Patterns In Phase Space Research Articles

Extended Weyl-Wigner phase-space framework for nonlinear systems: Typical and modified prey-predator-like dynamics.

The extension of the phase-space Weyl-Wigner quantum mechanics to the subset of Hamiltonians in the form of H(q,p)=K(p)+V(q) [with K(p) replacing single p^{2} contributions] is revisited. Deviations from classical and stationary profiles are identified in terms of Wigner functions and Wigner currents for Gaussian and gamma/Laplacian distribution ensembles. The procedure is successful in accounting for the exact pattern of quantum fluctuations when compared with the classical phase-space pattern. General results are then specialized to some specific Hamiltonians revealing nonlinear dynamics, and suggest a novel algorithm to treat quantum modifications mapped by Wigner currents. Our analysis shows that the framework encompasses, for instance, the quantized prey-predator-like scenarios subjected to statistical constraints.

Revealing inherent quantum interference and entanglement of a Dirac particle

Although originally predicted in relativistic quantum mechanics, Zitterbewegung can also appear in some classical systems, which leads to the important question of whether Zitterbewegung of Dirac particles is underlain by a more fundamental and universal interference behavior without classical analogs. We here reveal such an interference pattern in phase space, which underlies but goes beyond Zitterbewegung, and whose nonclassicality is manifested by the negativity of the phase space quasiprobability distribution, and the associated pseudospin-momentum entanglement. We confirm this discovery by numerical simulation and an on-chip experiment, where a superconducting qubit and a quantized microwave field respectively emulate the internal and external degrees of freedom of a Dirac particle. The measured quasiprobability negativities agree well with the numerical simulation. Besides being of fundamental importance, the demonstrated nonclassical effects are useful in quantum technology.

On the formation of lines in quantum phase space

We theoretically study the formation of lines in phase space using Wigner’s distribution In trapped quantum systems such lines form generically, crisscrossing phase space and they can have astonishing extent, reaching across the entire state. In classical systems this does not happen. We show that the formation of such straight line patterns is due to the formation of ‘randomized comb-states’. We establish their stability to perturbations, and that they are tied to coherences in configuration space. We additionally identify generic higher-order ‘eye’ patterns in phase space which occur less often since they arise from more specific symmetric comb-states; we show that the perturbation of eye patterns through their randomization tends to deform them into lines. Lines in phase space should give rise to large probability peaks in measurements.

The role of rotation on the formation of second generation stars in globular clusters

ABSTRACT By means of 3D hydrodynamic simulations, we explore the effects of rotation in the formation of second-generation (SG) stars in globular clusters (GC). Our simulations follow the SG formation in a first-generation (FG) internally rotating GC; SG stars form out of FG asymptotic giant branch (AGB) ejecta and external pristine gas accreted by the system. We have explored two different initial rotational velocity profiles for the FG cluster and two different inclinations of the rotational axis with respect to the direction of motion of the external infalling gas, whose density has also been varied. For a low (10−24 g cm−3) external gas density, a disc of SG helium-enhanced stars is formed. The SG is characterized by distinct chemo-dynamical phase space patterns: it shows a more rapid rotation than the FG with the helium-enhanced SG subsystem rotating more rapidly than the moderate helium-enhanced one. In models with high external gas density ($10^{-23}\, {\rm g\ cm^{-3}}$), the inner SG disc is disrupted by the early arrival of external gas and only a small fraction of highly enhanced helium stars preserves the rotation acquired at birth. Variations in the inclination angle between the rotation axis and the direction of the infalling gas and the velocity profile can slightly alter the extent of the stellar disc and the rotational amplitude. The results of our simulations illustrate the complex link between dynamical and chemical properties of multiple populations and provide new elements for the interpretation of observational studies and future investigations of the dynamics of multiple-population GCs.

Noncommutative phase-space Lotka-Volterra dynamics: The quantum analog.

