Linear canonical Wigner distribution and its application

Linear canonical transform (LCT) form a three-parameter family of intergral transforms with wide application in optics. In this paper, we investigate the linear canonical Wigner distribution (LCWD) which is based on the LCT and the classical Wigner distribution (WD). Firstly, the definition of LCWD is discussed. Moreover, the transformation law for the LCWD through a first-order optical system is derived. This new phase-space distribution provides analysis of signals in both space and LCT domains simultaneously. Then, the main properties of LCWD are investigated in detail. Finally, the application of the LCWD is presented. The LCWD is found to be the appropriate phase-space distribution function for light-beam characterization in first-order optical system. Moreover, the moment matrix formalism for beam characterization is studied.     

Effect of boundary treatments on quantum transport current in the Green’s function and Wigner distribution methods for a nano-scale DG-MOSFET

In this paper, we conduct a study of quantum transport models for a two-dimensional nano-size double gate (DG) MOSFET using two approaches: non-equilibrium Green’s function (NEGF) and Wigner distribution. Both methods are implemented in the framework of the mode space methodology where the electron confinements below the gates are pre-calculated to produce subbands along the vertical direction of the device while the transport along the horizontal channel direction is described by either approach. Each approach handles the open quantum system along the transport direction in a different manner. The NEGF treats the open boundaries with boundary self-energy defined by a Dirichlet to Neumann mapping, which ensures non-reflection at the device boundaries for electron waves leaving the quantum device active region. On the other hand, the Wigner equation method imposes an inflow boundary treatment for the Wigner distribution, which in contrast ensures non-reflection at the boundaries for free electron waves entering the device active region. In both cases the space-charge effect is accounted for by a self-consistent coupling with a Poisson equation. Our goals are to study how the device boundaries are treated in both transport models affects the current calculations, and to investigate the performance of both approaches in modeling the DG-MOSFET. Numerical results show mostly consistent quantum transport characteristics of the DG-MOSFET using both methods, though with higher transport current for the Wigner equation method, and also provide the current–voltage (I–V) curve dependence on various physical parameters such as the gate voltage and the oxide thickness.     

The wigner distribution function and Hamilton''s characteristics of a geometric-optical system

Four system functions have been defined for an optical system; each of these functions describes the system completely in terms of Fourier optics. From the system functions the Wigner distribution function of an optical system has been defined; although derived from Fourier optics, this Wigner distribution function can directly be interpreted in terms of geometrical optics. In the special case of a geometric-optical system, the Wigner distribution function clearly demonstrates the relations that exist between the four system functions and the four Hamilton's characteristics of the system.     

The Wigner distribution function applied to optical signals and systems

In this paper the Wigner distribution function has been introduced for optical signals and systems. The Wigner distribution function of an optical signal appears to be in close resemblance to the ray concept in geometrical optics. This resemblance reaches even farther: although derived from Fourier optics, the Wigner distribution functions of some elementary optical systems can directly be interpreted in terms of geometrical optics.     

Wigner distribution of a transducer beam pattern within a multiple scattering formalism for heterogeneous solids

Diffuse ultrasonic backscatter measurements have been especially useful for extracting microstructural information and for detecting flaws in materials. Accurate interpretation of experimental data requires robust scattering models. Quantitative ultrasonic scattering models include components of transducer beam patterns as well as microstructural scattering information. Here, the Wigner distribution is used in conjunction with the stochastic wave equation to model this scattering problem. The Wigner distribution represents a distribution in space and time of spectral energy density as a function of wave vector and frequency. The scattered response is derived within the context of the Wigner distribution of the beam pattern of a Gaussian transducer. The source and receiver distributions are included in the analysis in a rigorous fashion. The resulting scattered response is then simplified in the single-scattering limit typical of many diffuse backscatter experiments. Such experiments, usually done using a modified pulse-echo technique, utilize the variance of the signals in space as the primary measure of microstructure. The derivation presented forms a rigorous foundation for the multiple scattering process associated with ultrasonic experiments in heterogeneous media. These results are anticipated to be relevant to ultrasonic nondestructive evaluation of polycrystalline and other heterogeneous solids.     

