Wigner equation for general electromagnetic fields: The Weyl-Stratonovich transform

Abstract

Gauge-invariant Wigner theories are formulated in terms of the kinetic momentum, which-being a physical quantity-is conserved after a change of the gauge. These theories rely on a transform of the density matrix, originally introduced by Stratonovich, which generalizes the Weyl transform by involving the vector potential. We thus present an alternative derivation of the Weyl-Stratonovich transform, which bridges the concepts and notions used by the different, available gauge-invariant approaches and thus links physically intuitive with formal mathematical viewpoints. Furthermore, an explicit form of the Wigner equation, suitable for numerical analysis and corresponding to general, inhomogeneous, and time-dependent electromagnetic conditions, is obtained. For a constant magnetic field, the equation reduces to two models: in the case of a constant electric field, this is the ballistic Boltzmann equation, where classical particles are driven by local forces. The second model, derived for general electrostatic conditions, involves novel physics, where the magnetic field acts locally via the Liouville operator, while the electrostatics is determined by the manifestly nonlocal Wigner potential. A significant consequence of our work is the fact that now the constant magnetic field case can be treated with existing numerical approaches developed for the standard, scalar potential Wigner theory. Therefore, in order to demonstrate the feasibility of the approach, a stochastic method is applied to simulate a physically intuitive evolution problem.

Gate-Controlled Electron Quantum Interference Logic Devices

“…and corresponds to spatially dependent, electric fields slowly varying in time, so that magnetic effects are not involved. 26 Here, t is the time, m the electron mass, p the momentum, and r the spatial position; (p, r) spans the phase space.…”

Complex Systems in Phase Space

“…In extension to previous work, here we target a non-focusing potential well setup, inspired by [ 80 ]. We consider the evolution of an initial electron state described by the Wigner function in a two-dimensional phase space ( ) in the presence of a uniform magnetic field [ 81 ] and simulated by the Wigner EMC method using the signed-particle model via ViennaWD [ 82 ].…”

Wigner ensemble Monte Carlo simulation without splitting error of a GaAs resonant tunneling diode

“…We had to wait until the beginning of 2000, to have particle Monte Carlo (MC) solvers for the Wigner equation [20,25]. From that period until now, several papers have been published on this subject (see [27] for a review) and, recently, very interesting device simulations have been provided [4,19,22].…”