Phase-space Function Research Articles

Split-operator technique for propagating phase space functions: Exploring chaotic, dissipative and relativistic dynamics

We conduct a comprehensive analysis of the split-operator method for propagating phase space distribution functions in different scenarios of classical mechanics. A numerical method based on Fast Fourier Transform allows to propagate almost any sampled or exact localized initial state, as well as the direct calculation of current densities in phase space. In order to demonstrate the potential of the proposed numerical scheme some simulations involving chaotic, dissipative and relativistic dynamics are performed. In the conducted simulations, dynamical functions like autocorrelations as well as the detailed structures in phase space are discussed. We find that the split-operator technique demonstrates the effectiveness for studying time evolution of interacting one-dimensional classical systems.

Diophantine type of interval exchange maps

AbstractRoth type irrational rotation numbers have several equivalent arithmetical characterizations as well as several equivalent characterizations in terms of the dynamics of the corresponding circle rotations. In this paper we investigate how to generalize Roth-like Diophantine conditions to interval exchange maps. If one considers the dynamics in parameter space one can introduce two non-equivalent Roth type conditions, the first (condition (Z)) by means of the Zorich cocycle [Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier 46(2) (1996), 325–370], the second (condition (A)) by means of a further acceleration of the continued fraction algorithm by Marmi–Moussa–Yoccoz introduced in [The cohomological equation for Roth type interval exchange maps, J. Amer. Math. Soc. 18 (2005), 823–872]. A third very natural condition (condition (D)) arises by considering the distance between the discontinuity points of the iterates of the map. If one considers the dynamics of an interval exchange map in phase space then one can introduce the notion of Diophantine type by considering the asymptotic scaling of return times pointwise or with respect to uniform convergence (respectively conditions (R) and (U)). In the case of circle rotations all the above conditions are equivalent. For interval exchange maps of three intervals we show that (D) and (A) are equivalent and imply (Z), (U) and (R), which are equivalent among them. For maps of four intervals or more we prove several results; the only relation that we cannot decide is whether (Z) implies (R) or not.

Two-photon ladder climbing and transition to autoresonance in a chirped oscillator

The two-photon ladder climbing (successive two-photon Landau-Zener-type transitions) in a chirped quantum nonlinear oscillator and its classical limit (subharmonic autoresonance) are discussed. An isomorphism between the chirped one- and two-photon resonances in the system is used in calculating the threshold for the phase-locking transition in both the classical and quantum limits. The theory is tested by solving the Schr\"odinger equation in the energy basis and illustrated via the Wigner function in phase space.

Dirac prescription from BRST symmetry in FRW space-time

A procedure to define the Becchi-Rouet-Stora-Tyutin (BRST) charge from the Noether one in extended phase space is given. It is outlined how this prescription can be applied to a Friedmann-Robertson-Walker space-time with a differential gauge condition and it allows us to reproduce the results of Ref. [21]. Then, we discuss the cohomological classes associated with functions in extended phase space having ghost number one and we recover the frozen formalism for classical observables. Finally, we consider the quantization of Becchi-Rouet-Stora-Tyutin--closed states and we define a scalar product which implements the super-Hamiltonian constraint.

Entanglement Swapping Transform Rule of Entangled Wigner Operators

For mixed input fields quantum information processing, it is very convenient to investigate a specified protocol by employ quasi-probability functions and characteristic functions in phase space. In this work, considering a nonlocal swapping operation labelled by \(\hat{E}_{s}\), we derive the entanglement swapping transform rule for entangled Wigner operators. The same rule can be obtained by implementing this nonlocal swapping operation via two entangled pairs channels. And then we apply this rule to examine how does the Wigner function for output states change to demonstrate the entanglement swapping. As a result, this transform rule can be utilized to investigate swapping operation for any two-body entangled system.

Thermodynamic properties of liquid water from a polarizable intermolecular potential

Molecular dynamics simulation results are reported for the pressure, isothermal pressure coefficient, thermal expansion coefficient, isothermal and adiabatic compressibilities, isobaric and isochoric heat capacities, Joule-Thomson coefficient and speed of sound of liquid water using a polarizable potential [Li et al., J. Chem. Phys. 127, 154509 (2007)]. These properties were obtained for a wide range of temperatures and pressures at a common liquid density using the treatment of Lustig [J. Chem. Phys. 100, 3048 (1994)] and Meier and Kabelac [J. Chem. Phys. 124, 064104 (2006)], whereby thermodynamic state variables are expressible in terms of phase-space functions determined directly from molecular dynamics simulations. Comparison with experimental data indicates that the polarizable potential can be used to predict most thermodynamic properties with a very good degree of accuracy.

