Phase Space Perspective Research Articles

Noise filtering options for conically scanning Doppler lidar measurements with low pulse accumulation

Abstract. Doppler lidar (DL) applications with a focus on turbulence measurements sometimes require measurement settings with a relatively small number of accumulated pulses per ray in order to achieve high sampling rates. Low pulse accumulation comes at the cost of the quality of DL radial velocity estimates and increases the probability of outliers, also referred to as “bad” estimates or noise. Careful filtering is therefore the first important step in a data processing chain that begins with radial velocity measurements as DL output variables and ends with turbulence variables as the target variable after applying an appropriate retrieval method. It is shown that commonly applied filtering techniques have weaknesses in distinguishing between “good” and “bad” estimates with the sensitivity needed for a turbulence retrieval. For that reason, new ways of noise filtering have been explored, taking into account that the DL background noise can differ from generally assumed white noise. It is shown that the introduction of a new coordinate frame for a graphical representation of DL radial velocities from conical scans offers a different perspective on the data when compared to the well-known velocity–azimuth display (VAD) and thus opens up new possibilities for data analysis and filtering. This new way of displaying DL radial velocities builds on the use of a phase-space perspective. Following the mathematical formalism used to explain a harmonic oscillator, the VAD’s sinusoidal representation of the DL radial velocities is transformed into a circular arrangement. Using this kind of representation of DL measurements, bad estimates can be identified in two different ways: either in a direct way by singular point detection in subsets of radial velocity data grouped in circular rings or indirectly by localizing circular rings with mostly good radial velocity estimates by means of the autocorrelation function. The improved performance of the new filter techniques compared to conventional approaches is demonstrated through both a direct comparison of unfiltered with filtered datasets and a comparison of retrieved turbulence variables with independent measurements.

Phase space perspective on a model for isomerization in an optical cavity.

Explanation for the modification of rates and mechanism of reactions carried out in optical cavities still eludes us. Several studies indicate that the cavity-mediated changes in the nature of vibrational energy flow within a molecule may play a significant role. Here, we study a model polaritonic system, proposed and analyzed earlier by Fischer et al., J. Chem. Phys. 156, 154305 (2022), comprising a one-dimensional isomerization mode coupled to a single photon mode in a lossless cavity. We show that the isomerization probability in the presence of virtual photons, for specific cavity-system coupling strengths and cavity frequencies, can exhibit suppression or enhancement for different choices of the initial reactant vibropolariton wavepacket. We observe a qualitative agreement between the classical and quantum average isomerization probabilities in the virtual photon case. A significant part of the effects due to coupling to the cavity can be rationalized in terms of a "chaos-order-chaos" transition of the classical phase space and the phase space localization nature of the polariton states that dominantly participate in the quantum isomerization dynamics. On the other hand, for initial states with zero photons (i.e., a "dark cavity"), the isomerization probability is suppressed when the cavity frequency is tuned near to the fundamental frequency of the reactive mode. The classical-quantum correspondence in the zero photon case is unsatisfactory. In this simple model, we find that the suppression or enhancement of isomerization arises due to the interplay between cavity-system energy flow dynamics and quantum tunneling.

Two-dimensional transverse charge distributions of the Δ baryon: Interpolation between the nonrelativistic and ultrarelativistic limits

We investigate how the charge distributions of both the unpolarized and transversely polarized $\Delta$ baryon change as the longitudinal momentum~($P_{z}$) of the $\Delta$ baryon increases from $P_{z}=0$ to $P_{z}=\infty$ in a Wigner phase-space perspective. When the $\Delta$ baryon is longitudinally polarized, its two-dimensional charge distribution is kept to be spherically symmetric with $P_{z}$ varied, whereas when the $\Delta$ baryon is transversely polarized along the $x$-axis, the quadrupole contribution emerges at the rest frame ($P_{z}=0$). When $P_{z}$ grows, the electric dipole and octupole moments are induced. The induced dipole moment dominates over other higher multipole contributions and governs the deformation of the charge distribution of the $\Delta$ baryon.

Mesophases with chiral structure under constant electric field constrained by parallel plates with soft anchoring boundary conditions

In this work, we present a new approach based on Euler-Lagrange equilibrium equations analysed from the phase space (PS) perspective to investigate states of elastic deformation of Cholesteric Liquid Crystals (N*) confined, which present an intrinsic chiral structure; between parallel plates by imposing soft anchoring boundary conditions (SAC) under a constant electric field-induced perpendicular to the plates. By minimizing the Frank-Oseen (FO) energy density, we obtain a set of equilibrium state integral equations and corresponding boundary conditions. We establish planar conditions on the plates that force the optical axis to align with a specific direction at the plates. We verify the convergence between SAC to strong anchoring conditions (SSC) whether the ratio between surface and bulk energy tends to infinity, under this condition we examine the different configurations of N* by variating the parameters of control, including a variation of potential difference. We also give a detailed derivation of the threshold electric field value, we show that above this value, there is a transition from N to N*, and below this value transition, N* to focal conic texture is observed. We discuss our results and address our conclusions.