Reduced forms of the Wigner distribution function for the numerical analysis of rotationally symmetric synchrotron radiation

In an effort to provide a computationally convenient approach to the characterization of partially coherent synchrotron radiation in phase space, a thorough discussion of the minimum dimensionality of the Wigner distribution function for rotationally symmetric sources of arbitrary degrees of coherence is presented. It is found that perfectly coherent, perfectly incoherent and partially coherent sources may all be characterized by a three‐dimensional reduced Wigner distribution function, and some special cases are discussed in which a two‐dimensional reduced Wigner distribution function suffices. An application of the dimension‐reducing formalism to the case of partially coherent radiation from a planar undulator and a circularly symmetric electron beam as can be found in linear accelerators is demonstrated. The photon distribution is convolved over a realistic electron bunch, and how the beta function, emittance and energy spread of the bunch affect the total degree of coherence of the radiation is inspected. Finally the cross spectral density is diagonalized and the eigenmodes of the partially coherent radiation are recovered.     

夸克的半经典输运方程

对夸克的量子输运方程取半径典近似时保留到Wigner函数的一次微商项;在色空间和自旋空间展开这个半经典输运方程,得到了色单态自旋标量和色单态自旋矢量的输运方程:并把得到的结果和阿贝尔等离子体进行比较讨论了QGP的非阿见尔性质.     

Algebraic aspects of the wigner distribution in quantum mechanics

The algebraic structure underlying the method of the Wigner distribution in quantum mechanics and the Weyl correspondence between classical and quantum dynamical variables is analysed. The basic idea is to treat the operators acting on a Hilbert space as forming a second Hilbert space, and to make use of certain linear operators on them. The Wigner distribution is also related to the diagonal coherent state representation of quantum optics by this method.     

Transport Theory for Friedberg-Lee Model

By means of correlation density matrix theory and in terms of Wigner distribution function for quark, we obtain the transport equation for Friedberg-Lee model, which includes the coUision terms consis ten tly.     

Spin Hall effect and Berry phase of spinning particles

We consider the adiabatic evolution of the Dirac equation in order to compute its Berry curvature in momentum space. It is found that the position operator acquires an anomalous contribution due to the non-Abelian Berry gauge connection making the quantum mechanical algebra noncommutative. A generalization to any known spinning particles is possible by using the Bargmann–Wigner equation of motions. The noncommutativity of the coordinates is responsible for the topological spin transport of spinning particles similarly to the spin Hall effect in spintronic physics or the Magnus effect in optics. As an application we predict new dynamics for nonrelativistic particles in an electric field and for photons in a gravitational field.     

Solving the Boltzmann equation in a 2-D-configuration and2-D-velocity space for capacitively coupled RF discharges

A new kinetic scheme, the generalized Monte Carlo flux (GMCF) method, provides the electron particle distribution function in phase space, f(ν, μ, r, z, t) (ν: speed, μ: velocity angle, r: radial position, z: axial position, and t: time), for solving the Boltzmann equation in modeling capacitively coupled RP discharges. For a simulation with spatial- and temporal-varying fields in RF discharges, the GMCF method handles the collision terms of the Boltzmann equation by using one transition matrix to compute the collision transition between velocity space cells. An anti-diffusion flux transport scheme is developed to overcome the numerical diffusion in the velocity and configuration spaces. The major advantages of the GMCF method are the increase in resolution in the tail of distribution functions and the decrease of computation time. The GMCF calculation results in terms of microscopic electron distribution function and macroscopic quantities of density, electric field and ionization rate, are presented for RF discharges and compared with other kinetic and fluid simulation and experimental results. The effects of the induced radial electric field in the sheath close to the radial wall in a cylindrically symmetric parallel-plate geometry are discussed     

Universal invariants in quantum mechanics and physics of optical and particle beams

We give a brief review of the theory of quantum universal invariants and their counterparts in the physics of light and particle beams. The invariants concerned are certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators, or the transverse phase-space coordinates of the paraxial beams of light or particles. They are conserved in time (or along the beam axis) independently of the concrete form of the coefficients of the Schrödinger-like equations governing the evolution of the systems, provided that the effective Hamiltonian is either a generic quadratic form of the generalized coordinate-momenta operators or a linear combination of generators of some finite-dimensional algebra (in particular, any semisimple Lie algebra). Using the phase space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we elucidate the relation of the quantum (optical) invariants to the classical universal integral invariants of Poincaré and Cartan. The specific features of Gaussian beams are discussed as examples. The concept of the universal quantum integrals of motion is introduced, and examples of the “universal invariant solutions” to the Schrödinger equation, i.e., self-consistent eigenstates of the universal integrals of motion, are given.