The Wigner function was formulated in 1932 by Eugene Paul Wigner, at a time when quantum mechanics was in its infancy. In doing so, he brought phase space representations into quantum mechanics. However, its unique nature also made it very interesting for classical approaches and for identifying the deviations from classical behavior and the entanglement that can occur in quantum systems. What stands out, though, is the feature to experimentally reconstruct the Wigner function, which provides far more information on the system than can be obtained by any other quantum approach. This feature is particularly important for the field of quantum information processing and quantum physics. However, the Wigner function finds wide-ranging use cases in other dominant and highly active fields as well, such as in quantum electronics—to model the electron transport, in quantum chemistry—to calculate the static and dynamical properties of many-body quantum systems, and in signal processing—to investigate waves passing through certain media. What is peculiar in recent years is a strong increase in applying it: Although originally formulated 86 years ago, only today the full potential of the Wigner function—both in ability and diversity—begins to surface. This review, as well as a growing, dedicated Wigner community, is a testament to this development and gives a broad and concise overview of recent advancements in different fields.

CUDA, OpenCL, and OpenMP are popular programming models for the multicore architectures of CPUs and many-core architectures of GPUs or Xeon Phis. At the same time, computational scientists face the...

Confinement in small structures has required quantum mechanics, which has been known for a great many years. This leads to quantum transport. The field-effect transistor has had no need to be described by quantum transport over most of the century for which it has existed. But, this has changed in the past few decades, as modern versions tend to be absolutely controlled by quantum confinement and the resulting modifications to the normal classical descriptions. In addition, correlation and confinement lead to a need for describing the transport by quantum methods as well. In this review, we describe the quantum effects and the methods of treament through various approaches to quantum transport.

We provide a full set of growth rate coefficients to enable high-accuracy two- and three-dimensional simulations of dry thermal oxidation of 4H-silicon carbide. The available models are insufficient for the simulation of complex multi-dimensional structures, as they are unable to predict oxidation for arbitrary crystal directions because of the insufficient growth rate coefficients. By investigating time-dependent dry thermal oxidation kinetics, we obtain temperature-dependent growth rate coefficients for surfaces with different crystal orientations. We fit experimental data using an empirical relation to obtain the oxidation growth rate parameters. Time-dependent oxide thicknesses at various temperatures are taken from published experimental findings. We discuss the oxidation rate parameters in terms of surface orientation and oxidation temperature. Additionally, we fit the obtained temperature-dependent growth rate coefficients using the Arrhenius equation to obtain activation energies and pre-exponential factors for the four crystal orientations. The thereby obtained parameters are essential for enabling high-accuracy simulations of dry thermal oxidation and can be directly used to augment multi-dimensional process simulations.

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.