Phase space geometry of isolated to condensed chemical reactions.

The complexity of gas and condensed phase chemical reactions has generally been uncovered either approximately through transition state theories or exactly through (analytic or computational) integration of trajectories. These approaches can be improved by recognizing that the dynamics and associated geometric structures exist in phase space, ensuring that the propagator is symplectic as in velocity-Verlet integrators and by extending the space of dividing surfaces to optimize the rate variationally, respectively. The dividing surface can be analytically or variationally optimized in phase space, not just over configuration space, to obtain more accurate rates. Thus, a phase space perspective is of primary importance in creating a deeper understanding of the geometric structure of chemical reactions. A key contribution from dynamical systems theory is the generalization of the transition state (TS) in terms of the normally hyperbolic invariant manifold (NHIM) whose geometric phase-space structure persists under perturbation. The NHIM can be regarded as an anchor of a dividing surface in phase space and it gives rise to an exact non-recrossing TS theory rate in reactions that are dominated by a single bottleneck. Here, we review recent advances of phase space geometrical structures of particular relevance to chemical reactions in the condensed phase. We also provide conjectures on the promise of these techniques toward the designand control of chemical reactions.

The two-spin model of a tracking chamber: a phase-space perspective

We study the dynamics of classical localization in a simple, one-dimensional model of a tracking chamber. The emitted particle is represented by a superposition of Gaussian wave packets moving in opposite directions, and the detectors are two spins in fixed, opposite positions with respect to the central emitter. At variance with other similar studies, we give here a phase-space representation of the dynamics in terms of the Wigner matrix of the system. This allows a better visualization of the phenomenon and helps in its interpretation. In particular, we discuss the relationship of the localization process with the properties of entanglement possessed by the system.

Phase space structure and escape time dynamics in a Van der Waals model for exothermic reactions.

We study the phase space objects that control the transport in a classical Hamiltonian model for a chemical reaction. This model has been proposed to study the yield of products in an ultracold exothermic reaction. In this model, two features determine the evolution of the system: a Van der Waals force and a short-range force associated with the many-body interactions. In the previous work, small random periodic changes in the direction of the momentum were used to simulate the short-range many-body interactions. In the present work, random Gaussian bumps have been added to the Van der Waals potential energy to simulate the short-range effects between the particles in the system. We compare both variants of the model and explain their differences and similarities from a phase space perspective. To visualize the structures that direct the dynamics in the phase space, we construct a natural Lagrangian descriptor for Hamiltonian systems based on the Maupertuis action S_{0}=∫_{q_{i}}^{q_{f}}p·dq.

Charge Distributions of Moving Nucleons.

We introduce relativistic charge distributions for targets with arbitrary average momentum, providing a natural interpolation between the usual Breit frame and infinite-momentum frame distributions. Among the remarkable results, we find that Breit frame distributions can be interpreted from a phase-space perspective as internal charge quasidensities in the rest frame of a localized target, without any relativistic correction. Moreover, we show that the unexpected negative center observed in the unpolarized neutron infinite-momentum frame charge distribution results from a magnetization contribution generated by the Wigner rotation.

Teaching Entropy from Phase Space Perspective: Connecting the Statistical and Thermodynamic Views Using a Simple One-Dimensional Model

Connecting the thermodynamic definition of entropy, dS = dQ/T (Clausius’s equation), with the statistical definition, S = kB ln Ω (Boltzmann’s equation), has been a persistent challenge in chemical education at the undergraduate level. Not meeting this challenge results in students taking away the meaning of entropy in a vague and subjective way as a measure of “disorder” or increase in number of configurations without any meaningful way of connecting it to heat. To address this challenge, we present a simple model that connects these two definitions. This approach relies centrally on emphasizing that the number of configurations, Ω, includes configurations in both real space and momentum space, collectively known as the phase space. Without including momentum configurations (i.e., how fast the particles move), connecting heat to entropy change is not possible. We construct the phase space for an ensemble of simple one-dimensional systems at equilibrium and show that delivery of heat dQ to the system results in an increase in the number of momentum configurations and consequently an expansion of the phase space area by dΩ. Relating dQ to dΩ is the linchpin between the two views and, when integrated, leads to Boltzmann’s equation. We further show that understanding entropy in terms of volume of phase space removes common ambiguities in teaching the subject. Among other examples, we show that understanding adiabatic compression, if treated using the usual approach, results in contradictions that are resolved if a phase space view is adopted. We propose this approach at the undergraduate physical chemistry and physics level.

