Phase Space Area Research Articles

Comment on “Why the `coarse-graining' of Wigner function is always coarse” [Opt. Commun. 178 (2000) 147

Incorrect statements and assumptions undermining the validity of the results presented in the above paper are pointed out and shortly discussed. It is stressed that a proper `coarse graining' of a Wigner phase-space distribution function cannot be in general performed over finite phase-space cells and that any `smearing' over infinite phase-space areas cannot violate the respective uncertainty relations, which furnishes an answer to the question asked in the title.

Transverse linear dynamics in an axisymmetric ionization cooling channel

This paper outlines a formalism for the description of the linear transverse dynamics of charged particles in an axisymmetric ionization cooling channel. The particle trajectories in the absence of Coulomb scattering are described in terms of lattice functions \`a la Courant and Snyder, which depend only on the electric and magnetic fields in the channel. The process of multiple Coulomb scattering, which introduces stochastic terms into the particle equations of motion, is treated (in Gaussian approximation) by obtaining the distribution function in phase space as a solution of a Fokker-Planck equation. The distribution function is then used to obtain moment equations for the transverse variables and for combinations of variables such as the emittance and angular momentum. The distribution function is also used to obtain an expression for the peak four-dimensional phase space density and for the fraction of the beam that is within a certain area in phase space. The special case of a periodic channel is then considered and expressions for the asymptotic rms emittance and peak phase space density are obtained. Finally, the application of the general formalism to a numerical example, based on the reported design of a cooling channel for a neutrino source, is considered, and comparisons are made with numerical simulations of that channel.

Diffusion by extrinsic noise in the kicked Harper map.

A significantly improved analytic understanding of the extrinsically driven diffusion process is presented in a nonlinear dynamical system in which the phase space is divided into periodic two-dimensional tiles of regular motion, separated by a connected separatrix network (web) [previously studied by A. J. Lichtenberg and Blake P. Wood, Phys. Rev. Lett. 62, 2213 (1989)]. The system is represented by the usual "kicked Harper map" with added extrinsic noise terms. Three different diffusion regimes are found depending upon the strength of the extrinsic perturbation l relative to the web and regular motions. When the extrinsic noise is dominant over the intrinsic stochasticity and the regular rotation motions in the tile, diffusion obeys the random phase scaling l(2). When the extrinsic noise is dominant over the intrinsic stochasticity, but weaker than the regular rotation motion, the diffusion scales as lK(1/2), where K is the strength of the intrinsic kick. These findings agree well with numerical simulation results. When the extrinsic noise process is weaker than the stochastic web process, we analytically reproduce the well-known numerical result: The web diffusion is reduced by the ratio of phase-space areas of intrinsic to extrinsic stochasticity.

Reflected shock experiments on the equation-of-state properties of liquid deuterium at 100–600 GPa (1–6 Mbar)

Laser-driven shock experiments are used to study the equation-of-state (EOS) properties of liquid deuterium. Reflected shocks are utilized to increase the shock pressure, to expand the area of EOS phase space probed by the experiment, and to enhance the sensitivity to differences in compressibility. The results of these experiments differ substantially from the predictions of the Sesame EOS. EOS models showing large dissociation effects and large compressibility (up to a factor of 2) agree with the data. By use of independent techniques, this experiment offers the first confirmation of an earlier observation of enhanced compressibility in liquid deuterium.

Warm-fluid collective mode excitations in intense charged particle beams with nonlinear equilibrium self-fields: test particle simulations

This paper examines analytically and numerically the effects of self-consistent collective oscillations, excited in a charged particle beam with nonlinear equilibrium self-fields, on the motion of a test particle in the beam core and halo region. The infinite set of linearized eigenmodes of a waterbag equilibrium beam have been found in previous work (Sean Strasburg and R.C. Davidson, Phys. Lett. A 269 (2000) 40) using the smooth-focusing approximation and assuming axisymmetric perturbations. These eigenmodes, in combination with the nonlinear equilibrium charge-density and applied fields, cause areas of phase space to break into islands and, in the case of sufficiently large-amplitude perturbations and intense beams, to become stochastic. Nonlinear shifts in the transverse oscillation frequency are determined analytically. Using this frequency shift and the eigenmode frequencies, the location of resonant islands as a function of particle orbit amplitude, beam intensity, and mode number is predicted analytically and confirmed numerically. The dependence of island width on perturbative mode amplitude and beam intensity is explored numerically. Using the Poincaré technique, the particle phase space in the beam core and the beam halo region is investigated by numerically integrating the test particle equations over long periods.

