Adding Decoherence to the Wigner Equation

Abstract

Starting from the detailed description of the single-collision decoherence mechanism proposed by Adami, Hauray and Negulescu in Ref.[2], we derive a Wigner equation endowed with a decoherence term of a fairly general form. This equation is shown to contain well known decoherence models, such as the Wigner-Fokker-Planck equation, as particular cases. The effect of the decoherence mechanism on the dynamics of the macroscopic moments (density, current, energy) is illustrated by deriving the corresponding set of balance laws. The issue of large-time asymptotics of our model is addressed in the particular, although physically relevant, case of gaussian solutions. It is shown that the addition of a Caldeira-Legget friction term provides the asymptotic behaviour that one expects on the basis of physical considerations.

Wigner function with correlation damping

“…In order endow the WF formalism with a mechanism describing decoherence, a model was developed by Barletti, Frosali and Giovannini in [14], based on the rigorous results of Adami, Hauray and Negulescu [19]. The idea is to let the carriers, described by the WF formalism, undergo a number of collisions per unit time with a nominal background medium of light particles; each interaction is described by the model introduced in [19] and, in the limit of very small mass ratio, it amounts to the following transformation of the particle density matrix ρ(x, y, t 0 ) −→ I(x, y)ρ(x, y, t 0 ).…”

“…where ∆ λ and Γ are quantities which depend on the light particle wave function and on the scattering coefficients [14,19]. In particular, ∆ λ (η) describes the damping of the correlation for large values of x − y; it depends on the positive parameter λ which is the typical length of the correlation damping (we will often refer to it as "correlation length"), with λ → +∞ for the fully coherent system.…”

“…In addition, Γ is usually small [19] and we shall neglect its contribution. Assuming that collisions occur randomly with frequency 1/τ 0 , we obtain the following equation for the Wigner function (see [14] for the details)…”

“…This is an important factor, for example, in the description of semiconductor devices with the WF approach [5,6,9]. This issue has been addressed in [10][11][12][13][14]. In [10] the finite size of a semiconductor device was considered by modifying the correlation function across the device boundaries, allowing in fact for a finite correlation length, within the framework of the Schrödinger equation.…”

“…In this paper, we examine again the effect of the finite correlation length on a onedimensional scattering problem by solving numerically the evolution equation for the Wigner function proposed in [14]. A gaussian wave packet, supposed free at t → −∞, enters a region where a gaussian shaped potential is present, and the final state is observed in terms of average quantities (momentum and position), transmission coefficient and density profiles.…”