Vibration Signature Extraction of High-Voltage Circuit Breaker by Frequency and Chaotic Analysis

This paper aims at presenting a novel approach for the analysis of vibration signals detected from circuit breaker (CB) opening or closing operations and to mechanical fault identification. An adaptive signal decomposition approach and chaotic nonlinear dynamic technique are used for extracting fault sensitive features of CB. The fault sensitive frequency band is derived using variational mode decomposition (VMD) combining with Hilbert marginal spectrum (HMS). The obtained sensitive frequency band(s) are employed for the reconstruction of the dynamical attractor (chaotic dynamics viewed from a phase space perspective, namely, the attractor-based perspective) following the Taken's embedding theorem. The invariant measures and ergodic quantities such as the largest Lyapunov exponent, correlation dimension, and Kolmogorov entropy which can be estimated on the reconstructed attractor are presented as the fault sensitive features. The CB fault is estimated in a chaotic space using these sensitive features. The effectiveness of the developed method is evaluated by using data sets recorded from several vacuum CBs with the same type.

Hamiltonian flow over saddles for exploring molecular phase space structures.

Despite using potential energy surfaces, multivariable functions on molecular configuration space, to comprehend chemical dynamics for decades, the real happenings in molecules occur in phase space, in which the states of a classical dynamical system are completely determined by the coordinates and their conjugate momenta. Theoretical and numerical results are presented, employing alanine dipeptide as a model system, to support the view that geometrical structures in phase space dictate the dynamics of molecules, the fingerprints of which are traced by following the Hamiltonian flow above saddles. By properly selecting initial conditions in alanine dipeptide, we have found internally free rotor trajectories the existence of which can only be justified in a phase space perspective.This article is part of the theme issue 'Modern theoretical chemistry'.

Roaming: A Phase Space Perspective.

In this review we discuss the recently described roaming mechanism for chemical reactions from the point of view of nonlinear dynamical systems in phase space. The recognition of the roaming phenomenon shows the need for further developments in our fundamental understanding of basic reaction dynamics, as is made clear by considering some questions that cut across most studies of roaming: Is the dynamics statistical? Can transition state theory be applied to estimate roaming reaction rates? What role do saddle points on the potential energy surface play in explaining the behavior of roaming trajectories? How do we construct a dividing surface that is appropriate for describing the transformation from reactants to products for roaming trajectories? How should we define the roaming region? We show that the phase space perspective on reaction dynamics provides the setting in which these questions can be properly framed and answered. We illustrate these ideas by considering photodissociation of formaldehyde. The phase-space formulation allows an unambiguous description of all possible reactive events, which also allows us to uncover the phase space mechanism that explains which trajectories roam, as opposed to evolving toward a different reactive event.

A matter of style: Adolf Lohmann’s approach to optical phase space

Phase-space optics was one of Adolf Lohmann’s favorite research topics and appears frequently in his work. With a unique style he applied the phase-space perspective to a variety of optical problems, with a focus on more comprehensible problem solutions and a particular interest to analyze optical signals and systems in terms of their space–bandwidth product requirements. In his work, optical phase space is promoted as a hybrid of paraxial wave optics and geometrical optics. In contrast to more rigorous notions of phase space, Lohmann preferred a heuristic approach that avoids almost entirely the mathematical formalism associated with the Wigner distribution function. This approach is introduced for the first time as a distinct way to apply phase-space optics. The application of Lohmann’s phase-space optics to paraxial optical cloaking, the Lohmann–Alvarez lens, and holography is not only used to exemplify this unique form of phase-space optics, but also to highlight Lohmann’s unique and creative way of problem solving.

Phase Space Structures Explain Hydrogen Atom Roaming in Formaldehyde Decomposition.

We re-examine the prototypical roaming reaction--hydrogen atom roaming in formaldehyde decomposition--from a phase space perspective. Specifically, we address the question "why do trajectories roam, rather than dissociate through the radical channel?" We describe and compute the phase space structures that define and control all possible reactive events for this reaction, as well as provide a dynamically exact description of the roaming region in phase space. Using these phase space constructs, we show that in the roaming region, there is an unstable periodic orbit whose stable and unstable manifolds define a conduit that both encompasses all roaming trajectories exiting the formaldehyde well and shepherds them toward the H2···CO well.

