Quantum Transport in the Phase Space, the Wigner Equation

Fabrice Debbasch

A relativistic Wigner function for free Discrete Time Quantum Walks (DTQWs) on the square 2D space-time lattice is defined. The transport equation obeyed by the relativistic Wigner function is derived in terms of discrete derivatives and degenerates in the continuum limit into the transport equation obeyed by the Wigner function of 2D Dirac fermions. The first corrections to the continuous equation induced by the discreteness of the lattice are also determined and the similarities with non-quantum relativistic transport in phase space are discussed further.

GERALD J. IAFRATE

J. Rosato

The photon picture of radiation transport is examined. Because the photon has no mass and spin one, a position operator for a photon particle cannot be defined properly. This lack leads to ambiguity if the radiative transfer equation is associated with a quantum transport equation for photons. The issue of photon nonlocalizability is discussed and illustrated through a wavefunction-based approach. It is next shown that the phase space formalism due to Wigner provides a suitable framework for a proper derivation of a photon transport equation from first principles.

Shirun Ho, Ken Yamaguchi

Paolo Bordone, Andrea Bertoni, ROSSELLA BRUNETTI +1 more

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.