Jauch-Piron system of imprimitivities for phonons. I. Localizability in discrete space

This paper is devoted to a discussion of the notion of localizability for phonons, i.e., quasiparticles arising from the harmonic vibrations of a system ofn atoms bound to one another by elastic forces. The natural tools for the analysis of localizability are the projection operatorsÊ() acting on the Hilbert space of one-phonon states, where is an arbitrary subset of the set that consists ofn vectors specifying the equilibrium positions ofn atoms. The expectation value ofÊ() is the probability that the phonon belongs to the atoms whose equilibrium positions are characterized by the elements of . For a strongly localizable phonon all of the projection operatorsÊ() commute with one another, whereas in the case of a weakly localizable phonon the operatorsÊ(₁) andÊ(₂) do not commute when ₁ and ₂ overlap. With the aid of the Jauch-Piron quantum theory of localization in space, the present paper describes the method of obtainingÊ() and also shows that if in the system ofn atoms there exist normal modes of zero frequency, then the phonon is only weakly localizable. Given the explicit expression forÊ(), one can define the number-of-phonons operator as well as the quasiparticle analogue (given in a companion paper) of the Wigner distribution function.     

Spectrum of the parametric down converted radiation calculated in the Wigner function formalism

We continue the study of parametric down conversion within the framework of the Wigner representation, by using a Maxwellian approach developed in a recent paper [A. Casado et al., Eur. Phys. J. D 11, 465 (2000)]. This gives a mechanism, inside the crystal, for the production of the down-converted radiation. We obtain the electric field to second order in the coupling constant by using the Green's function method, and compare our treatment with the standard Hamiltonian approach. The spectrum of the down-converted radiation is calculated as a function of the parameters of the nonlinear crystal (in particular the length) and the radius of the pumping beam. Received 15 May 2000     

The Wigner distribution function

In this letter, the relationship between the characteristic function for two arbitrary noncommuting observables and a generalized Wigner distribution function is established. This distribution function is shown to have no simple interpretation in the sense of probability theory but, in lieu of its special properties, can be used directly for calculating the expectation values of observables.     

Wigner函数的简单引入

通过结合坐标表象及动量表象完备性的纯高斯积分形式及Wigner函数的物理意义,在量子统计的意义下简单的引入了Wigner算符及Wigner函数     

Wigner function for twisted photons

A comprehensive theory of the Weyl-Wigner formalism for the canonical pair angles-angular momentum is presented, with special emphasis in the implications of rotational periodicity and angular-momentum discreteness.     

Quantum electronics of optical phonons. II

Optical maser research develops new quantum-electronics phenomena in alpha-quartz, including optical phonon-masers, second- and first-order ultraviolet lasers and diffraction of ultraviolet light by optical phonons     

The probability interpretation of Wigner function by SIC-POVM

This paper will give the probability interpretation of Wigner function. We will show that Wigner function is linear function of the probability associated with special POVM (positive operator valued measure). This special POVM is called symmetric informationally complete POVM (SIC-POVM). Therefore, Wigner function can be described by physical measurement system.     

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.     

A formalism for the investigation of algebraically special metrics. II

The equations of the formalism developed in part I are simplified by specialisation of the basic pair of null directions. The auxiliary vectors previously introduced are shown to have intrinsic geometric properties which are directly related to the complexity of the differential operators of the formalism. An Ansatz based on these properties is introduced and the calculation of the metric so defined is used to display the connection between the coordinate-free and coordinate-dependent solutions.     

The Wigner function in the relativistic quantum mechanics

A detailed study is presented of the relativistic Wigner function for a quantum spinless particle evolving in time according to the Salpeter equation.     

The Wigner distribution function applied to laser radiation

The Wigner distribution function (WDF) is investigated analytically and then calculated numerically and represented graphically for Laguerre-Gaussian modes and for a laser with annular gain medium. The derivation of the normalized beam qualityM ² from the WDF on the basis of second order moments is discussed, andM ² is determined for the examples mentioned above.     

The Wigner distribution function—50th birthday

Foundations of Physics - We discuss the profound influence which the Wigner distribution function has had in many areas of physics during its fifty years of existence.     

Wigner function and phase properties for a two-qubit field system under pure phase noise

We investigate an analytical solution for a two-qubit field system in the dispersive regime with a reservoir. We analyze the influence of the phase damping on the Wigner function and the phase properties. We found that the phase damping destroys the phase probability of the global system for the coherent state and even coherent state. The phase damping leads to decay of the Wigner function for the coherent state.     

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.     

On the Wigner distribution function for an oscillator

We present two new derivations of the Wigner distribution function for a simple harmonic oscillator Hamiltonian. Both methods are facilitated using a formula which expresses the Wigner function as a simple trace. The first method of derivation then utilizes a modification of a theorem due to Messiah. An alternative procedure makes use of the coherent state representation of an oscillator. The Wigner distribution function gives a semiclassical joint probability for finding the system with given coordinates and momenta, and the joint probability is factorable for the special case of an oscillator. An important application of this result occurs in the theory of nuclear fission for calculating the probability distributions for the masses, kinetic energies, and vibrational energies of the fission fragments at infinite separation.     

Wigner function for highly convergent three-dimensional wave fields

The angle-impact Wigner function for highly convergent three-dimensional scalar wave fields is derived directly by use of the three-dimensional generalized optical transfer function rather than from a six-dimensional Wigner function. The angle-impact Wigner function is a real four-dimensional function from which the intensity at any point in space is readily determined.