Wigner Distribution Function Research Articles

Convolution based fractional Wigner distribution and ambiguity function: theory and applications

Convolution based fractional Wigner distribution and ambiguity function: theory and applications

Simulation of diffraction and scattering using the Wigner distribution function.

We present a new, to the best of our knowledge, method to simulate diffraction images accounting for both coherent and incoherent effects, based on the Wigner distribution function of the exit wave. This permits the simulation of wave and particle effects simultaneously and simulates images photon by photon. It is motivated by artifacts observed in x ray phase-contrast images after phase retrieval, present as noise in the low spatial frequency range, which can make analysis of such images challenging. Classical simulations have so far not been able to reproduce these artifacts. We hypothesize that these artifacts are due to incoherent scatter present in the images, hence the interest in developing a simulator that permits the simulation of both diffraction and incoherent scattering. Here, we give a first demonstration of the method by simulating the Gaussian double-slit experiment. We demonstrate the capability of combining diffraction and incoherent scattering, as well as simulating images for any propagation distance.

Nonseparable wave evolution equations in quantum kinetics

A nonseparable wave-like integro-differential equation for the time evolution of the Wigner distribution function in phase space is educed from the corresponding separable kinetic equation. By employing the quantum hydrodynamical description, a non-local evolution wave equation is also derived by synthesizing the Hamilton‐Jacobi equation with that of continuity, which predicts the generation of nonlocal and quadrupole quantum phenomena in the propagation of the spatial probability density.

Statistical description of interacting multistream quantum systems

In this research, the electrostatically coupled multistream quasiparticle excitations are studied in the framework of the Wigner distribution function. It is remarked that the Wigner distribution of coupled multistream collective quantum excitations satisfies a simple Liouville-like evolution equation from which a generalized distribution function for multistream quasiparticle excitations is deduced. The phase-space structure of collective quantum excitations in counter-stream electron and two-stream electron–positron gas with their evolution is calculated and electron/positron hole formation due to the onset of quantum stream instability is studied in connection with the energy band structure of the multistream quantum system, for the first time. The quantum stream instabilities in symmetric and asymmetric stream systems are studied and compared. It is found that the presence of opposite-charge streams leads to overall stability due to lowering the interaction potential effect. The generalized Wigner theory is also applied to study the electron transport in a one-dimensional periodic lattice using the concept of virtual streams. Current generalized statistical formalism may be used to model different quantum phenomena in the linear excitations limit with collective electrostatic interactions. The applications extend to the stream instability in quantum charge transport in metals, semiconductors, plasmonic devices, phase-space structure of charge carriers in periodic lattices interacting with the external potential of arbitrary shape and the dynamic evolution of dense electron–positron jets in active galactic nuclei or within the extremely dense astrophysical objects.

Wigner distribution and entropy of partially coherent light generated by perfect optical vortices.

We developed analytical expressions for the Wigner distribution function of partially coherent fields generated by the scattering of beams with a particular phase structure, namely perfect optical vortex beams. In addition, we provide the modal decomposition of the field correlations and evaluate the evolution of Shannon entropy associated with the partially coherent field.

Novel two-dimensional Wigner distribution and ambiguity function in the framework of the two-dimensional nonseparable linear canonical transform

Novel two-dimensional Wigner distribution and ambiguity function in the framework of the two-dimensional nonseparable linear canonical transform

New quadratic phase Wigner distribution and ambiguity function with applications to LFM signals

New quadratic phase Wigner distribution and ambiguity function with applications to LFM signals

Fault identification, classification, and localization in microgrids using superimposed components and Wigner distribution function