A New Way to Generate High-Dimensional Hyperchaos

In this paper we described t he process of generating a high-dimensional hyperchaos in signal communication and computer chaotic encryption. We found out f our new high-dimensional complex hyperchaotic systems and three regular ways to generat e high-dimensional hyperchaotic systems. Then we revealed t he relationship between the low -dimensional chaos and high -dimensional hyperchaos and studied t he features of phase space maps and Lyapunov exponents maps of high-dimensional complex hyperchaotic system. The results of theoretical analysis and experiment show that the new high-dimensional complex hyperchaotic system hav e strong chaotic features.

量子光学中的一种新变换

We find a new kind of integration transformation in quantum optics,which can reflect the Baker-Hausdorff formula as well as establish the transformation between phase space basis and the function space of entangled state.This transformation is invertible and satisfies Parseval-like theorem.Combining Wigner transformation and this new transformation in phase space,We can acquire a new relation between phase space function h(p,q) and corresponding Weyl-Wigner operator,which can benefit the development of quantum phase space theory.

Antenna showers with hadronic initial states

We present an antenna shower formalism including contributions from initial-state partons and corresponding backwards evolution. We give a set of phase-space maps and antenna functions for massless partons which define a complete shower formalism suitable for computing observables with hadronic initial states. We focus on the initial-state components: initial–initial and initial–final antenna configurations. The formalism includes comprehensive possibilities for uncertainty estimates. We report on some preliminary results obtained with an implementation in the Vincia antenna-shower framework.

Electron cyclotron maser emission mode coupling to the z-mode on a longitudinal density gradient in the context of solar type III bursts

A beam of super-thermal, hot electrons was injected into maxwellian plasma with a density gradient along a magnetic field line. 1.5D particle-in-cell simulations were carried out which established that the EM emission is produced by the perpendicular component of the beam injection momentum. The beam has a positive slope in the distribution function in perpendicular momentum phase space, which is the characteristic feature of a cyclotron maser. The cyclotron maser in the overdense plasma generates emission at the electron cyclotron frequency. The frequencies of generated waves were too low to propagate away from the injection region, hence the wavelet transform shows a pulsating wave generation and decay process. The intensity pulsation frequency is twice the relativistic cyclotron frequency. Eventually, a stable wave packet formed and could mode couple on the density gradient to reach frequencies of the order of the plasma frequency that allowed for propagation. The emitted wave is likely to be a z-mode wave. The total electromagnetic energy generated is of the order of 0.1% of the initial beam kinetic energy. The proposed mechanism is of relevance to solar type III radio bursts, as well as other situations, when the injected electron beam has a non-zero perpendicular momentum, e.g., magnetron.

RF knock-out extraction in HITFiL

A compact facility for cancer therapy has been designed and is presently under construction. A slow beam extraction system using the RF-Knock Out method and 3rd-order resonance is adopted in the synchrotron of this facility. Eight sextupoles are used, four of them are for correcting the chromaticity and the rest for driving the 3rd-order resonance. In order to save the aperture of vacuum chamber, a 3-magnet bump is adopted during the extraction process. The extraction phase space map and the last 3 turns' particle trajectory before extraction are given. The matching betatron functions with HEBT (high energy beam transport) are also presented.

