Wigner-Weyl Formalism Research Articles

Phase space propagation of waves in nonhomogeneous media: corrections beyond the optical geometry limit

We investigate the corrections to the optical geometry approximation for waves traveling in non homogeneous media. We model the wave propagation in dispersive and non dispersive materials in terms of the phase space Wigner–Weyl formalism. The ray tracing optical geometry limit is introduced by numerical tests. We solve the exact Wigner propagation equation for 1D non dispersive materials. We discuss the connection of the Wigner–Weyl description of waves with the particle-wave duality phenomenon in quantum mechanics.

Wigner-Weyl description of light absorption in disordered semiconductor alloys using the localization landscape theory

The presence of disorder in semiconductors can dramatically change their physical properties. Yet, models faithfully accounting for it are still scarce and computationally inefficient. We present a mathematical and computational model able to simulate the optoelectronic response of semiconductor alloys of several tens of nanometer sidelength, while at the same time accounting for the quantum localization effects induced by the compositional disorder at the nano-scale. The model is based on a Wigner-Weyl analysis of the structure of electron and hole eigenstates in phase space made possible by the localization landscape theory. After validation against eigenstates-based computations in 1D and 2D, our model is applied to the computation of light absorption in 3D InGaN alloys of different compositions. We obtain the detailed structures of the absorption tail below the average bandgap and the Urbach energies of all simulated compositions. Moreover, the Wigner-Weyl formalism allows us to define and compute 3D maps of the effective locally absorbed power at all frequencies. Finally the proposed approach opens the way to generalize this method to all energy-exchange processes such as radiative and non-radiative recombination in realistic devices.

Wigner-Weyl calculus in Keldysh technique

We discuss the non-equilibrium dynamics of condensed matter/quantum field systems in the framework of Keldysh technique. In order to deal with the inhomogeneous systems we use the Wigner-Weyl formalism. Unification of the mentioned two approaches is demonstrated on the example of Hall conductivity. We express Hall conductivity through the Wigner transformed two-point Green's functions. We demonstrate how this expression is reduced to the topological number in thermal equilibrium at zero temperature. At the same time both at finite temperature and out of equilibrium the topological invariance is lost. Moreover, Hall conductivity becomes sensitive to interaction corrections.

Wigner–Weyl formalism for the lattice models

We discuss application of Wigner–Weyl formalism to the lattice models of condensed matter physics and relativistic quantum field theory. For the noninteracting models our technique relates Wigner transformation of the fermionic Green’s function with the Weyl symbol [Formula: see text] of lattice Dirac operator. We take as an example of the model defined on rectangular lattice the model of Wilson fermions. It represents the regularization of relativistic quantum field theory and, in addition, describes qualitatively certain topological materials in condensed matter physics. In this model we derive expression for [Formula: see text] in the presence of the most general case of external [Formula: see text] gauge field. Next, we solve the Groenewold equation to all orders in powers of the derivatives of [Formula: see text].

Quasi-probability distributions in loop quantum cosmology

In this paper, we introduce a complete family of parametrized quasi-probability distributions in phase space and their corresponding Weyl quantization maps with the aim to generalize the recently developed Wigner–Weyl formalism within the loop quantum cosmology (LQC) program. In particular, we intend to define those quasi-distributions for states valued on the Bohr compactification of the real line in such a way that they are labeled by a parameter that accounts for the ordering ambiguity corresponding to non-commutative quantum operators. Hence, we notice that the projections of the parametrized quasi-probability distributions result in marginal probability densities which are invariant under any ordering prescription. We also note that, in opposition to the standard Schrödinger representation, for an arbitrary character the quasi-distributions determine a positive function independently of the ordering. Further, by judiciously implementing a parametric-ordered Weyl quantization map for LQC, we are able to recover in a simple manner the relevant cases of the standard, anti-standard, and Weyl symmetric orderings, respectively. We expect that our results may serve to analyze several fundamental aspects within the LQC program, in special those related to coherence, squeezed states, and the convergence of operators, as extensively analyzed in the quantum optics and in the quantum information frameworks.

Topological invariant in terms of the Green functions for the Quantum Hall Effect in the presence of varying magnetic field

Recently the Wigner–Weyl formalism has been applied to the lattice models of solid state physics and to the lattice regularized quantum field theory. This allows to demonstrate that the electric current of intrinsic Anomalous Quantum Hall effect is expressed through the momentum space topological invariant composed of the Green functions both for the two-and the three-dimensional systems. Here we extend this consideration to the case of the Quantum Hall Effect existing in the presence of arbitrarily varying external magnetic field. The corresponding electric current appears to be proportional to the topological invariant in phase space composed of the Wigner transformed Green function that depends both on coordinates and momenta.