Phase-space foundations of electron holography

We present a unified formalism for describing various forms of electron holography in quantum mechanical phase space including their extensions to quantum-state reconstructions. The phase-space perspective allows for taking into account partial coherence as well as the quantum mechanical detection process typically hampering the unique reconstruction of a wave function. We elaborate on the limitations imposed by the electron optical elements of the transmission electron microscope as well as the scattering at the target. The results provide the basis for vastly extending the scope of electron holographic techniques towards analyzing partially coherent signals such as inelastically scattered electrons or electron pulses used in ultrafast transmission electron microscopy.

A Phase Space Perspective on Electron Holography - Building Bridges Between Inline-, Off-axis Holography, Differential Phase Contrast and Diffractive Imaging

Journal Article A Phase Space Perspective on Electron Holography - Building Bridges Between Inline-, Off-axis Holography, Differential Phase Contrast and Diffractive Imaging Get access Axel Lubk, Axel Lubk Triebenberg Laboratory, Institute of Structure Physics, TU Dresden, 01062 Dresden, Germany Search for other works by this author on: Oxford Academic Google Scholar Falk Roder, Falk Roder Triebenberg Laboratory, Institute of Structure Physics, TU Dresden, 01062 Dresden, Germany Search for other works by this author on: Oxford Academic Google Scholar Hannes Lichte Hannes Lichte Triebenberg Laboratory, Institute of Structure Physics, TU Dresden, 01062 Dresden, Germany Search for other works by this author on: Oxford Academic Google Scholar Microscopy and Microanalysis, Volume 21, Issue S3, 1 August 2015, Pages 2307–2308, https://doi.org/10.1017/S1431927615012313 Published: 23 September 2015

Classical-quantum correspondence in a model for conformational dynamics: Connecting phase space reactive islands with rare events sampling

The role of intramolecular vibrational energy redistribution in conformational reactions has been a topic of continued interest since low barrier conformational reactions are examples of systems that are intrinsically non-statistical in nature. Nearly three decades ago, De Leon and Berne investigated the dynamics of a model isomerization system in great detail. Interestingly, this early study has bearings on a topic of current interest – the sampling of rare events. In this work we bring out this connection in terms of phase space structures known as reactive islands and thereby provide the anatomy of a rare event. En route, we obtain insights into the transition path sampling method for harvesting rare trajectories from a phase space perspective. We show that quantum dynamics is sensitive to the reactive islands, thereby answering a question raised by De Leon and Berne. Our classical-quantum correspondence studies raise the possibility of defining a quantum rare event.

Transport of intensity phase retrieval and computational imaging for partially coherent fields: The phase space perspective

The well-known transport of intensity equation (TIE) allows the phase of a coherent field to be retrieved non-interferometrically given positive defined intensity measurements and appropriate boundary conditions. However, in many cases like the optical microscopy, the imaging systems often involve extended and polychromatic sources for which the effect of the partial coherence is not negligible. In this work, we present a phase-space formulation for the TIE for analyzing phase retrieval under partially coherent illumination. The conventional TIE is reformulated in the joint space-spatial frequency domain using Wigner distribution functions. The phase-space formulation clarifies the physical meaning of the phase of partially coherent fields, and enables explicit account of partial coherence effects on phase retrieval. The correspondence between the Wigner distribution function and the light field in geometric optics limit further enables TIE to become a simple yet effective approach to realize high-resolution light field imaging for slowly varying phase specimens, in a purely computational way.

Multiple transition states and roaming in ion–molecule reactions: A phase space perspective

We provide a dynamical interpretation of the recently identified ‘roaming’ mechanism for molecular dissociation reactions in terms of geometrical structures in phase space. These are NHIMs (Normally Hyperbolic Invariant Manifolds) and their stable/unstable manifolds that define transition states for ion–molecule association or dissociation reactions. The associated dividing surfaces rigorously define a roaming region of phase space, in which both reactive and non reactive trajectories can be trapped for arbitrarily long times.

Single-particle dissipation in a time-dependent Hartree-Fock approach studied from a phase-space perspective

We study dissipation and relaxation processes within the time-dependent Hartree-Fock approach using the Wigner distribution function. On the technical side we present a geometrically unrestricted framework which allows us to calculate the full six-dimensional Wigner distribution function. With the removal of geometrical constraints, we are now able to extend our previous phase-space analysis of heavy-ion collisions in the reaction plane to unrestricted mean-field simulations of nuclear matter on a three-dimensional Cartesian lattice. From the physical point of view we provide a quantitative analysis on the stopping power in TDHF. This is linked to the effect of transparency. For the medium-heavy ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ system we examine the impact of different parametrizations of the Skyrme force, energy-dependence, and the significance of extra time-odd terms in the Skyrme functional.