Phase Space Representation Research Articles

A study of complex scaling transformation using the Wigner representation of wavefunctions

The complex scaling operator exp(-θ ̂x̂p/ℏ), being a foundation of the complex scaling method for resonances, is studied in the Wigner phase-space representation. It is shown that the complex scaling operator behaves similarly to the squeezing operator, rotating and amplifying Wigner quasi-probability distributions of the respective wavefunctions. It is disclosed that the distorting effect of the complex scaling transformation is correlated with increased numerical errors of computed resonance energies and widths. The behavior of the numerical error is demonstrated for a computation of CO(2+) vibronic resonances.

Point sources and rays in the phase-space representation of random electromagnetic fields

Random stationary electromagnetic fields can be modeled in terms of point sources disposed onto two distinct layers, named the radiant and the virtual layers. It allows describing the propagation of the power and the states of spatial coherence and polarization of the field by means of separate phase-space ray-maps. Spatial coherence of the field, correlation of the polarization angles and achievement of the Fresnel–Arago laws within the structured spatial coherence supports are the necessary conditions that determine the electromagnetic point sources in the virtual layer. The contributions of these sources produce two types of modulations on the electromagnetic field at the observation plane, one of them of scalar type is the interferometric power modulation, and the other one of vector type is de modulation of the polarization state provided by the radiant point sources, according to the Stokes' parameters. Some examples are presented.

Phase-Space Wave Functions of Harmonic Oscillator in Nanomaterials

In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator in nanomaterials within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the harmonic oscillator. We also discuss the character of wave function and the “Fourier-like” projection transformations in phase space.

Phase-Space Wave Functions of Diatomic System in One-Dimensional Nanomaterials

The exact solutions of the stationary Schrödinger equations for the diatomic system with an empirical potential function in one-dimensional nanomaterials are solved within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The wave functions in position and momentum representations can be obtained through the Fourier-like projection transformation from the phase-space wave functions.

Phase-space path integral and Brownian motion

We formulate and examine a phase-space path integral representation of a quantum time evolution in terms of Fock–Bargmann spaces and stochastic line integrals. The Fock–Bargmann space \documentclass[12pt]{minimal}\begin{document}${{\cal H}_{\rm FB}}$\end{document}H FB is a closed subspace of \documentclass[12pt]{minimal}\begin{document}$L^2({\cal M})$\end{document}L2(M), where \documentclass[12pt]{minimal}\begin{document}${\cal M}$\end{document}M is the phase space. Let EFB be the orthogonal projection onto \documentclass[12pt]{minimal}\begin{document}${{\cal H}_{\rm FB}}$\end{document}H FB , and h the classical Hamiltonian, then H = EFBhEFB is understood as a quantization of h. The quantum time evolution e−itH is asymptotically represented by the solution of a variant of diffusion equation with the imaginary potential ih. Thus the quantum time evolution can be represented stochastically, in terms of Brownian motions and stochastic integrals.

RECURRENCE-BASED ESTIMATION OF TIME-DISTORTION FUNCTIONS FOR ERP WAVEFORM RECONSTRUCTION

We introduce an approach to compensate for temporal distortions of repeated measurements in event-related potential research. The algorithm uses a combination of methods from nonlinear time-series analysis and is based on the construction of pairwise registration functions from cross-recurrence plots of the phase-space representations of ERP signals. The globally optimal multiple-alignment path is approximated by hierarchical cluster analysis, i.e. by iteratively combining pairs of trials according to similarity. By the inclusion of context information in form of externally acquired time markers (e.g. reaction time) into a regularization scheme, the extracted warping functions can be guided near paths that are implied by the experimental procedure. All parameters occurring in the algorithm can be optimized based on the properties of the data and there is a broad regime of parameter configurations where the algorithm produces good results. Simulations on artificial data and the analysis of ERPs from a psychophysical study demonstrate the robustness and applicability of the algorithm.

Phase-space representation of non-classical behaviour of scalar wave-fields

The modelling of optical fields by using radiant and virtual point sources for the spatial coherence wavelets in the phase-space representation evidences some effects, conventionally attributed to non-classical correlations of light, although such type of correlations are not explicitly included in the model. Specifically, a light state is produced that has similar morphology to the Wigner Distribution Function of the well-known quantum Schrödinger cat and squeezed states.

Optimal pulses for arbitrary dispersive media

A variational procedure is given for finding the pulses for which the initial temporal rms width and the rate of increase of this width are jointly minimized for propagation in non-absorbing media with arbitrary dispersive properties. We show that, while in linearly dispersive media the optimal pulses are Gaussian, in other situations such as a hollow metallic waveguide or for purely cubic dispersion departures from Gaussian behavior become evident. An interpretation of the results in terms of suitable phase-space representations is also given.