Causal signal transmission by quantum fields. V: Generalised Keldysh rotations and electromagnetic response of the Dirac sea

The connection between real-time quantum field theory (RTQFT) [see, e.g., A. Kamenev and A. Levchenko, Adv. Phys. 58 (2009) 197] and phase-space techniques [E. Wolf and L. Mandel, Optical Coherence and Quantum Optics (Cambridge, 1995)] is investigated. The Keldysh rotation that forms the basis of RTQFT is shown to be a phase-space mapping of the quantum system based on the symmetric (Weyl) ordering. Following this observation, we define generalised Keldysh rotations based on the class of operator orderings introduced by Cahill and Glauber [K.E. Cahill, R.J. Glauber, Phys. Rev. 177 (1969) 1882]. Each rotation is a phase-space mapping, generalising the corresponding ordering from free to interacting fields. In particular, response transformation [L.I. Plimak, S. Stenholm, Ann. Phys. (N.Y.) 323 (2008) 1989] extends the normal ordering of free-field operators to the time-normal ordering of Heisenberg operators. Structural properties of the response transformation, such as its association with the nonlinear quantum response problem and the related causality properties, hold for all generalised Keldysh rotations.Furthermore, we argue that response transformation is especially suited for RTQFT formulation of spatial, in particular, relativistic, problems, because it extends cancellation of zero-point fluctuations, characteristic of the normal ordering, to interacting fields. As an example, we consider quantised electromagnetic field in the Dirac sea. In the time-normally-ordered representation, dynamics of the field looks essentially classical (fields radiated by currents), without any contribution from zero-point fluctuations. For comparison, we calculate zero-point fluctuations of the interacting electromagnetic field under orderings other than time-normal. The resulting expression is physically inconsistent: it does not obey the Lorentz condition, nor Maxwell’s equations.

Multifunctional solid/solid phononic crystal

A two-dimensional, solid/solid phononic crystal (PC) comprised a square array of steel cylinders in epoxy is shown to perform a variety of spectral, wave vector, and phase-space functions. Over a range of operating frequencies, the PC’s elastic band structure shows uniquely shaped equifrequency contours that are only accessible to excitations of longitudinal polarization. Under this condition, the PC is shown to behave as (1) an acoustic wave collimator, (2) a defect-less wave guide, (3) a directional source for elastic waves, (4) an acoustic beam splitter, (5) a phase-control device, and (6) a k-space multiplexer. Wave vector diagrams and finite-difference time-domain simulations are employed to authenticate the above mentioned capabilities.

Efficient grid-based Bayesian estimation of nonlinear low-dimensional systems with sparse non-Gaussian PDFs

Bayesian estimation strategies represent the most fundamental formulation of the state estimation problem available, and apply readily to nonlinear systems with non-Gaussian uncertainties. The present paper introduces a novel method for implementing grid-based Bayesian estimation which largely sidesteps the severe computational expense that has prevented the widespread use of such methods. The method represents the evolution of the probability density function (PDF) in phase space, px(x′,t), discretized on a fixed Cartesian grid over all of phase space, and consists of two main steps: (i) between measurement times, px(x′,t) is evolved via numerical discretization of the Kolmogorov forward equation, using a Godunov method with second-order corner transport upwind correction and a total variation diminishing flux limiter; (ii) at measurement times, px(x′,t) is updated via Bayes’ theorem. Computational economy is achieved by exploiting the localized nature of px(x′,t). An ordered list of cells with non-negligible probability, as well as their immediate neighbors, is created and updated, and the PDF evolution is tracked only on these active cells.

Fractional diffusion in a periodic potential: Overdamped and inertia corrected solutions for the spectrum of the velocity correlation function

Anomalous diffusion of a particle in a cosine periodic potential is treated using fractional diffusion equations in both phase and configuration space. Exact solutions of two distinct forms of the fractional Klein-Kramers (Fokker-Planck) equation for the distribution function in phase space are obtained via matrix continued fractions yielding the average velocity, the velocity autocorrelation function, its spectrum, etc. In the overdamped limit, the results yielded by both equations agree with those from a fractional probability density diffusion equation in configuration space. A simple analytic solution for the spectrum of the velocity correlation function is also given using the effective eigenvalue approximation. The results represent generalizations of the conventional solutions for the normal diffusion of a Brownian particle in a cosine potential to fractional dynamics (giving rise to anomalous diffusion).

