Wigner function of a two-dimensional electron gas in a magnetic field

The Wigner function of a two-dimensional electron gas in an arbitrary magnetic field perpendicular to the plane in which the electrons are confined is constructed rigorously. The function is useful in taking various statistical averages and illuminates the roles played by the hyperbolic functions of the field which appear in the expressions of the susceptibility and other physical quantities.     

Reconstruction of the Wigner transform of a rotationally symmetric two-dimensional beam from the Wigner transform of the beam's one-dimensional sample

A coherent rotationally symmetric two-dimensional beam is essentially one-dimensional in content: It is fully determined by the one-dimensional sample along a diagonal of the circularly symmetric field distribution in a transverse plane. The linear transform that reconstructs the four-dimensional Wigner distribution of the full two-dimensional beam from the two-dimensional Wigner distribution of the one-dimensional sample is presented.     

Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics

In quantum theory, symmetry has to be defined necessarily in terms of the family of unit rays, the state space. The theorem of Wigner asserts that a symmetry so defined at the level of rays can always be lifted into a linear unitary or an antilinear antiunitary operator acting on the underlying Hilbert space. We present two proofs of this theorem which are both elementary and economical. Central to our proofs is the recognition that a given Wigner symmetry can, by post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis often focuses on the behaviour of certain two-dimensional subspaces of the Hilbert space under the action of a given Wigner symmetry, but the relevance of this behaviour to the larger picture of the whole Hilbert space is made transparent at every stage.     

A new kind of representations on noncommutative phase space

We introduce new representations to formulate quantum mechanics on noncommutative phase space, in which both coordinate-coordinate and momentum-momentum are noncommutative. These representations explicitly display entanglement properties between degrees of freedom of different coordinate and momentum components. To show their potential applications, we derive explicit expressions of Wigner function and Wigner operator in the new representations, as well as solve exactly a two-dimensional harmonic oscillator on the noncommutative phase plane with both kinetic coupling and elastic coupling.     

Ray picture of diffraction gratings

We elaborate on a geometrical picture of diffraction gratings. Exact paraxial propagation including coherence effects is obtained by allowing the geometrical rays to transport Wigner function instead of simply specific intensity. We apply this formalism to perfect and less than perfect gratings, illuminated by plane and Gaussian waves.     

Symplectic Tomography of De Broglie Wave

We consider tomograms and quasidistribution functions like the Wigner functions that violate the standard normalization condition and obtain the conditions under which a reconstruction of the density matrix using these tomograms and quasidistribution functions is possible. Then we study an example of the de Broglie plane wave.     

Wigner distribution function of Gaussian beam through an apertured and misaligned ABCD optical system

By expanding the hard aperture function into a finite sum of complex Gaussian functions, approximate analytical expressions of Wigner distribution function for fundamental mode Gaussian beam through an apertured and misaligned ABCD optical system are derived. Compared with the aligned case, results show that the misaligned optical system will introduce in the output Wigner distribution function a coordinate shift in space-frequency plane. To examine the analytical results, some numerical simulations are carried out and the absolute errors between analytical results and numerical integral ones are presented. It is shown that the approximate analytical results are proper and ascendant.     

Interpretation of the hydrodynamical formalism of quantum mechanics

The hydrodynamical formalism for the quantum theory of a nonrelativistic particle is considered, together with a reformulation of it which makes use of the methods of kinetic theory and is based on the existence of the Wigner phase-space distribution. It is argued that this reformulation provides strong evidence in favor of the statistical interpretation of quantum mechanics, and it is suggested that this latter could be better understood as an almost classical statistical theory. Moreover, it is shown how, within this context, the Wigner and the Margenau-Hill functions are not equivalent, and that the latter is essentially unsatisfactory, as well as the associated symmetrization rule. Arguments in favor of a stochastic picture of the phenomena at the microscopic level are also presented.     