The integration of Distributed Energy Resources (DERs) into distribution grids has become increasingly feasible and sustainable due to the development of microgrids. However, the development of an effective protection strategy remains a challenge in the implementation of microgrids. To address this challenge, this paper presents a simple and novel microgrid protection method based on superimposed components, Wigner distribution function (WDF) and alienation index-based. The proposed method develops a new fault detection index (FDI) by applying the alienation coefficient and WDF on a superimposed current signal to detect faulty events in the microgrid. The scheme is inherently phase segregated because the FDI is obtained for each phase individually. In addition, the proposed strategy introduces a new fault zone identification method based on the superimposed positive sequence reactive power (SPSQ). After obtaining the complete fault information, a relevant trip signal is generated to isolate the faulty section from the rest of the grid. The proposed methodology is evaluated through simulations using MATLAB/SIMULINK software. Various fault types, with varying parameters are simulated to validate the proposed approach. The results indicate that the proposed methodology is capable of recognizing, classifying, and locating all fault types in both grid-connected and islanded modes of operation.

Radiance-field holography for high-quality 3D reconstruction

Conventional light field-based computer-generated holograms, such as holographic stereograms, can provide nature parallax and accurate occlusion relation but are restricted by the inherent tradeoff concerning the angular and spatial resolution of the reconstructed optical field. In contrast, Wigner distribution function based light field generated holograms, also called non-hogel-based holograms, release this limitation by using basic ray-to-wave transformation frameworks. However, at present, this transformation framework is accurate only for generating holograms from orthogonal light field images of 3D scenes with a limited depth range and size. Thus, the radiance field holography method is proposed in this paper to generate high-quality 3D holograms for arbitrary scenes that can achieve high-resolution reconstruction within a deep depth range. In our method, a proposed radiance field is constructed from the perspective light field, which consists of multiple reference planes to resample and redistribute the light rays to represent the diffracted optical field of the 3D objects according to our developed sampling criterion. The complex optical field of the 3D object can be synthesized from these reference planes based on inverse Wigner transform and Fresnel propagation. As a result, both the numerical simulations and the optical experiments are implemented, and it is demonstrated that the high-quality 3D reconstruction of the synthesized hologram is achieved, which reflects the highest reconstruction resolution among the existing methods, the large depth range with accurate refocus and defocus, and the natural view-dependent effects.

Distribution of intensity and M2 factor for a partially coherent flat-topped beam in bidirectional turbulent atmosphere and plasma connection

This study investigates the bidirectional transmission of a partially coherent flat-topped beam in a turbulent atmosphere and plasma. Analytical formulas for the intensity distribution and M2 factor are derived based on the optical transmission matrix, Collins formula, and second moment theory with Wigner distribution function. Numerical results show that the beam order and transverse spatial coherence width can be selected appropriately to mitigate turbulence and plasma induced evolution properties. The partially coherent flat-topped beam propagation through a turbulent atmosphere and plasma of the forward transmission effect on the intensity distribution and M2 factor are smaller than that of the reverse transmission. Under the same conditions, the M2 factor of a partially coherent flat-topped beam is smaller than the Gaussian beam in bidirectional transmission. Our results can be used in long-distance free-space optical communications.

The Wigner function of a semiconfined harmonic oscillator model with a position-dependent effective mass

We propose a phase-space representation concept in terms of the Wigner function for a quantum harmonic oscillator model that exhibits the semiconfinement effect through its mass varying with the position. The new method is used to compute the Wigner distribution function exactly for such a semiconfinement quantum system. This method suppresses the divergence of the integrand in the definition of the quantum distribution function and leads to the computation of its analytical expressions for the stationary states of the semiconfined oscillator model. For this quantum system, both the presence and absence of the applied external homogenous field are studied. Obtained exact expressions of the Wigner distribution function are expressed through the Bessel function of the first kind and Laguerre polynomials. Furthermore, some of the special cases and limits are discussed in detail.