Elastic Deformations and Wigner–Weyl Formalism in Graphene

We discuss the tight-binding models of solid state physics with the Z 2 sublattice symmetry in the presence of elastic deformations in an important particular case—the tight binding model of graphene. In order to describe the dynamics of electronic quasiparticles, the Wigner–Weyl formalism is explored. It allows the calculation of the two-point Green’s function in the presence of two slowly varying external electromagnetic fields and the inhomogeneous modification of the hopping parameters that result from elastic deformations. The developed formalism allows us to consider the influence of elastic deformations and the variations of magnetic field on the quantum Hall effect.

Wigner–Weyl formalism and the propagator of Wilson fermions in the presence of varying external electromagnetic field

We develop Wigner–Weyl formalism for the lattice models. For the definiteness we consider Wilson fermions in the presence of U(1) gauge field. The given technique reduces calculation of the two point fermionic Green function to solution of the Groenewold equation. It relates Wigner transformation of the Green function with the Weyl symbol QW of Wilson Dirac operator. We derive the simple expression for QW in the presence of varying external U(1) gauge field. Next, we solve the Groenewold equation to all orders in powers of the derivatives of QW. Thus the given technique allows to calculate the fermion propagator in the lattice model with Wilson fermions in the presence of arbitrary background electromagnetic field. The generalization of this method to the other lattice models is straightforward.

Scale magnetic effect in quantum electrodynamics and the Wigner-Weyl formalism

The Scale Magnetic Effect (SME) is the generation of electric current due to conformal anomaly in external magnetic field in curved spacetime. The effect appears in a vacuum with electrically charged massless particles. Similarly to the Hall effect, the direction of the induced anomalous current is perpendicular to the direction of the external magnetic field $\bf B$ and to the gradient of the conformal factor $\tau$, while the strength of the current is proportional to the beta function of the theory. In massive electrodynamics the SME remains valid, but the value of the induced current differs from the current generated in the system of massless fermions. In the present paper we use the Wigner--Weyl formalism to demonstrate that in accordance with the decoupling property of heavy fermions the corresponding anomalous conductivity vanishes in the large-mass limit with $m^2 \gg |e {\bf B}|$ and $m \gg |\nabla \tau|$.

Magnetic susceptibility of Dirac fermions, Bi-Sb alloys, interacting Bloch fermions, dilute nonmagnetic alloys, and Kondo alloys

Wide ranging interest in Dirac Hamiltonian is due to the emergence of novel materials, namely, graphene, topological insulators and superconductors, the newly-discovered Weyl semimetals, and still actively-sought after Majorana fermions in real materials. We give a brief review of the relativistic Dirac quantum mechanics and its impact in the developments of modern physics. The quantum band dynamics of Dirac Hamiltonian is crucial in resolving the giant diamagnetism of bismuth and Bi-Sb alloys. Quantitative agreement of the theory with the experiments on Bi-Sb alloys has been achieved, and physically meaningful contributions to the diamagnetism has been identified. We also treat relativistic Dirac fermion as an interband dynamics in uniform magnetic fields. For the interacting Bloch electrons, the role of translation symmetry for calculating the magnetic susceptibility avoids any approximation to second order in the field. The expressions for magnetic susceptibility of dilute nonmagnetic alloys give a firm theoretical foundation of the empirical formulas used in fitting experimental results. The unified treatment of all the above calculations is based on the lattice Weyl-Wigner formulation of discrete phase-space quantum mechanics. For completeness, the magnetic susceptibility of Kondo alloys is also given since Dirac fermions in conduction band and magnetic impurities exhibit Kondo effect.

Semiclassical spectral function for matter waves in random potentials

An ℏ-expansion is presented for the ensemble-averaged spectral function of noninteracting matter waves in random potentials. We obtain the leading quantum corrections to the deep classical limit at high energies by the Wigner–Weyl formalism. The analytical results are checked with success against the numerical data for Gaussian and laser speckle potentials with Gaussian spatial correlation in two dimensions.

Wigner model for quantum transport in graphene

The single graphene layer is a novel material consisting of a flat monolayer of carbon atoms packed in a two-dimensional honeycomb lattice, in which the electron dynamics is governed by the Dirac equation. A pseudo-spin phase-space approach based on the Wigner–Weyl formalism is used to describe the ballistic transport of electrons in graphene including quantum effects. Our two-band quantum mechanical representation of the particles reveals itself to be particularly close to the classical description of the particle motion. We analyze the Klein tunneling and correction to the total current in graphene induced by this phenomenon. The equations of motion are analytically investigated and some numerical tests are presented. The temporal evolution of the electron–hole pairs in the presence of an external electric field and rigid potential step is investigated. The connection of our formalism with the Berry phase approach is also discussed.

