coherence length. A new dynamical equation is found, corresponding to move the Wigner momentum off the real axis, and a numerical analysis is performed for the case of study of the onedimensional potential barrier.
Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.
A new numerical approach for the calculation of quantum time correlation functions in the canonical and micro-canonical ensembles has been developed within the Wigner formulation of quantum statistical mechanics. For electrons in disordered systems of scatterers the numerical results for the tensor of static electrical conductivity have been obtained according to the quantum Kubo formula in the one-electron approximation in the canonical ensemble. The second quantity considered is the flux position time correlation function characterizing the momentum flow for tunnelling particles at the barrier crossing from one potential well to another. The difficulties with studies of exponentially small effects in the temperature averaging for the canonical ensemble has been overcome by a transition to the micro-canonical ensemble with a fixed total energy. The related integral Wigner-Liouville type equations have been obtained and a numerical approach combining both molecular dynamics and Monte Carlo methods for solving this equation has also been developed. Numerical results have been obtained for the momentum flow of a particle tunnelling through the barrier dividing two potential wells in two- and three-dimensional cases.