The Lotka-Volterra (LV) dynamics is investigated in the framework of the Weyl-Wigner (WW) quantum mechanics extended to one-dimensional Hamiltonian systems, H(x,k) constrained by the ∂^{2}H/∂x∂k=0 condition. Supported by the Heisenberg-Weyl noncommutative algebra, where [x,k]=i, the canonical variables x and k are interpreted in terms of the LV variables, y=e^{-x} and z=e^{-k}, eventually associated with the number of individuals in a closed competitive dynamics: the so-called prey-predator system. The WW framework provides the ground for identifying how classical and quantum evolution coexist at different scales and for quantifying quantum analog effects. Through the results from the associated Wigner currents, (non-)Liouvillian and stationary properties are described for thermodynamic and Gaussian quantum ensembles in order to account for the corrections due to quantum features over the classical phase-space pattern yielded by the Hamiltonian description of the LV dynamics. In particular, for Gaussian statistical ensembles, the Wigner flow framework provides the exact profile for the quantum modifications over the classical LV phase-space trajectories so that Gaussian quantum ensembles can be interpreted as an adequate Hilbert space state configuration for comparing quantum and classical regimes. The generality of the framework developed here extends the boundaries of the understanding of quantumlike effects on competitive microscopical biosystems.

Generalized phase-space description of nonlinear Hamiltonian systems and Harper-like dynamics

Phase-space features of the Wigner flow for generic one-dimensional systems with a Hamiltonian, $H^{W}(q,\,p)$, constrained by the $\partial ^2 H^{W} / \partial q \partial p = 0$ condition are analytically obtained in terms of Wigner functions and Wigner currents. Liouvillian and stationary profiles are identified for thermodynamic (TD) and Gaussian quantum ensembles to account for exact corrections due to quantum modifications over a classical phase-space pattern. General results are then specialized to the Harper Hamiltonian system which, besides working as a feasible test platform for the framework here introduced, admits a statistical description in terms of TD and Gaussian ensembles, where the Wigner flow properties are all obtained through analytical tools. Quantum fluctuations over the classical regime are therefore quantified through probability and information fluxes whenever the classical Hamiltonian background is provided. Besides allowing for a broad range of theoretical applications, our results suggest that such a generalized Wigner approach works as a probe for quantumness and classicality of Harper-like systems, in a framework which can be extended to any quantum system described by Hamiltonians in the form of $H^{W}(q,\,p) = K(p) + V(q)$.

Investigation of rotating detonation fueled by a methane–hydrogen–carbon dioxide mixture under lean fuel conditions

Rotating detonation combustor (RDC) operation is investigated under the main composition of the coal and supercritical water gasification process (a methane–hydrogen–carbon dioxide mixture), and air is selected as an oxidizer. Experiments are performed under four flow rates (79.1, 101.2, 125.8, and 158.2 g/s), and the equivalence ratio (ER) ranges from 0.23 to 0.87. The detonation delay time and propagation characteristics of the rotating detonation wave (RDW) are investigated, and the nonlinear time series analysis method is applied to the pressure signal using the wavelet entropy algorithm and phase space reconstruction. It is demonstrated that a decrease in the flow rate can increase the detonation delay time near the lean fuel boundary; however, the detonation delay time is affected only slightly when the ER is greater than 0.4. The ER significantly affects the wave velocity but is affected only slightly by the flow rate. Contrary to the wave velocity performance, the pressure of the RDW increases with the flow rate. The lean fuel boundary becomes wider when the flow rate increases. Five combustion modes are investigated: failure, fast deflagration, sawtooth wave, single-wave, and two-counter rotating wave modes. The fast deflagration and sawtooth wave modes may occur near the detonation operating boundary, and a higher flow rate tends to increase the occurrence probability of a multiwave mode. The phase space pattern and distribution of wavelet entropy can be used to distinguish the combustion modes intuitively.

