Wigner Distribution Function Research Articles

Complexity of quantum motion and quantum-classical correspondence: A phase-space approach

We discuss the connection between the out-of-time-ordered correlator and the number of harmonics of the phase-space Wigner distribution function. In particular, we show that both quantities grow exponentially for chaotic dynamics, with a rate determined by the largest Lyapunov exponent of the underlying classical dynamics, and algebraically—linearly or quadratically—for integrable dynamics. It is then possible to use such quantities to detect in the time domain the integrability-to-chaos crossover in many-body quantum systems.Received 5 January 2020Accepted 12 October 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.043178Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum chaosQuantum-to-classical transitionSemiclassical physicsNonlinear DynamicsGeneral Physics

Multi-hyperbolic sine-correlated beams and their statistical properties in turbulent atmosphere.

We introduce a partially coherent beam, called a multi-hyperbolic sine-correlated (MHSC) beam, by employing a multi-hyperbolic sine function to modulate the spectral degree of coherence. Based on the extended Huygens-Fresnel principle and second-order moments of the Wigner distribution function, we derive the analytical expressions for the spectral intensity, the root-mean-square (rms) angular width and the M2 factor in turbulent atmosphere. Numerical results show that the intensity profile, which keeps the dark-hollow invariant in free space, will be gradually destroyed by the turbulence along the propagation distance. We believe that the MHSC beams have significant advantage over the hyperbolic sine-correlated beams for reducing the turbulence-induced degradation, especially for the MHSC beams with a higher beam order N. The effects of the beams parameters and the turbulent atmosphere on the beam quality are analyzed in detail.

Wigner Time-Delay and Distribution for Polarization Interaction in Strongly Coupled Semiclassical Plasmas

The quantum effect on the Wigner time-delay and distribution for the polarization scattering in a semiclassical dense plasma is explored. The partial wave analysis is applied for a partially ionized dense plasma to derive the phase shift for the polarization interaction. The Wigner time-delay and the Wigner distribution are derived for the electron–atom polarization interaction including the effects of quantum-mechanical characteristic and plasma screening. In this work, we show that the Wigner time-delay and the Wigner distribution for the polarization interaction can be suppressed by the quantum effect. The Wigner time-delay and the Wigner distribution are also significantly suppressed by the increase of plasma shielding. The variation of the Wigner time-delay and the Wigner distribution function due to quantum screening is discussed.

A Hybrid Fault Recognition Algorithm Using Stockwell Transform and Wigner Distribution Function for Power System Network with Solar Energy Penetration

Penetration level of solar photovoltaic (PV) energy in the utility network is steadily increasing. This changes the fault level and causes protection problems. Furthermore, multi-tapped structure of distribution network deployed to integrate solar PV energy to the grid and supplying loads at the same time also raised the protection challenges. Hence, this manuscript is aimed at introducing an algorithm to identify and classify the faults incident on the network of utilities where penetration level of the solar PV energy is high. This fault recognition algorithm is implemented in four steps: (1) calculation of Stockwell transform-based fault index (STFI) (2) calculation of Wigner distribution function-based fault index (WDFI) (3) calculation of combined fault index (CFI) by multiplying STFI and WDFI (4) calculation of index for ground fault (IGF) used to recognize the involvement of ground in a fault event. The STFI has the merits that its performance is least affected by the noise associated with the current signals and it is effective in identification of the waveform distortions. The WDFI employs energy density of the current signals for estimation of the faults and takes care of the current magnitude. Hence, CFI has the merit that it considers the current magnitude as well as waveform distortion for recognition of the faults. The classification of faults is achieved using the number of faulty phases. An index for ground fault (IGF) based on currents of zero sequence is proposed to classify the two phase faults with and without the ground engagement. Investigated faults include phase to ground, two phases fault without involving ground, two phases fault involving ground and three phase fault. Fault recognition algorithm is tested for fault recognition with the presence of noise, various angles of fault incidence, different impedances involved during faulty event, hybrid lines consisting of overhead line (OHL) and underground cable (UGC) sections, and location of faults on all nodes of the test grid. Fault recognition algorithm is also tested to discriminate the transients due to switching operations of feeders, loads and capacitor banks from the faulty transients. Performance of the fault recognition algorithm is compared with the algorithms based on discrete wavelet transform (DWT), Stockwell transform (ST) and hybrid combination of alienation coefficient and Wigner distribution function (WDF). Effectiveness of the fault recognition algorithm is established using a detailed study on the IEEE-13 nodes test feeder modified to incorporate solar PV plant of capacity 1 MW in MATLAB/Simulink. Algorithm is also validated on practical utility grid of Rajasthan State of India.