Quantum interference as phase space filtering

Summary A new interpretation of the quantum interference, as a filtering in the phase space, is proposed. This filtering is applied to the quantum wavefunction, considered as an amplitude probability distribution, and is due to the mismatch of the phase space areas of the quantum system and the experimental set-up. The quantum particle is considered to have physical reality and is distinguished from a classical particle through the inherent impossibility of assigning to it a sharp phase space localizability.

Classical Fluctuations and Semiclassical Matrix Elements

We discuss the fluctuation properties of diagonal matrix elements in the semiclassical limit in chaotic systems. For extended observables, covering a phase space area of many times Planck's constant, both classical and quantal distributions are Gaussian. If the observable is a projection onto a single state or an incoherent projection onto several states classical and quantal distribution differ, but the mean and the variance are still obtainable from classical considerations.

Characterization of high charge electron beams induced by excimer laser irradiation

This work reports on electron beam characteristics obtained from a Cu photocathode utilizing an UV excimer laser of 222 nm for the first time. The values of the current, emittance, and brightness of the electron beams generated are reported. A suitable slit–slit emittance meter was developed to assess the electron beam emittance. This instrument allowed us to determine the electron beam phase-space areas by slicing the beam in small beamlets and measuring the beamlet direction distributions. Small Faraday cups placed after the slits were utilized to this reason. The laser beam was focused in a 4 mm2 spot and the maximum energy laser was established to 0.5 mJ. The highest current was 410 mA and the corresponding normalized emittance value was 10 π mm mrad corresponding to 80% of the total points. From these data the beam quality, expressed as normalized beam brightness, resulted in 4.1×109 A[π m rad]−2 and the quantum efficiency was 4.6×10−5. By increasing the KrCl laser spot and its energy a very charge electron beam was extracted of 16.4 A.

Phase space conservation and selection rules for astigmatic mode converters

An astigmatic mode converter consists of a pair of cylindrical lenses which, when properly located and oriented, can transform a Hermite–Gauss mode (HG nm) with rectangular symmetry into a Laguerre–Gauss mode (LG nm) with cylindrical symmetry (see E. Abramochkin, V. Volstnikov, Opt. Commun. 83 (1991) 123; M.W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, J.P. Woerdman, Opt. Commun. 96 (1993) 123; M.J. Padgett, J. Arlt, N. Simpson, Am. J. Phys. 64 (1996) 77; and J. Courtial, M.J. Padgett, Opt. Commun. 159 (1999) 13). In this paper we discuss how the conservation of phase space area determines the selection rules on which mode conversions are possible. A discussion of the beam quality factor M 2 is also given for both Hermite–Gauss and Laguerre–Gauss modes.

Experimental evidence for the role of cantori as barriers in a quantum system

We investigate the effect of cantori on momentum diffusion in a quantum system. Ultracold caesium atoms are subjected to a specifically designed periodically pulsed standing wave. A cantorus separates two chaotic regions of the classical phase space. Diffusion through the cantorus is classically predicted. Quantum diffusion is only significant when the classical phase-space area escaping through the cantorus per period greatly exceeds Planck's constant. Experimental data and a quantum analysis confirm that the cantori act as barriers.

Experimental Study of Quantum Dynamics in a Regime of Classical Anomalous Diffusion