二维Sinai台球系统的量子混沌研究

研究了二维Sinai台球系统的经典与量子的对应关系,运用定态展开法和Gutzwiller的周期轨道理论对Sinai台球系统的态密度经傅里叶变换得到的量子长度谱进行分析,并把量子长度谱中峰的位置与其所对应的经典体系的周期轨道长度做对比,发现两者之间存在很好的对应关系.观察到了一些量子态局域在短周期轨道附近形成量子scarred态或量子superscarred态.还研究了同心与非同心Sinai台球系统的能级最近邻间距分布,发现同心Sinai台球系统是近可积的,非同心Sinai台球系统在θ=3π/8下,随两中心间距离的增加,能级最近邻间距分布将由近可积向维格那分布过渡.     

The Wigner function for two dimensional tori: Uniform approximation and projections

The Wigner representation of a quantum state, corresponding to a classically integrable Hamiltonian, has been shown to be intimately tied to a classical phase space torus of the same energy. The fact that the semiclassical approximation of the Wigner function there derived turns out to be singular on the torus, as well as on the “Wigner caustic” which contains it, is due to well known limitations of the stationary phase method. The uniform approximation, here derived, does indeed ascribe to the Wigner function a high amplitude along the Wigner caustic, but this is modulated by rapid oscillations except at the torus itself. Asymptotic expansion away from the torus leads back to the semiclassical approximation. Close to the torus the Wigner function is described by a simple transitional approximation which can be resolved into a product of Wigner functions corresponding to one dimensional tori. These results permit one to explicitly project the Wigner function onto any (Lagrangian) coordinate plane so as to obtain the corresponding wave intensity.     

Nonclassical properties of odd and even elliptical states

As a generalization of the optical circular states, elliptical states which are quantum superposition of coherent states on an ellipse in the α plane are constructed. The statistical properties of the states are investigated by using sub-Poissonian photon statistics, quadrature squeezing, Wigner function and phase distribution. It is shown that the elliptical states exhibit stronger quadrature squeezing. The interference fringes between the coherent states form the elliptic annuli of Fock states in the Wigner function picture. The phase distribution is no longer uniform as the circular states. An experimental scheme is proposed for generating equidistant coherent-state superpositions on an ellipse for the motion of the center of mass of a trapped ion.     

Transition from two-dimensional to three-dimensional classical artificial atoms

The transition from two-dimensional (2D) to three-dimensional (3D) classical artificial atoms is studied. As a function of the confinement strength perpendicular to the plane, first- and second-order structural transitions are observed, which change the 2D ring structure into a layered structure and finally into the 3D shell structure in the case of a spherical confinement potential. A clear structural analogy exists between the infinite bilayer Wigner crystal (BLWC) and our finite system when the confinement is such that a bilayer structure exists in the central part of the atom.     

The Wigner distribution functions of coherent and partially coherent Bessel-Gaussian beams

Based on the integral representation of the Bessel functions and the generating function of the Tricomi function,an analytical expression of the Wigner distribution function(WDF) for a coherent or partially coherent Bessel-Gaussian beam is presented.The reduced two-dimensional WDFs are also demonstrated graphically,which reveals the dependence of the reduced WDFs on the beam parameters.     

The Wigner distribution functions of coherent and partially coherent Bessel-Gaussian beams

Based on the integral representation of the Bessel functions and the generating function of the Tricomi function, an analytical expression of the Wigner distribution function (WDF) for a coherent or partially coherent Bessel-Gaussian beam is presented. The reduced two-dimensional WDFs are also demonstrated graphically, which reveals the dependence of the reduced WDFs on the beam parameters.     

Determination of the interactions in confined macroscopic Wigner islands: theory and experiments

Macroscopic Wigner islands present an interesting complementary approach to explore the properties of two-dimensional confined particles systems. In this work, we characterize theoretically and experimentally the interaction between their basic components, viz., conducting spheres lying on the bottom electrode of a plane condenser. We show that the interaction energy can be approximately described by a decaying exponential as well as by a modified Bessel function of the second kind. In particular, this implies that the interactions in this system, whose characteristics are easily controllable, are the same as those between vortices in type-II superconductors.     