Wigner distribution function and alienation coefficient‐based transmission line protection scheme

This study presents an algorithm for detection, classification, and location of transmission line faults. A fault index based on features extracted from current signals using the alienation coefficient and Wigner distribution function has been proposed for the detection and classification of faults. Double line and double line to ground faults have been classified from each other using ground fault index based on negative sequence current. Statistical relations are proposed for the estimation of fault location using peak values of the proposed fault index. The results of different case studies established the effectiveness of the algorithm. The algorithm is found to be effective for providing protection to transmission line against various faults. This is achieved using current signals recorded on one terminal of the line. This makes the protection scheme less complex, fast and more economic due to the elimination of the requirement of communication channel and global positioning system synchronisation. The proposed protection scheme is also validated on a real-time network of transmission utility. The effectiveness of the algorithm is established by comparing performance with reported algorithms.

Alienation Coefficient and Wigner Distribution Function Based Protection Scheme for Hybrid Power System Network with Renewable Energy Penetration

The rapid growth of grid integrated renewable energy (RE) sources resulted in development of the hybrid grids. Variable nature of RE generation resulted in problems related to the power quality (PQ), power system reliability, and adversely affects the protection relay operation. High penetration of RE to the utility grid is achieved using multi-tapped lines for integrating the wind and solar energy and also to supply loads. This created considerable challenges for power system protection. To overcome these challenges, an algorithm is introduced in this paper for providing protection to the hybrid grid with high RE penetration level. All types of fault were identified using a fault index (FI), which is based on both the voltage and current features. This FI is computed using element to element multiplication of current-based Wigner distribution index (WD-index) and voltage-based alienation index (ALN-index). Application of the algorithm is generalized by testing the algorithm for the recognition of faults during different scenarios such as fault at different locations on hybrid grid, different fault incident angles, fault impedances, sampling frequency, hybrid line consisting of overhead (OH) line and underground (UG) cable sections, and presence of noise. The algorithm is successfully tested for discriminating the switching events from the faulty events. Faults were classified using the number of faulty phases recognized using FI. A ground fault index (GFI) computed using the zero sequence current-based WD-index is also introduced for differentiating double phase and double phase to ground faults. The algorithm is validated using IEEE-13 nodes test network modelled as hybrid grid by integrating wind and solar energy plants. Performance of algorithm is effectively established by comparing with the discrete wavelet transform (DWT) and Stockwell transform based protection schemes.