Self-consistent models of quasi-relaxed rotating stellar systems

Two new families of self-consistent axisymmetric truncated equilibrium models for the description of quasi-relaxed rotating stellar systems are presented. The first extends the spherical King models to the case of solid-body rotation. The second is characterized by differential rotation, designed to be rigid in the central regions and to vanish in the outer parts, where the energy truncation becomes effective. The models are constructed by solving the nonlinear Poisson equation for the self-consistent mean-field potential. For rigidly rotating configurations, the solutions are obtained by an asymptotic expansion on the rotation strength parameter. The differentially rotating models are constructed by means of an iterative approach based on a Legendre series expansion of the density and the potential. The two classes of models exhibit complementary properties. The rigidly rotating configurations are flattened toward the equatorial plane, with deviations from spherical symmetry that increase with the distance from the center. For models of the second family, the deviations from spherical symmetry are strongest in the central region, whereas the outer parts tend to be quasi-spherical. The relevant parameter spaces are explored and the intrinsic and projected structural properties are described. Special attention is given to the effect of different options for the truncation of the distribution function in phase space. Models in the moderate rotation regime are best suited to applications to globular clusters. For general interest in stellar dynamics, at high values of the rotation strength the differentially rotating models exhibit a toroidal core embedded in a quasi-spherical configuration. Physically simple analytical models of the kind presented here provide insights into dynamical mechanisms and may be a basis for more realistic investigations with the help of N-body simulations.

Integro-differential equation for joint probability density in phase space associated with continuous-time random walk

We derive an integro-differential equation for the joint probability density function in phase space associated with the continuous-time random walk, with generic waiting time probability density function and external force. This equation permits us to investigate whole diffusion processes covering initial-, intermediate-, and long-time ranges, which can distinguish the evolution details for systems having the same behavior in the long-time limit with different initial- and intermediate-time behaviors. Moreover, we obtained analytic solutions for probability density functions both in velocity and phase spaces, and interesting dynamic behaviors are discovered.

Master equation in phase space applied to the quantum Brownian motion in a tilted periodic potential

Quantum effects in the Brownian motion in a tilted cosine potential are treated in the high temperature and weak bath–particle coupling limit via the semiclassical master equation for the time evolution of the reduced Wigner function in phase space. The differential recurrence relation generated from the master equation by expanding the periodic Wigner function expressed as a function of wave number k in Fourier series via Floquet's theorem is solved for bounded periodic initial conditions using matrix-continued fractions yielding both the time-independent and the time-dependent bounded periodic solutions. Just as the classical problem, the non-periodic solution is determined from the recurrence relation by integrating w.r.t. k over the first Brillouin zone with the initial conditions generated by the non-periodic Wigner distribution corresponding to zero probability current. The time-independent periodic (locked) solution is used to estimate quantum effects in the dc current–voltage characteristic of a Josephson junction including the capacitance, while the time-dependent non-periodic (running) solution is used to determine the dynamic structure factor and ultimately via integration over the first Brillouin zone the lifetime of the zero-voltage state. The current–voltage characteristics displayed as average velocity versus tilt reproduce in the high damping limit those from the semiclassical Smoluchowski equation, while the decrease in the lifetime of the zero-voltage state due to (the barrier lowering) quantum effects appears to be very sensitive to the tilt.

Phase Space Evolution and Discontinuous Schrödinger Waves

The problem of Schrödinger propagation of a discontinuous wavefunction – diffraction in time – is studied under a new light. It is shown that the evolution map in phase space induces a set of affine transformations on discontinuous wavepackets, generating expansions similar to those of wavelet analysis. Such transformations are identified as the cause for the infinitesimal details in diffraction patterns. A simple case of an evolution map, such as SL(2) in a two-dimensional phase space, is shown to produce an infinite set of space-time trajectories of constant probability. The trajectories emerge from a breaking point of the initial wave.

Quantum dynamics in ultracold atomic physics

We review recent developments in the theory of quantum dynamics in ultra-cold atomic physics, including exact techniques, but focusing on methods based on phase-space mappings that are appli- cable when the complexity becomes exponentially large. These phase-space representations include the truncated Wigner, positive-P and general Gaussian operator representations which can treat both bosons and fermions. These phase-space methods include both traditional approaches using a phase-space of classical dimension, and more recent methods that use a non-classical phase-space of increased dimensionality. Examples used include quantum EPR entanglement of a four-mode BEC, time-reversal tests of dephasing in single-mode traps, BEC quantum collisions with up to 106 modes and 105 interacting particles, quantum interferometry in a multi-mode trap with nonlinear absorp- tion, and the theory of quantum entropy in phase-space. We also treat the approach of variational optimization of the sampling error, giving an elementary example of a nonlinear oscillator.