Rotating detonation mode recognition using non-intrusive vibration sensing

Rotating detonation engine can be integrated with gas turbines to realize a high-performance power unit. We propose a novel method of non-intrusive vibration sensing for rotating detonation mode recognition. The conventional method is intrusive and exposes the pressure sensor to the high-temperature and high-pressure detonation wave, which limits the sensor life. In the proposed method, the vibration sensor is installed on the outer wall of the combustor, thus ensuring its safety and longevity. A comparison of the vibration sensing results at two installation sites shows that the vibration sensor must be installed on the propagation path of the detonation wave and downstream of the pre-detonator. In rotating detonation of hydrogen-air mixture, vibration sensing can recognize three rotating detonation modes, namely, stable, unstable, and failure. The frequency spectrum of vibration is consistent with that of pressure, and the discontinuities in the unstable case can be recognized in the time-frequency spectrum. The calculated phase difference between vibration and pressure agrees well with the actual installation angle. The phase space patterns and distributions of wavelet entropy are found to differ in the three rotating detonation modes. Vibration sensing is demonstrated to be a potential non-intrusive method for the mode recognition of a rotating detonation wave.

Elemental Abundance Distributions in the (R, Vϕ) Plane with LAMOST DR5 and Gaia DR2

Abstract Since Gaia DR2 was released, many velocity structures in the disk have been revealed, such as large-scale ridge-like patterns in phase space. Both kinematic information and stellar elemental abundances are needed to reveal their evolution history. We have used labels from the APOGEE survey to predict elemental abundances for a huge amount of low-resolution spectra from the Large Sky Area Multi-Object Fibre Spectroscopic Telescope survey. Deep learning with artificial neural networks can automatically draw on physically sensible features in the spectrum for their predictions. Abundances of 12 individual elements, [C/Fe], [N/Fe], [O/Fe], [Mg/Fe], [Al/Fe], [Si/Fe], [S/Fe], [Cl/Fe], [Ca/Fe], [Ti/Fe], [Mn/Fe], and [Ni/Fe], along with basic stellar labels T eff, log g, metallicity ([M/H] and [Fe/H]), and [α/M] for 1,063,386 stars have been estimated. Then, those stars were cross-matched with Gaia DR2 data to obtain kinematic parameters. We present distributions of chemical abundances in the V ϕ versus R coordinate. Our results extend the chemical characterization of the ridges in the (R, V ϕ ) plane to about R = 13 kpc toward the anticenter direction. In addition, radial elemental abundance gradients for disk stars with abs(z) < 0.5 kpc are investigated, and we fitted a line for median abundance values of bins of stars with galactocentric distance between R > 7.84 kpc and R < 15.84 kpc. The radial metallicity gradients for disk stars are, respectively, −0.0475 ± 0.0015 for R ≥ 13.09 kpc and −0.0173 ± 0.0028 for R < 13.09 kpc. Gradients for other elemental abundances are also obtained for disk stars: the [α/M] gradient is 0.0030 ± 0.0002; the [Al/Fe] gradient is 0.0030 ± 0.0002; and the [Mn/Fe] gradient is −0.0078 ± 0.0005.

Simulating cosmological substructure in the solar neighbourhood

ABSTRACT We explore the predictive power of cosmological, hydrodynamical simulations for stellar phase-space substructure and velocity correlations with the auriga simulations and aurigaia mock Gaia catalogues. We show that at the solar circle the auriga simulations commonly host phase-space structures in the stellar component that have constant orbital energies and arise from accreted subhaloes. These structures can persist for a few Gyr, even after coherent streams in position space have been erased. We also explore velocity two-point correlation functions and find this diagnostic is not deterministic for particular clustering patterns in phase space. Finally, we explore these structure diagnostics with the aurigaia catalogues and show that current catalogues have the ability to recover some structures in phase space but careful consideration is required to separate physical structures from numerical structures arising from catalogue generation methods.

Three-dimensional classical imaging of a pattern localized in a phase space

In coventional imaging experiments, objects are localized in a position space and such optically responsive objects can be imaged with a convex lens and can be seen by a human eye. In this paper, we introduce an experiment on a three-dimensional imaging of a pattern which is localized in a three-dimesional phase space. The phase space pattern can not be imaged with a lens in a conventional way and it can not be seen by a human eye. In this experiment, a phase space pattern is produced from object transparancies and imprinted onto the phase space of an atomic gaseous medium, of doppler broadened absorption profile at room temperature, by utilizing velocity selective hole burning in the absorption profile. The pattern is localized in an unique three dimensional phase space which is a subspace of the six dimensional phase space. Imaging of the localized phase space pattern is performed at different momentum locations. In addition, imaging of the imprinted pattern of an object of nonuniform transmittance is presented.