The correspondence between a quantum system and the underlying classical dynamics is a topic of fundamental importance. The paradigm system for the study of this correspondence is the kicked rotor, because of the simplicity of its equations of motion and the wealth of knowledge available on the classical system. One particularly interesting aspect of the kicked rotor is the existence of accelerator modes, which lead to Levy flights in generic phase-space trajectories [1,2]. These Levy flights can have a strong influence on the global transport properties of a system, and have been recently employed in the understanding of a subrecoil laser cooling scheme for atoms [3] and the motion of particles in a nonuniform fluid flow [4]. In the classical kicked rotor, the effects of the accelerator modes become most important in the long-time limit, because they account for a relatively small area in phase space. However, recent theoretical work has shown that these structures can have a dramatic effect in the quantum case, because of the nonlocal nature of the wave functions [5]. In this Letter, we study an experimental realization of the quantum kicked rotor, where cold cesium atoms are “kicked” by a periodically pulsed standing wave of fardetuned light. Previous work with sodium atoms has established this system as an excellent setting for the study of quantum chaos [6,7]. In this previous work, we observed dynamical localization, which is the quantum suppression of the classical chaotic momentum diffusion; the hallmark of this effect is a localized momentum distribution with an exponential profile. Here, we report the experimental study of the atomic momenta as a function of the pulse amplitude and period. We observe the oscillations in the momentum distribution widths that are expected from theory. We also observe that for certain kick amplitudes where accelerator modes are present in the classical phase space, the momentum distributions do not have the expected exponential form over the time scale of our experiment [8]. These results, which form the main focus of this Letter, suggest a correlation between classical anomalous diffusion and the observed quantum dynamics.

Nonadiabatic geometric phase for the cyclic evolution of a time-dependent Hamiltonian system

The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of squeezed state. It is shown that the cyclic and quasicyclic squeezed states correspond to the periodic and quasiperiodic solutions of an effective Hamiltonian defined on an extended phase space, respectively. The geometric phase of the cyclic squeezed state is found to be a phase-space area swept out by a periodic orbit. Furthermore, a class of cyclic states are expressed as a superposition of an infinte number of squeezed states. Their geometric phases are found to be independent of $\hbar$, and equal to $-(n+1/2)$ times the classical nonadiabatic Hannay angle.

Emission and emittance measurements of electron beams generated from Cu and diamond photocathodes

We report on the electron emission from Cu and diamond film cathodes irradiated with two ultraviolet excimer laser beams working at 308 nm XeCl, and 222 nm KrCl. The beam emittance measurements were performed using a new experimental setup composed of two movable slit arrays. The electron beam phase space areas were determined by splitting the beam in small beamlets and measuring the beamlet direction spread via small cups and slits. The highest emission was obtained at the lowest laser wavelength. With a 4 mm2 beam spot and 0.5 mJ laser energy the maximum currents from the Cu and diamond film cathode were 370 and 410 mA, respectively, and the corresponding emittances were 18 and 27 (π mm mrad). From these values the normalized beam brightness was estimated to be 4.6×109 A(π m rad)−2 for the Cu cathode and 2.3×109 A(π m rad)−2 for the diamond cathode. By increasing the laser energy and the spot beam the maximum extracted current was 16.4 A from the Cu cathode with the KrCl laser.

Effect of an ac electric field on chaotic electronic transport in a magnetic superlattice

We study classical transport properties of a two-dimensional electron gas subject to a spatially periodic magnetic field and to a time-dependent (ac) electric field. We investigate the case of a strong one-dimensional magnetic modulation and show how the chaotic electronic motion affects the conductivity of the sample. The dependence of the longitudinal conductivity on the amplitude of the ac field clearly reflects the arising of coupled chaotic phase-space areas as well as features of anomalous diffusion of the charge carriers.

The Restricted Hill Four-Body Problem with Applications to the Earth–Moon–Sun System

Starting from the four-body problem a generalization of both the restricted three-body problem and the Hill three-body problem is derived. The model is time periodic and contains two parameters: the mass ratio ν of the restricted three-body problem and the period parameter m of the Hill Variation orbit. In the proper coordinate frames the restricted three-body problem is recovered as m → 0 and the classical Hill three-body problem is recovered as ν → 0. This model also predicts motions described by earlier researchers using specific models of the Earth–Moon–Sun system. An application of the current model to the motion of a spacecraft in the Sun perturbed Earth–Moon system is made using Hill's Variation orbit for the motion of the Earth–Moon system. The model is general enough to apply to the motion of an infinitesimal mass under the influence of any two primaries which orbit a larger mass. Using the model, numerical investigations of the structure of motions around the geometric position of the triangular Lagrange points are performed. Values of the parameter ν range in the neighborhood of the Earth–Moon value as the parameter m increases from 0 to 0.195 at which point the Hill Variation orbit becomes unstable. Two families of planar periodic orbits are studied in detail as the parameters m and ν vary. These families contain stable and unstable members in the plane and all have the out-of-plane stability. The stable and unstable manifolds of the unstable periodic orbits are computed and found to be trapped in a geometric area of phase space over long periods of time for ranges of the parameter values including the Earth–Moon–Sun system. This model is derived from the general four-body problem by rigorous application of the Hill and restricted approximations. The validity of the Hill approximation is discussed in light of the actual geometry of the Earth–Moon–Sun system.