A Protection Scheme for a Power System with Solar Energy Penetration

As renewable energy (RE) penetration has a continuously increasing trend, the protection of RE integrated power systems is a critical issue. Recently, power networks developed for grid integration of solar energy (SE) have been designed with the help of multi-tapped lines to integrate small- and medium-sized SE plants and simultaneously supplying power to the loads. These tapped lines create protection challenges. This paper introduces an algorithm for the recognition of faults in the grid to which a solar photovoltaic (PV) system is integrated. A fault index (FI) was introduced to identify faults. This FI was calculated by multiplying the Wigner distribution (WD) index and Alienation (ALN) index. The WD-index was based on the energy density of the current signal evaluated using Wigner distribution function. The ALN-index was evaluated using sample-based alienation coefficients of the current signal. The performance of the algorithm was validated for various scenarios with different fault types at various locations, different fault incident angles, fault impedances, sampling frequencies, hybrid line consisting of overhead (OH) line and underground (UG) cable sections, different types of transformer windings and the presence of noise. Two phase faults with and without the involvement of ground were differentiated using the ground fault index based on the zero sequence current. This study was performed on the IEEE-13 nodes test network to which a solar PV plant with a capacity of 1 MW was integrated. The performance of the algorithm was also tested on the western part of utility grid in the Rajasthan State in India where solar PV energy integration is high. The performance of the algorithm was effectively established by comparing it with the discrete Wavelet transform (DWT), Wavelet packet transform (WPT) and Stockwell transform-based methods.

Wigner distribution and Boltzmann-Shannon entropy for a diatomic molecule under polarized electric field interaction

We study the classical chaos appearing in a diatomic molecules BeO, CO and CN due to the interaction with a circularly polarized electric field, and its signature in Quantum Mechanics through the Wigner distribution function and the Boltzmann-Shannon entropy. We found a motion out of the center of the quantum phase space defined by Wigner function when the classical system becomes chaotic, and we found a jumping behavior of the average Boltzmann-Shannon entropy with respect the electric field strength when the classical system becomes chaotic, indicating a sudden increasing in the disorder (or sudden lost of information) in the quantum system.

Decoherence of a Damped Anisotropic Harmonic Oscillator under Magnetic Field Effects in a Two-Dimensional Noncommutative Phase-Space

In this paper, decoherence of a damped anisotropic harmonic oscillator in the presence of a magnetic field is studied in the framework of the Lindblad theory of open quantum systems in noncommutative phase-space. General fundamental conditions that should follow our quantum mechanical diffusion coefficients appearing in the master equation are kindly derived. From the master equation, the expressions of density operator, the Wigner distribution function, the expectation and variance with respect to coordinates and momenta are obtained. Based on these quantities, the total energy of the system is evaluated and simulations show its dependency to phase-space structure and its improvement due to noncommutativity effects and the environmental temperature as well. In addition, we also evaluate the decoherence time scale and show that it increases with noncommutativity phase-space effects as compared to the commutative case. It turns out from simulations that this time scale is significantly improved under magnetic field effects.

Twisted elliptical multi-Gaussian Schell-model beams and their propagation properties.

We introduce a new kind of partially coherent source whose cross-spectral density (CSD) function is described as the incoherent superposition of elliptical twisted Gaussian Schell-model sources with different beam widths and transverse coherence widths, named twisted elliptical multi-Gaussian Schell-model (TEMGSM) beams. Analytical expression for the CSD function propagating through a paraxial ABCD optical system is derived with the help of the generalized Collins formula. Our results show that the TEMGSM beam is capable of generating a flat-topped elliptical beam profile in the far field, and the beam spot during propagation exhibits clockwise/anti-clockwise rotation with respect to its propagation axis. In addition, the analytical expressions for the orbital angular momentum (OAM) and the propagation factor are also derived by means of the Wigner distribution function. The influences of the twisted factor and the beam index on the OAM and the propagation factor are studied and discussed in detail.

Statistical properties of Gaussian Schell-model beams propagating through anisotropic hypersonic turbulence

Hypersonic turbulence, causing the environment disturbance, is an important factor influencing optical propagation and optical communication. Based on the extended Huygens-Fresnel principle and second-order moments of the Wigner distribution function, we have derived the analytical expressions of the average intensity, the normalized propagation factor, and the normalized mean-squared beam width of Gaussian Schell-model beams in anisotropic hypersonic turbulence. The evolution properties demonstrate the severe beam expansion and beam quality degradation, and the influences of the source and turbulent parameters are discussed in detail. The comparison between anisotropic hypersonic turbulence and atmosphere turbulence is given, and it is proved that the influence of anisotropic hypersonic turbulence is much stronger. Our results play a guiding role in optical signal transmission in the turbulent environment induced by hypersonic aircraft.