Dynamics of quadruple systems composed of two binaries: stars, white dwarfs, and implications for Ia supernovae

[Abridged]We investigate the long-term secular dynamics and Lidov-Kozai(LK) eccentricity oscillations of quadruple systems composed of two binaries at quadrupole and octupole order in the perturbing Hamiltonian. We show that the fraction of systems reaching high eccentricities is enhanced relative to triple systems, over a broader range of parameter space. We show that this fraction grows with time, unlike triple systems evolved at quadrupole order. This is fundamentally because with their additional degrees of freedom, quadruple systems do not have a maximal set of commuting constants of the motion, even in secular theory at quadrupole order. We discuss these results in the context of star-star and white dwarf-white dwarf(WD) binaries, with emphasis on WD-WD mergers and collisions relevant to the Type Ia supernova problem. For star-star systems, we find that more than 30% of systems reach high eccentricity within a Hubble time, potentially forming triple systems via stellar mergers or close binaries. For WD-WD systems, taking into account general relativistic and tidal precession and dissipation, we show that the merger rate is enhanced in quadruple systems relative to triple systems by a factor of 3.5-10, and that the long-term evolution of quadruple systems leads to a delay-time distribution ~1/t for mergers and collisions. In GW-driven mergers of compact objects, we classify the mergers by their evolutionary patterns in phase space and identify a regime in ~8% of orbital shrinking mergers, where eccentricity oscillations occur on the general relativistic precession timescale, rather than the much longer LK timescale. Finally, we generalize previous treatments of oscillations in the inner binary eccentricity to eccentric mutual orbits. We assess the merger rate in quadruple and triple systems and the implications for their viability as progenitors of stellar mergers and SNe Ia.

Disentangling weak and strong interactions in Brightarrow K^{*}(rightarrow Kpi )pi Dalitz-plot analyses

Dalitz-plot analyses of Brightarrow Kpi pi decays provide direct access to decay amplitudes, and thereby weak and strong phases can be disentangled by resolving the interference patterns in phase space between intermediate resonant states. A phenomenological isospin analysis of Brightarrow K^*(rightarrow Kpi )pi decay amplitudes is presented exploiting available amplitude analyses performed at the BaBar, Belle and LHCb experiments. A first application consists in constraining the CKM parameters thanks to an external hadronic input. A method, proposed some time ago by two different groups and relying on a bound on the electroweak penguin contribution, is shown to lack the desired robustness and accuracy, and we propose a more alluring alternative using a bound on the annihilation contribution. A second application consists in extracting information on hadronic amplitudes assuming the values of the CKM parameters from a global fit to quark flavour data. The current data yields several solutions, which do not fully support the hierarchy of hadronic amplitudes usually expected from theoretical arguments (colour suppression, suppression of electroweak penguins), as illustrated from computations within QCD factorisation. Some prospects concerning the impact of future measurements at LHCb and Belle II are also presented. Results are obtained with the CKMfitter analysis package, featuring the frequentist statistical approach and using the Rfit scheme to handle theoretical uncertainties.

Fixation probabilities of evolutionary coordination games on two coupled populations.

Evolutionary forces resulted from competitions between different populations are common, which change the evolutionary behavior of a single population. In an isolated population of coordination games of two strategies (e.g., s_{1} and s_{2}), the previous studies focused on determining the fixation probability that the system is occupied by only one strategy (s_{1}) and their expectation times, given an initial mixture of two strategies. In this work, we propose a model of two interdependent populations, disclosing the effects of the interaction strength on fixation probabilities. In the well-mixing limit, a detailed linear stability analysis is performed, which allows us to find and to classify the different equilibria, yielding a clear picture of the bifurcation patterns in phase space. We demonstrate that the interactions between populations crucially alter the dynamic behavior. More specifically, if the coupling strength is larger than some threshold value, the critical initial density of one strategy (s_{1}) that corresponds to fixation is significantly delayed. Instead, the two populations evolve to the opposite state of all (s_{2}) strategy, which are in favor of the red queen hypothesis. We delineate the extinction time of strategy (s_{1}) explicitly, which is an exponential form. These results are validated by systematic numerical simulations.