Excited hadron channels in hadronization

The proper treatment of hadronic resonances plays an important role for many aspects of heavy ion collisions. We expect this to be the case also for hadronization, due to the large degeneracies of excited states, and the abundant production of hadrons from their decays. We show how a comprehensive treatment of excited meson states can be incorporated into quark recombination, and in extension, into Hybrid Hadronization. We discuss in detail the quantum mechanics of forming excited states, utilizing the Wigner distribution functions of angular momentum eigenstates of isotropic 3-D harmonic oscillators. We describe how resonance decays can be handled, based on a set of minimal assumptions, by creating an extension of hadron decays in PYTHIA 8. Finally, we present a study of hadron production by jets using PYTHIA and Hybrid Hadronization with excited mesons up to orbital angular momentum L = 4. We find that states up to L = 2 are produced profusely by quark recombination.

Angular width and beam quality of a partially coherent standard Laguerre-Gaussian vortex beam in turbulent plasma.

Analytical formulas for the angular width and propagation factor of a partially coherent standard Laguerre-Gaussian (sLG) vortex beam through anisotropic turbulent plasma were derived based on the extended Huygens-Fresnel integral and the second-order moments of the Wigner distribution function. The evolution properties of the angular width and propagation factor of partially coherent sLG vortex beams propagating in anisotropic turbulent plasma were investigated numerically. The numerical results demonstrate the influence of the source and turbulence parameters on the normalized angular width and normalized propagation factor of the partially coherent sLG vortex beams. It can be observed that in a partially coherent sLG vortex beam with a lower beam order, topological charge, and wavelength, or a higher correlation coefficient of the source plane, and with increasing propagation distance and refractive index fluctuation variance or decreasing anisotropy parameters, the outer and inner scales of the turbulent plasma have a large angular width and propagation factor (i.e.,the beam quality is worse). The results of this study will be beneficial for applications in remote sensing and optical communications.

Wigner quasi-probability distribution of a resonator coherent field interacting with a flux qubit via two-photon coupling

The Wigner quasi-probability distribution function is an important tool for estimating the quantumness of quantum states. The Wigner distribution function (WDF) is related monotonically to the coherent field state density operator and their probability distributions. In this paper, we analyze the time-dependent resonator coherent field states produced by a flux qubit’s interaction with a resonator-field through a two-photon coupling. The WDF dynamics of the coherent field states is investigated under the effects of the qubit-resonator detuning and the intrinsic decoherence. The WDF non-classicality associated with the mixedness resonator entropy dynamics can be controlled by varying the qubit-resonator interaction, detuning as well as the decoherence. The intrinsic decoherence clearly affects the WDF non-classicality, identified by the negative values of the WDF. Finally, by fixing one or both of the complex phase space components, we are able to explore the dynamics of the Wigner-distribution functions.

Dynamical entropic measure of nonclassicality of phase-dependent family of Schrödinger cat states

The phase-space approach based on the Wigner distribution function is used to study the quantum dynamics of the three families of the Schrödinger cat states identified as the even, odd, and Yurke–Stoler states. The considered states are formed by the superposition of two Gaussian wave packets localized on opposite sides of a smooth barrier in a dispersive medium and moving towards each other. The process generated by this dynamics is analyzed regarding the influence of the barrier parameters on the nonclassical properties of these states in the phase space below and above the barrier regime. The performed analysis employs entropic measure resulting from the Wigner–Rényi entropy for the fixed Rényi index. The universal relation of this entropy for the Rényi index equal one half with the nonclassicality parameter understood as a measure of the negative part of the Wigner distribution function is proved. This relation is confirmed in the series of numerical simulations for the considered states. Furthermore, the obtained results allowed the determination of the lower bound of the Wigner–Rényi entropy for the Rényi index greater than or equal to one half.

