A discrete formulation of the Wigner transport equation

Abstract

Discrete Wigner function by symmetric informationally complete positive operator valued measureA discrete formulation of the Wigner distribution function ͑WDF͒ and the Wigner transport equation ͑WTE͒ is proposed, where the "discreteness" of the WDF and WTE is not just a practical, mathematical feature of discretization for the possible computations, but reveals a fundamental physics regarding the maximum correlation length of potentials ͑an essential quantum-mechanical feature of the WTE͒: it is set by the positional uncertainty due to the discrete values of momentum in evaluating the discrete WDF. Our formulation also shows that the weighting function to the potential-correlation term can be derived naturally from a mathematical necessity related to the antiperiodicity of the discrete density operator. In addition, we propose a mutually independent discretization scheme for the diagonal and cross-diagonal coordinates of the density operator, which results in a numerically effective discrete WTE in that it requires much less computational resources without significant loss in accuracy.

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

“…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 .…”

An efficient numerical scheme for the discrete Wigner transport equation via the momentum domain narrowing

“…Then, the discrete Wigner function can be obtained simply by the discrete Fourier transform of the discretized density matrix. 24 For the discretization of the density matrix, we scale q and r with two respective constants ∆ q and ∆ r , obtainingq = l(l = 0, 1, . .…”

Open Quantum Dynamics Theory on the Basis of Periodical System-Bath Model for Discrete Wigner Function

“…A central issue in the development of a computational simulation for a quantum system described in a phase space distribution is the instability of the numerical integration of a kinetic equation in time, which depends upon a discretization scheme of the coordinate and momentum [1][2][3][4][5]. In this paper, we introduce a new approach to construct a Wigner distribution function (WDF) for an open quantum dynamics system on the basis of a finite dimensional quantum mechanics developed by Schwinger [6].…”