Stochastic simulations of fermionic dynamics with phase-space representations

A Gaussian operator basis provides a means to formulate phase-space simulations of the real- and imaginary-time evolution of quantum systems. Such simulations are guaranteed to be exact while the underlying distribution remains well-bounded, which defines a useful simulation time. We analyse the application of the Gaussian phase-space representation to the dynamics of the dissociation of an ultra-cold molecular gas. We show how the choice of mapping to stochastic differential equations can be used to tailor the stochastic behaviour, and thus the useful simulation time. In the phase-space approach, it is only averages of stochastic trajectories that have a direct physical meaning. Whether particular constants of the motion are satisfied by individual trajectories depends on the choice of mapping, as we show in examples.

Efficient numerical calculation of interference and diffraction of optical fields in any state of spatial coherence in the phase-space representation

The extended use of ray maps in the phase-space representation of optical fields in any state of spatial coherence has promoted the search of efficient approaches for their computation. In this paper, those maps are calculated via a matrix-based approach that i) allows accounting for complex degrees of spatial coherence of any shape and value, ii) models transmittances with real dimensions and not just point objects, iii) accounts for Fresnel phases in the modeled fields, and iv) consumes reasonably short computation time (few seconds for relatively big matrices). Examples illustrate the mentioned features.

Master Equation in Phase Space for a Spin in an Arbitrarily Directed Uniform External Field

The time evolution equation for the probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for the nonaxially symmetric problem of a quantum paramagnet subjected to a uniform magnetic field of arbitrary direction. This is accomplished by first rotating the coordinate system into one in which the polar axis is collinear with the field vector, then writing the reduced density matrix equation in the new coordinate system as an explicit inverse Wigner-Stratonovich transformation so that the phase space master equation may be derived just as in the axially symmetric case [Yu.P. Kalmykov et al., J. Stat. Phys. 131:969, 2008]. The properties of this equation, resembling the corresponding Fokker-Planck equation, are investigated. In particular, in the large spin limit, S→∞, the master equation becomes the classical Fokker-Planck equation describing the magnetization dynamics of a classical paramagnet in an arbitrarily directed uniform external field.

From phase space representation to amplitude equations in a pattern-forming experiment

We describe and demonstrate a method to reconstruct an amplitude equation from the nonlinear relaxation dynamics in the neighbourhood of the Rosensweig instability. A flat layer of a ferrofluid is cooled such that the liquid has a relatively high viscosity. Consequently, the dynamics of the formation of the Rosensweig pattern becomes very slow. By a sudden switching of the magnetic induction, the system is pushed to an arbitrary point in the phase space spanned by the pattern amplitude and magnetic induction. Afterwards, it is allowed to relax to its equilibrium point. From the dynamics of this relaxation, we reconstruct the underlying fully nonlinear equation of motion of the pattern amplitude. The measured nonlinear dynamics helps us to select the best weakly nonlinear expansion that describes this hysteretic transition.

Quantitative assessment of somatosensory-evoked potentials after cardiac arrest in rats: Prognostication of functional outcomes*

High incidence of poor neurologic sequelae after resuscitation from cardiac arrest underscores the need for objective electrophysiological markers for assessment and prognosis. This study aims to develop a novel marker based on somatosensory evoked potentials (SSEPs). Normal SSEPs involve thalamocortical circuits suggested to play a role in arousal. Due to the vulnerability of these circuits to hypoxic-ischemic insults, we hypothesize that quantitative SSEP markers may indicate future neurologic status. Laboratory investigation. University Medical School and Animal Research Facility. : Sixteen adult male Wistar rats. None. SSEPs were recorded during baseline, during the first 4 hrs, and at 24, 48, and 72 hrs postasphyxia from animals subjected to asphyxia-induced cardiac arrest for 7 or 9 mins (n = 8/group). Functional evaluation was performed using the Neurologic Deficit Score (NDS). For quantitative analysis, the phase space representation of the SSEPs-a plot of the signal vs. its slope-was used to compute the phase space area bounded by the waveforms recorded after injury and recovery. Phase space areas during the first 85-190 mins postasphyxia were significantly different between rats with good (72 hr NDS >or=50) and poor (72 hr NDS <50) outcomes (p = .02). Phase space area not only had a high outcome prediction accuracy (80-93%, p < .05) during 85-190 mins postasphyxia but also offered 78% sensitivity to good outcomes without compromising specificity (83-100%). A very early peak of SSEPs that precedes the primary somatosensory response was found to have a modest correlation with the 72 hr NDS subscores for thalamic and brainstem function (p = .066) and not with sensory-motor function (p = .30). Phase space area, a quantitative measure of the entire SSEP morphology, was shown to robustly track neurologic recovery after cardiac arrest. SSEPs are among the most reliable predictors of poor outcome after cardiac arrest; however, phase space area values early after resuscitation can enhance the ability to prognosticate not only poor but also good long-term neurologic outcomes.

Radiant, virtual, and dual sources of optical fields in any state of spatial coherence

A novel description of interference and diffraction with fields in arbitrary states of spatial coherence is introduced in the framework of the phase-space representation. The field is modeled as produced by radiant and virtual point sources. The first ones emit the radiant power of the field, independently of its spatial coherence state, and the second ones emit the modulating energy in strong dependence on such state. This energy can take on positive and negative values that produce the interference and diffraction patterns after adding them to the radiant energy. Radiant and virtual point sources at a given plane can be arranged over two distinct layers, which can be brought together to provide a unified structure of point sources for the field at such plane. So, the coincidence of specific radiant and virtual sources at the same point induces a further type: the dual point source. Descriptions of diffraction arrangements, Young's experiment with diffraction effects, and some implications of this model are discussed.