Localization corrections to charge and spin conductivity in ferromagnetic layered structures

Localization corrections to charge and spin conductivity of two-dimensional ferromagnetic systems with spin–orbit interaction are studied theoretically. The corrections lead to negative magnetoresistance — also in the presence of spin–orbit scattering. Magnitude of the corrections depends on the magnetization orientation with respect to the plane of the system. The corrections to spin conductivity are shown to be small.     

Strong scintillation spectra behind a phase screen containing large-scale anisotropic irregularities

Using the approximation of an anisotropic statistically-homogeneous phase screen, we consider spectra of strong scintillations. Numerical calculations are made for the model of large-scale anisotropic inhomogeneities typical of the Earth’s stratosphere. The spectrum transformation is studied for the transition from weak scintillations to the asymptotic regime of strong scintillations. We show that with increasing level of the scintillations, their spectra rapidly broaden to the region of large wave numbers which exceed both the inverse internal scale of the irregularities and the inverse radius of the Fresnel zone by orders of magnitude. Notable deviations of the two-dimensional spectra from the predictions based on perturbation theory are shown to occur for scintillation variance exceeding 0.1. The obtained two-dimensional spectra of scintillations give a complete picture of the behavior of one-dimensional spectra which can be retrieved from satellite observations made for different angles between the orbit plane and the direction to the source. Vertical and horizontal one-dimensional spectra are studied in detail. Approximate algebraic formulas are derived and their validity is proved by applying them to the calculation of spectra of strong scintillations for a wide (several decades) range of the wave number values. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 50, No. 9, pp. 747–765, September 2007.     

Water sorption characteristics of carbon nanostructures calculated on the basis of the Ornstein–Zernike equation

Molecular systems in contact with the bounding surfaces are examined. New apparatus describing adequately quasi–two-dimensional structures whose cross sectional dimensions are comparable with the correlation length is suggested. It is based on the modified method of the Wigner distribution functions and the system of Ornstein–Zernike equations generalized for one- and two-particle distribution functions. Modeling of the physical gas sorption process in carbon nanofibers and nanotubes allows local microstructure, sorption characteristics, effective sizes, and optimal technological filling parameters to be calculated.     

Quantum spin Hall effect in graphene

We study the effects of spin orbit interactions on the low energy electronic structure of a single plane of graphene. We find that in an experimentally accessible low temperature regime the symmetry allowed spin orbit potential converts graphene from an ideal two-dimensional semimetallic state to a quantum spin Hall insulator. This novel electronic state of matter is gapped in the bulk and supports the transport of spin and charge in gapless edge states that propagate at the sample boundaries. The edge states are nonchiral, but they are insensitive to disorder because their directionality is correlated with spin. The spin and charge conductances in these edge states are calculated and the effects of temperature, chemical potential, Rashba coupling, disorder, and symmetry breaking fields are discussed.     

Pure state condition for the semi-classical Wigner function

The Wigner function W(p,q) is a symmetrized Fourier transform of the density matrix ρ(q₁,q₂), representing quantum-mechanical states or their statistical mixture in phase space. Identification of these two alternatives in the case of density matrices depends on the projection identity ρ² = ρ; its Wigner correspondence is the pure state condition. This criterion is applied to the Wigner functions obtained from standard semiclassical wave functions, determining as pure states those whose classical invariant tori satisfy the generalized Bohr-Sommerfeld conditions. Superpositions of eigenstates are then examined and it is found that the Wigner function corresponding to Gaussian random wave functions are smoothed out in the manner of mixed-state Wigner functions. Attention is also given to the pure-state condition in the case where an angular coordinate is used.