TH-CD-BRA-05: Proton Beam Deflection in MRI Fields: Implications for MRI-Guided Proton Therapy

Purpose: To simulate the delivery of proton beams to the treatment zone inside a split-bore MRI-guided proton therapy system. Methods: Field maps from a split-bore 1 T MRI system are used as input to Monte Carlo simulations which model the trajectory of proton beams towards isocentre. Both inline (along the MRI bore) and perpendicular (through the split-bore gap) orientations are simulated. Monoenergetic diverging beams of energy 90 MeV, 195 MeV and 300 MeV starting from 1.5 m above isocentre were modelled. A phase space file detailing a 2D calibration pattern is used to set the particle starting positions, and their spatial location as they cross isocentre recorded. Results: Inline Orientation: The radial symmetry of the solenoidal style fringe field acts to rotate the protons around the beam’s central axis. For protons starting at 1.5 m from isocentre this rotation is 17° (90 MeV), and 8° (300 MeV). Perpendicular Orientation: Isocentre shifts of 135 mm (90 MeV) and 65 mm (300 MeV) were observed in the direction perpendicular to the main imaging field. Off-axis protons are also slightly deflected towards or away from the central axis in the direction perpendicular to the main deflection direction. This leads to a distortion of the phase space pattern, not just a shift. The distortion increases from zero at the central axis to 10 mm (90 MeV) and 5 mm (300 MeV) for a proton 150 mm off-axis. Conclusion: The complexity and energy-dependent nature of the magnetic deflection and distortion indicates the pencil beam scanning method will be the only choice for delivering a therapeutic proton beam inside a potential MRI-guided proton therapy system. Significant correction strategies will be required to account for the MRI fringe fields. The authors acknowledge funding from NHMRC Program Grant No. 1036078 and ARC Discovery Grant No. DP120100821.

Proton beam deflection in MRI fields: Implications for MRI-guided proton therapy.

This paper investigates, via magnetic modeling and Monte Carlo simulation, the ability to deliver proton beams to the treatment zone inside a split-bore MRI-guided proton therapy system. Field maps from a split-bore 1 T MRI-Linac system are used as input to geant4 Monte Carlo simulations which model the trajectory of proton beams during their paths to the isocenter of the treatment area. Both inline (along the MRI bore) and perpendicular (through the split-bore gap) orientations are simulated. Monoenergetic parallel and diverging beams of energy 90, 195, and 300 MeV starting from 1.5 and 5 m above isocenter are modeled. A phase space file detailing a 2D calibration pattern is used to set the particle starting positions, and their spatial location as they cross isocenter is recorded. No beam scattering, collimation, or modulation of the proton beams is modeled. In the inline orientation, the radial symmetry of the solenoidal style fringe field acts to rotate the protons around the beam's central axis. For protons starting at 1.5 m from isocenter, this rotation is 19° (90 MeV) and 9.8° (300 MeV). A minor focusing toward the beam's central axis is also seen, but only significant, i.e., 2 mm shift at 150 mm off-axis, for 90 MeV protons. For the perpendicular orientation, the main MRI field and near fringe field act as the strongest to deflect the protons in a consistent direction. When starting from 1.5 m above isocenter shifts of 135 mm (90 MeV) and 65 mm (300 MeV) were observed. Further to this, off-axis protons are slightly deflected toward or away from the central axis in the direction perpendicular to the main deflection direction. This leads to a distortion of the phase space pattern, not just a shift. This distortion increases from zero at the central axis to 10 mm (90 MeV) and 5 mm (300 MeV) for a proton 150 mm off-axis. In both orientations, there is a small but subtle difference in the deflection and distortion pattern between protons fired parallel to the beam axis and those fired from a point source. This is indicative of the 3D spatially variant nature of the MRI fringe field. For the first time, accurate magnetic and Monte Carlo modeling have been used to assess the transport of generic proton beams toward a 1 T split-bore MRI. Significant rotation is observed in the inline orientation, while more complex deflection and distortion are seen in the perpendicular orientation. The results of this study suggest that due to the complexity and energy-dependent nature of the magnetic deflection and distortion, the pencil beam scanning method will be the only choice for delivering a therapeutic proton beam inside a potential MRI-guided proton therapy system in either the inline or perpendicular orientation. Further to this, significant correction strategies will be required to account for the MRI fringe fields.