Laser Beam Quality of Airy Beam in the Jet Engine Exhaust Induced Turbulence

Theoretical predictions of laser beam interactions with jet-engine-exhaust-induced turbulence are of importance for optimization of various optical systems, including LIDARs and airborne laser measurement systems. This paper is based on the extended Huygens–Fresnel diffraction integral and the Wigner distribution function, combined with the recently proposed power spectral density function of atmospheric turbulence induced by jet engine exhaust, to derive the formula for the quality factor of a partially coherent Airy beam in turbulent media induced by jet engine exhaust. Numerical calculations based on the analytical formula show that the smaller the structure constant and outer scale of the atmospheric turbulence induced by jet engine exhaust, the smaller the quality factor of a partially coherent Airy beam, indicating better laser beam quality. At the same time, the larger the generalized parameter and inner scale of the jet-engine-exhaust-induced turbulence and the longer the wavelength of the laser beam, the smaller the corresponding quality factor. The numerical results also show that reducing the coherence length of a partially coherent Airy beam and increasing its characteristic width are beneficial for reducing the quality factor in the atmospheric turbulence induced by jet engines. However, the numerical results show that increasing the coherence length of a partially coherent Airy beam and reducing its characteristic width are beneficial for reducing the quality factor in vacuum. This situation demonstrates significant differences between the interaction mechanisms of an Airy beam in vacuum and in the jet-engine-exhaust-induced turbulence. The results of our discussion contribute to clarifying the mechanism of interaction between an Airy laser beam and random media, as well as the application of a curved Airy laser beam in practical situations.

Analysis and 3D Imaging of Multidimensional Complex THz Fields and 3D Diagnostics Using 3D Visualization via Light Field

We present a numerical platform for 3D imaging and general analysis of multidimensional complex THz fields. A special 3D visualization is obtained by converting electromagnetic (EM) radiation to a light field via the Wigner distribution function, which is known for discovering (revealing) hidden details. This allows for 3D diagnostics using the simple techniques of geometrical optics, which significantly facilitates the whole analysis. This simulation was applied to a complex field composed of complex beams emitted as ultra-narrow femtosecond pulses. A method was developed for the generation of phase–amplitude and spectral characteristics of complex multimode radiation in a free-electron laser (FEL) operating under various parameters. The tool was successful at diagnosing an early design of the transmission line (TL) of an innovative accelerator at the Schlesinger Family Center for Compact Accelerators, Radiation Sources, and Applications.

Phase Spaces, Parity Operators, and the Born–Jordan Distribution

Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution.

Wigner distribution function approach to analyze MIMO communication within a waveguide

Multiple-input-multiple-output (MIMO) communication is a technology to create high capacity wireless links. The main aim of this paper is to provide a foundation to mathematically model wireless chip to chip communication within complex enclosures. This paper mainly concentrates on modelling wave propagation between transmit and receive antennas through a phase space approach which exploits the relationship between the field-field correlation function (CF) and the Wigner distribution function (WDF). A reliable model of wireless chip-to-chip (C2C) communication helps mitigate the information bottleneck caused due to the wired connections between chips, thus, help improve the efficiency of electronic devices of the future. Placing complex sources such as printed circuit board (PCB) inside a cavity or enclosure results in multi-path interference and hence makes the prediction of signal propagation more difficult. Thus, the CFs can be propagated based on a ray transport approach that predicts the average radiated density, but not the significant fluctuations that occur about it. Hence, the WDF approach can be extended to problems in finite cavities that incorporates reflections as well. Phase space propagators based on classical multi-reflection ray dynamics can be obtained by considering the high-frequency asymptotics.

On the independent significance of generalizations of the Wigner distribution function.

The Wigner distribution function (WDF) is a significant time-frequency analysis tool in, e.g.,the theory of optical coherence and signal processing. Recently, various generalizations of the WDF associated with linear canonical transforms have been proposed to improve and broaden its applications. It is useful to identify which of these novel distributions have independent significance for further investigation. We plot these distributions for a test signal using symbolic integration to find which distributions are linear coordinate transforms of the WDF or have unique features. Five distributions are determined to be linear coordinate transforms of the WDF. Two distributions show unique characteristics. We focus on the mathematical interpretation, properties, and possible applications of those two distributions. We demonstrate how one of them can be used in the analysis of partially coherent systems.