Wigner Distribution of Accelerated TripartiteW-state

In this contribution, we introduce an analytical form of the Wigner distribution function for any tripartite quantum system. These forms are extended to the accelerated tripartite quantum systems of W-state type. The quantum and classical correlations are predicted by negative/ positive values, respectively of the Wigner function. It is shown that for the non-accelerated W-state, the quantum correlations are predicted at large values of the distribution angles. However, for the accelerated W-state, the quantum correlations are predicted at different intervals of distribution angle and different initial acceleration. Due to the acceleration the quantum correlation decays, where the decay rate increases as the number of the accelerated qubits increases. We show that, the local noisy channels; bit and phase bit channels curb the decoherence of the quantum correlation. The ability of the phase-bit channel to defend the quantum correlation is stronger than that shown for the bit-channel, where it decreases if all the three qubits are accelerated. Moreover, we quantify, the amount of the survival quantum correlation by using the volume of the negativity. Our results show that, the volume of negativity doesn’t vanish even at large acceleration, which means thatW-state keeps some of its quantum correlation during the acceleration process.

Analytic Wigner distribution function for a split potential well

The analytic Wigner function for a single well with infinite walls is extended to the configuration where half of the well is raised and the weighting of the wave functions is in accordance with a thermal Fermi-Dirac distribution. This requires a method for determining the energy eigenstates particularly near the height of the higher half-well. The methods provide a means for constructing analytic and accurate trajectories that can be used to model tunneling times. The fidelity of the trajectory model is assessed with respect to absorption of quanta related to changes within the allowable eigenstates.

Modal decomposition for the fiber beams with arbitrary degree of coherence based on the Wigner distribution function.

Modal decomposition (MD) plays an increasingly important role in characterizing fiber beams. Several promising MD techniques have been proposed in literature, all of which are based on a common hypothesis that the modal field is coherently superposed by transverse modes. However, the partially coherent conditions have to be expected in general. In order to take account of this ordinary case, a novel MD scheme employing the Wigner distribution function (WDF) is introduced, which allows the decomposition of fiber beams without any restrictions regarding their degree of coherence. The four-dimensional (4D) WDF (two spatial and two spatial frequency dimensions) of the 2D fiber beam is reconstructed using the coded aperture technique. Based on the measured WDF and orthogonal property of transverse modes, the modal coefficients as well as the mutual modal degree of coherence will be determined unambiguously. The validity and reliability of the proposed approach are illustrated with numerical examples.

Excitation and depression of coherent state of the simple harmonic oscillator

Othman and Yevick [Int. J. Theor. Phys. 57, 2293 (2018)] introduced a new class of states defined as “near” coherent states attached to the simple harmonic oscillator. Such states can be expressed as superposition of a standard coherent state and a derivative state, which are neither completely quantum nor completely classical. Here, we introduce photon-added (-depleted) near coherent states [PA(D)NCS] through “m” times application of creation (annihilation) operators â†(â) to the near coherent state. A general analysis of nonclassical properties of the PA(D)NCS, such as sub-Poissonian statistics and squeezing effect, is given analytically and numerically in the context of the conventional quantum optics. We also derive the Wigner distribution function of the PA(D)NCS over phase space which may bear negative values, which is a good indication of their nonclassical properties. Finally, an experimental procedure for generating the PA(D)NCSs is established.