Neuman-Dirichlet tadpoles as new string states and quantum mechanical particle-wave duality from world-sheet t-duality

We consider new objects in bosonic open string theory — ND tadpoles, which have N(euman) boundary conditions at one end of the world-sheet and D(irichlet) at the other, must exist due to s − t duality in a string theory with both NN strings and D-branes. We demonstrate how to interpolate between N and D boundary conditions. In the case of mixed boundary conditions the action for a quantum particle is induced on the boundary. Quantum-mechanical particle-wave duality, a dual description of a quantum particle in either the coordinate or the momentum representation, is induced by world-sheet T-duality. The famous R = α R relation is equivalent to the quantization of the phase space area of a Planck cell ∮ pdx = 2 π h ̷ . We also introduce a boundary operator Z − a “Zipper” which changes the boundary condition from N into D and vice versa.

Transport properties of one-dimensional Kronig - Penney models with correlated disorder

Transport properties of one-dimensional Kronig - Penney models with binary correlated disorder are analysed using an approach based on classical Hamiltonian maps. In this method, extended states correspond to bound trajectories in the phase space of a parametrically excited linear oscillator, while the on-site potential of the original model is transformed to an external force. We show that in this representation the two-probe conductance takes a simple geometrical form in terms of evolution areas in phase space. We also analyse the case of a general N-mer model.

Equivalence of energy-time and momentum-position phase spaces of the solar system

In the first step, equivalence of energy-time and momentum-position phase spaces is discussed for simple periodic systems having a central force. The discussion concerns the phasespace areas taken as a whole, as well as their fragments. It is demonstrated that some fragments of the phase space areas can fit together with a considerable accuracy. This suggests similar equivalence of the corresponding fragments of the phase-space areas of the solar system. The hypothesis is examined in respect to the planet orbits and the orbits of some planet satellites.

Phase space approach for optimizing grid representations: The mapped Fourier method.

The representation of a quantum system by an evenly spaced Fourier grid is examined. This grid faithfully represents wave functions whose projection is contained in a rectangular phase space. This is mathematically equivalent to a band limited function with finite support. In general, wave packets decay exponentially in classically forbidden regions of phase space. This idea is then used first to optimize the rectangular shape of the Fourier grid, leading to exponential convergence. Nevertheless, in most cases the representation is suboptimal. The representation efficiency can then be extremely enhanced by mapping the coordinates. The mapping procedure reshapes the wave function to fit into the rectangular Fourier shape such that the wasted phase space area is minimal. It is shown that canonical transformations, which rescale the coordinates, improve the representation dramatically. A specific scaling transformation enables the representation of the notoriously difficult Coulomb potentials. The scaling transformation enables one to extract almost as many converged eigenstate energies as there are grid points. The method is extendible to more than one dimension, which is demonstrated by the study of the ${\mathrm{H}}_{2}^{+}$ problem. This scaling transformation can bridge the gap between quantum chemistry and quantum molecular dynamics by enabling the treatment of electronic problems in the vicinity of Coulomb potentials by grid methods developed for molecular dynamics. \textcopyright{} 1996 The American Physical Society.

A nonlinear probabilistic method for predicting vessel capsizing in random beam seas

The capsizing of vessels in random beam seas is investigated using a single degree-of-freedom model which is limited to the roll motion. Several factors, including sea wave spectrum, nonlinear righting moment characteristics, and nonlinear damping, are taken into account in the analysis. A nonlinear probabilistic method is developed by combining ideas from modern nonlinear dynamics (specifically, the Melnikov function and the phase space area flux) and random vibrations. Conditions for the onset of vessel capsizing are obtained in terms of the sea state characteristics (significant wave height and characteristic wave period) and the vessel parameters (damping and stiffness coefficients). Extensive numerical simulations are carried out to demonstrate the validity of the analytical results. It is found that there exists an excellent correlation between the rate of phase space flux and the probability of capsizing.