Deformation quantization of noncommutative quantum mechanics and dissipation

We review the main features of the Weyl-Wigner formulation of noncommutative quantum mechanics. In particular, we present a ∗-product and a Moyal bracket suitable for this theory as well as the concept of noncommutative Wigner function. The properties of these quasi-distributions are discussed as well as their relation to the sets of ordinary Wigner functions and positive Liouville probability densities. Based on these notions we propose criteria for assessing whether a commutative regime has emerged in the realm of noncommutative quantum mechanics. To induce this noncommutative-commutative transition, we couple a particle to an external bath of oscillators. The master equation for the Brownian particle is deduced.

The relationship between the Wigner-Weyl kinetic formalism and the complex geometrical optics method

The relationship between two different asymptotic techniques, namely, the Wigner-Weyl kinetic formalism and the complex geometrical optics method, is addressed within the framework of semiclassical theory of wave propagation. More specifically, in correspondence to appropriate boundary conditions, the solution of the wave kinetic equation, relevant to the Wigner-Weyl formalism, is obtained in terms of the corresponding solution of the complex geometrical optics equations. In particular, this implies that the two considered techniques yield the same wavefield intensity. Such a result is also discussed on the basis of the analytical solution of the wave kinetic equation specific to Gaussian beams of electromagnetic waves propagating in a “lens-like” medium for which the complex geometrical optics solution is already available.

Publisher's Note: “The relationship between the Wigner-Weyl kinetic formalism and the complex geometrical optics method” [J. Math. Phys. 46, 083510 (2005)

First Page

DEFORMATION QUANTIZATION AND WIGNER FUNCTIONS

We review the Weyl-Wigner formulation of quantum mechanics in phase space. We discuss the concept of Narcowich-Wigner spectrum and use it to state necessary and sufficient conditions for a phase space function to be a Wigner distribution. Based on this formalism we analize the modifications introduced by the presence of boundaries. Finally, we discuss the concept of environment-induced decoherence in the context of the Weyl-Wigner approach.

Wigner-Weyl Formalisms for Toroidal Geometries

The phase-space formulation of quantum mechanics due to Weyl and Wigner is based on the assumption that the space in which the particles move is Euclidean. In the standard formulation of quantum mechanics, to which this formalism is equivalent, the wave functions are therefore assumed to satisfy natural boundary conditions (sufficiently fast decay at infinity). However. these conditions are not satisfied when, for physical reasons or due to the approximations employed, the underlying geometry is that of a torus (periodic boundary conditions). We discuss the corresponding modifications of the Wigner-Weyl formalism by comparing the following three cases: (A) motion on the real line, (B) motion on the circle or on an infinite one-dimensional lattice, and (C) motion on a finite set of points. For each of these three different situations we first list the basic equations of an appropriate phase-space formalism. We then relate our approach to previous derivations of this formalism, if such exist in literature, and compare our formalism to equivalent but different schemes that were proposed for this type of geometry in the past. Finally, we discuss to what extent one scheme may be considered as an approximation of another one. The mathematical tools used in this paper are operator algebras and their bases.

The Wigner-Weyl formalism and the relativistic semi-classical approximation

The relativistic semi-classical approximation for a free massive particle is studied using the Wigner-Weyl formalism. A noncovariant Wigner function is proposed using the Newton-Wigner position operator. The perturbative solution for the time evolution is found. Causality is found to be perturbatively respected.

Lattice Weyl-Wigner formulation of exact many-body quantum-transport theory and applications to novel solid-state quantum-based devices.

The use of the lattice-space Weyl-Wigner formalism of the quantum dynamics of particles in solids, coupled with nonequilibrium Green's-function techniques, provides a rigorous and straightforward derivation of an exact integral form of the equation for a quantum distribution function in many-body quantum-transport theory. We show that with the present formalism more realistic calculations, both numerical (particularly, highly transient simulations) and analytical (fully gauge-invariant calculations), which include full quantum effects and many-body effects, can be carried out in a straightforward, elegant, and physically meaningful manner. This is demonstrated by new results based on a more-accurate numerical simulation procedure and novel applications in terms of ``quantum particle trajectories'' for resonant tunneling diodes, and by a straightforward and fully gauge-invariant formulation of the exact quantum-transport equation of a uniform electron-phonon scattering system in high electric fields, which, for the first time, do not involve any gradient expansion.

Formalism of distribution-function method in impurity screening

Using the Weyl-Wigner formulation of quantum mechanics, a rigorous quantum-mechanical basis of the distribution-function method in potential screening, both with and without an externally applied magnetic field, is given. An expansion of the local particle density, up to order ${\ensuremath{\hbar}}^{2}$, Leads to a quantum correction of the Thomas-Fermi self-consistent potential approximation and to the so-called "quasiclassical approximation" for nonzero magnetic field. The important results are nonlinear in the potential characteristic of the distribution-function method. They are compared with the results of Horing, based on linear response and random-phase approximation using Green's function, for the low- and very-high-field regime, thus, clarifying the exact nature of the "quasiclassical approximation" as a leading term (zero order in h) in a very-high-magnetic-field approximation. The formalism sheds some light on some of the unfamiliar aspects of the results based on linear response.