A Novel Method Based on OMPGW Method for Feature Extraction in Automatic Music Mood Classification

Music mood is useful for music-related applications such as music retrieval or recommendation, which represents the inherent emotional expression of music signals. In this paper, a novel technique is proposed for music signal analysis in the view of emotions, which is based on the orthogonal matching pursuit, Gabor functions, and the Wigner distribution function. The technique, called the OMPGW method, consists of three-level schemes: the low-level, the middle-level and the high-level schemes. For the low-level schemes, the orthogonal matching pursuit combined with Gabor functions is proposed to provide an adaptive time-frequency decomposition of music signals. Compared with other algorithms for signal analysis, the proposed algorithm can achieve a higher spatial and temporal resolution and give a better interpret of the music signal structures. In the middle-level schemes, the Wigner distribution function is applied to obtain the time-frequency energy distribution of the results from the low-level schemes. High-level schemes are used to describe the modeling of audio features, the procedure of music mood classification. A classifier based on support vector machines is utilized to model the extracted features with the proposed technique regarding the emotion models. Several experiments are conducted with four datasets, and better results are achieved with the proposed method. In music mood classification experiments, music clips are classified into different kinds of mood clusters, and mean accuracy of 69.53 percent on our dataset can be achieved using the OMPGW method.

The propagation parameters of a Lommel–Gaussian beam in atmospheric turbulence

Analytical formulas of the propagation parameters (including the angular width and propagation factor) of the Lommel–Gaussian beams in atmospheric turbulence are derived on the basis of extended Huygens–Fresnel integral and second-order moments of Wigner distribution function. Evolution properties of the propagation parameters of Lommel–Gaussian beams propagating in atmospheric turbulence are investigated numerically. The results show that the Lommel–Gaussian beams are less affected with smaller structure constant, as well as bigger orbital angular momentum quantum number, wavelength, and inner scale. Furthermore, the beam propagation is independent of the modulus of the asymmetry parameter. The results may be useful for the practical application of the Lommel–Gaussian beam in free space optical communications.

Torres-Vega distribution function in the extended phase space

Classical physics gives us our everyday perception of the world. However, the break down of this classical intuition could be explained by quantum mechanics. Hence, quantum corrections to the classical statistical mechanics have always remained an important issue, especially, in optics and laser physics for which the correspondence between the classical limit of a large number of photons and the quantum regime is of most importance. We could, therefore, employ distribution functions to find this relation between the quantum and classical solutions. One of these distribution functions which is also of interest in quantum optics and ultrafast laser physics is Wigner distribution function. Interestingly, it could be transferred to the other probability distributions such as the positive Torres-Vega distribution function. In this paper, for the first time, we obtain the Torres-Vega distribution function directly from the extended phase space. We also by means of a set of unitary transformations in the phase space extract the extended Hamiltonian for the Torres-Vega distribution.

Machine learning modeling of Wigner intracule functionals for two electrons in one-dimension.

In principle, many-electron correlation energy can be precisely computed from a reduced Wigner distribution function (W), thanks to a universal functional transformation (F), whose formal existence is akin to that of the exchange-correlation functional in density functional theory. While the exact dependence of F on W is unknown, a few approximate parametric models have been proposed in the past. Here, for a dataset of 923 one-dimensional external potentials with two interacting electrons, we apply machine learning to model F within the kernel Ansatz. We deal with over-fitting of the kernel to a specific region of phase-space by a one-step regularization not depending on any hyperparameters. Reference correlation energies have been computed by performing exact and Hartree-Fock calculations using discrete variable representation. The resulting models require W calculated at the Hartree-Fock level as input while yielding monotonous decay in the predicted correlation energies of new molecules reaching sub-chemical accuracy with training.

Analytic Wigner distribution function for tunneling and trajectory models

The Wigner function is assembled from analytic wave functions for a one-dimensional closed system (well with infinite barriers). A sudden change in the boundary potentials allows for the investigation of time-dependent effects in an analytically solvable model. A trajectory model is developed to account for tunneling when the barrier is finite. The behavior of the density (the zeroth moment of the Wigner function) after an abrupt change in potential shows net accumulation and depletion over time for a weighting of energy levels characteristic of the supply function in field emission. However, for a closed system, the methods have application to investigations of tunneling and transmission associated with field and photoemission at short time scales.