Number-phase Wigner representation for efficient stochastic simulations

Phase-space representations based on coherent states (P, Q, Wigner) have been successful in the creation of stochastic differential equations (SDEs) for the efficient stochastic simulation of high dimensional quantum systems. However many problems using these techniques remain intractable over long integrations times. We present a number-phase Wigner representation that can be unraveled into SDEs. We demonstrate convergence to the correct solution for an anharmonic oscillator with small dampening for significantly longer than other phase space representations. This process requires an effective sampling of a non-classical probability distribution. We describe and demonstrate a method of achieving this sampling using stochastic weights.

Recapitulating Quantum Phase Space Representation by Virtue of Normally Ordered Gaussian Form of Wigner Operator

Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector |q⟩λ,ν of linear combination of the coordinate and momentum operator, (λQ + νP), where λ, ν are real numbers, and |q⟩λν is complete, then the projector |q⟩λν⟨q| must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.

Phase-space representation of partially coherent imaging systems using the Cohen class distribution

We develop a phase-space representation (termed phase-space imager) suitable for analysis and design of imaging systems that use large illumination apertures (partially coherent illumination). The representation, developed from the transmission cross-coefficient model, falls in the general Cohen class of distributions and elegantly captures the bilinear nature of image formation in partially coherent systems. It uses only the requisite number of dimensions and leads to an efficient algorithm for calculating images. Computed partially coherent images of a double slit and a two-dimensional sinusoidal spoke are presented.

Comment on “Full quantum reconstruction of vortex states”

It is pointed out that the bona fide Wigner distribution recently introduced [Phys. Rev. A 78, 060101(R) (2008)] is in fact the rotational Wigner function earlier derived and thoroughly studied [Phys. Rev. A 49, 3255 (1994); 71, 069901(E) (2005)], so the latter remains indeed the natural phase-space representation for the rotation-angle and angular-momentum pair.

Application of MJO Simulation Diagnostics to Climate Models

Abstract The ability of eight climate models to simulate the Madden–Julian oscillation (MJO) is examined using diagnostics developed by the U.S. Climate Variability and Predictability (CLIVAR) MJO Working Group. Although the MJO signal has been extracted throughout the annual cycle, this study focuses on the boreal winter (November–April) behavior. Initially, maps of the mean state and variance and equatorial space–time spectra of 850-hPa zonal wind and precipitation are compared with observations. Models best represent the intraseasonal space–time spectral peak in the zonal wind compared to that of precipitation. Using the phase–space representation of the multivariate principal components (PCs), the life cycle properties of the simulated MJOs are extracted, including the ability to represent how the MJO evolves from a given subphase and the associated decay time scales. On average, the MJO decay (e-folding) time scale for all models is shorter (∼20–29 days) than observations (∼31 days). All models are able to produce a leading pair of multivariate principal components that represents eastward propagation of intraseasonal wind and precipitation anomalies, although the fraction of the variance is smaller than observed for all models. In some cases, the dominant time scale of these PCs is outside of the 30–80-day band. Several key variables associated with the model’s MJO are investigated, including the surface latent heat flux, boundary layer (925 hPa) moisture convergence, and the vertical structure of moisture. Low-level moisture convergence ahead (east) of convection is associated with eastward propagation in most of the models. A few models are also able to simulate the gradual moistening of the lower troposphere that precedes observed MJO convection, as well as the observed geographical difference in the vertical structure of moisture associated with the MJO. The dependence of rainfall on lower tropospheric relative humidity and the fraction of rainfall that is stratiform are also discussed, including implications these diagnostics have for MJO simulation. Based on having the most realistic intraseasonal multivariate empirical orthogonal functions, principal component power spectra, equatorial eastward propagating outgoing longwave radiation (OLR), latent heat flux, low-level moisture convergence signals, and vertical structure of moisture over the Eastern Hemisphere, the superparameterized Community Atmosphere Model (SPCAM) and the ECHAM4/Ocean Isopycnal Model (OPYC) show the best skill at representing the MJO.

Use of phase space representation to investigate points of geodynamical interest

The kinematical behavior of points on an area of geodynamical interest is analyzed in a low — dimensional Riemann phase space in contrast to classic approaches that operate in time or frequency domains or in physical space. The phase space is reconstructed from series derived from regularly repeated GPS measurements that were transformed into a unified terrestrial frame. For the reconstruction the time-delay method was used, a concept in nonlinear time series analysis as developed by Packard (Packard et al. 1980) and proved by Takens (Takens 1981). The underlying dynamical model is a Hamiltonian motion equation so the reconstructed space is extended according to holonomic conjugated Hamiltonian coordinates. The GPS measurements are selected from a small area of geodynamical interest after its investigation based on analysis of raw geoidal signals (Doufexopoulou et al. 2006). Points from a neutral zone are used also for comparison purposes.