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arXiv:1305.6527

Wigner transport equation with finite coherence length

Carlo Jacoboni, Paolo Bordone

Published May 28, 2013• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1305.6527

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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.
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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.
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.
We employ a recently developed formalism for describing open quantum systems, to device efficient ab initio numerical techniques for the time propagation of the Wigner function. The propagator's complexity is O(Nlog⁡N)O(N\log N)O(NlogN), where N is the length of the array containing the Wigner function. We illustrate the developed method by propagating single particle systems undergoing decoherence and draw a faithful comparison with the classical evolution governed by both the Fokker-Planck and the Koopman-von Neumann equations. These simulations allow us to witness the emergence of the classical world by observing how the Wigner function converges to the solution of the Fokker-Planck equation with sufficient classical diffusion. As a second example we simulate for first time a numerically challenging quantum system made of two coupled particles and observe the loss of decoherence on one of the particles as a result of continuous measurements performed on the other one, therefore obtaining quantitative predictions for the loss of coherence of both particles as a function of time.
Traditional theories of electron transport in crystals are based on the Boltzmann equation and do not capture physics arising from quantum coherence. We introduce a transport formalism based on orbital Wigner functions, which accurately captures quantum coherent physics in multiband fermionic systems. We illustrate the power of this approach compared with traditional semiclassical transport theory by testing it numerically against microscopic simulations of one-dimensional, noninteracting, two-band systems—the simplest systems capable of exhibiting interorbital coherence. We show that orbital Wigner functions accurately capture strongly nonequilibrium features of electron dynamics that lie beyond conventional Boltzmann theory, such as the ballistic transport of a relative phase between microscopic orbitals and topological Thouless pumping of charge, both at nonzero temperature and away from the adiabatic limit. Our approach is motivated in part by modern ultracold atom experiments that can prepare and measure far-from-equilibrium charge transport and phase coherence in multiband fermionic systems, calling for correspondingly precise theories of transport. The quantitative accuracy exhibited by our approach, together with its capacity to capture nontrivial physics even at the ballistic scale, establishes orbital Wigner functions as an ideal starting point for developing a fully systematic theory of transport in crystals.
We examine the effect of the decoherence-induced reduction of correlation length on a one-dimensional scattering problem by solving numerically the evolution equation for the Wigner function with decoherence proposed by Barletti et al. [J. Comput. Theor. Transp. 47, 209 (2018)2332-430910.1080/23324309.2018.1520732]. The numerical solution is achieved by the splitting-scheme algorithm, suitably modified to include the decoherence term. Three cases are examined, corresponding to a reflection-dominated regime, a transmission-dominated regime, and an intermediate one. The dynamic evolution of the Wigner function is followed until the separation process of the reflected and of the transmitted packets is complete and it is observed for three different values of the correlation length. The outcomes show a broadening and flattening of the Wigner function which becomes progressively more pronounced as the correlation length is decreased. This results in a reduced reflection at low energies and in a reduced transmission at high energies.