Wigner analysis of three dimensional pupil with finite lateral aperture.

A three dimensional (3D) pupil is an optical element, most commonly implemented on a volume hologram, that processes the incident optical field on a 3D fashion. Here we analyze the diffraction properties of a 3D pupil with finite lateral aperture in the 4-f imaging system configuration, using the Wigner Distribution Function (WDF) formulation. Since 3D imaging pupil is finite in both lateral and longitudinal directions, the WDF of the volume holographic 4-f imager theoretically predicts distinct Bragg diffraction patterns in phase space. These result in asymmetric profiles of diffracted coherent point spread function between degenerate diffraction and Bragg diffraction, elucidating the fundamental performance of volume holographic imaging. Experimental measurements are also presented, confirming the theoretical predictions.

Adiabatic formation of high-Q modes by suppression of chaotic diffusion in deformed microdiscs

Resonant modes with high-Q factors in a two-dimensional deformed microdisc cavity are analyzed by using a dynamical and semiclassical approach. The analysis focuses particularly on the ultra-small cavity regime, where the scale of a resonant free-space wavelength is comparable with that of the microdisc size. Although the deformed microcavity has strongly chaotic internal ray dynamics, modes with high-Q factors in this regime show unexpectedly regular distributions in configuration space and adiabatic features in phase space. By tracing the evolution process of such high-Q modes through the deformation from a circular cavity, it is uncovered that the high-Q modes are formed adiabatically on cantori. Due to the openness of microcavities, such adiabatic formation of high-Q modes around cantori is enabled, in spite of strong chaos in ray dynamics. Since the cantori are in close contact with short periodic orbits, their influence on the modes, such as localization patterns in phase space, can be also clarified. In order to quantitatively analyze the spectral range where high-Q modes appear, the phase space section of the deformed microcavity is partitioned by partial barriers of short periodic orbits, and the semiclassical quantization scheme is applied to the partitioned areas and their action fluxes. The derived spectral ranges for the high-Q modes show a good agreement with a numerically observed spectrum. In the course of semiclassical quantization, it is shown that the chaotic diffusion in the system that we investigate can be resolved by the scale of a quarter effective Planck's constant, and the topological structure of the manifolds in phase space allows for this resolution higher than a Planck constant scale. By analyzing flux Farey trees, the role of short periodic orbits in chaotic diffusion and their connection to cantori are verified.

Persistent patterns and multifractality in fluid mixing

Persistent patterns in periodically driven dynamics have been reported in a wide variety of contexts ranging from table-top and ocean-scale fluid mixing systems to the weak quantum-classical transition in open Hamiltonian systems. We illustrate a common framework for the emergence of these patterns by considering a simple measure of structure maintenance provided by the average radius of the scalar distribution in transform space. Within this framework, scaling laws related to both the formation and persistence of patterns in phase space are presented. Further, preliminary results linking the scaling exponents associated with the persistent patterns to the multifractal nature of the advective phase-space geometry are shown.

Phase-space patterns of quantum transport on ordered and disordered networks

In this paper, we consider the quantum-mechanical phase space patterns on ordered and disordered networks. For ordered networks in which each node is connected to its $2m$ nearest neighbors ($m$ on either side), the phase space quasiprobability of Wigner function shows various patterns. In the long time limit, on even-numbered networks, we find an asymmetric quasiprobability between the node and its opposite node. This asymmetry depends on the network parameters and specific phase space positions. For disordered networks in which each edge is rewired with probability $p>0$, the phase space displays regional localization on the initial node.