A new method for solving the quantum hydrodynamic equations of motion

Summary form only given. Wave packets (WPs) has recently emerged as a very active research field in quantum optics. We show here that the evolution of WPs may result in new class of large-scale interference effects: the formation of highly regular spatio-temporal patterns in the QM probability, |/spl Psi/|/sup 2/, with the key mechanism for this being pair interference between the eigen-modes of the system and the degeneracy of this interference. The new patterns have been discovered in the probability density |/spl Psi/|/sup 2/(x,t) of a particle in an infinitely deep square well, by numerical simulations of the evolution of a Gaussian WP; later work by a few groups used other initial conditions and media (e.g. Bose-Einstein condensate and angular WPs). In all these cases, the QM distribution shows a net of canals (minima), and ridges (maxima), running as straight lines in space-time (x,t). Similar patterns in the near field of a diffraction grating, and in matter waves in atom optics reveals a new aspects of the Talbot effect. So far the numerical exploration of these patterns has relied on the specific choice (i) of the initial conditions and (ii) of the square well; no physical explanation of the patterns has been suggested. Recently, we observed a quantum carpet under more general conditions, offered its universal explanation, and predicted distinct curved patterns in general case of an anharmonic potential.

PurposeThe paper explores the spectral solution of a partial differential equation governing nonlinear transmission lines (NLTLs). It aims to present an efficient simulation technique for a nonlinear fractional-order transmission line model and to show the effect of fractional derivatives on the evolution of solitons and wave packets of increasing spatial frequency.Design/methodology/approachThe paper is concerned with the solution of the equation governing NLTLs for both integer and fractional time derivatives. The spectral method shall be considered with Hermite and Malmquist-Takenaka orthogonal functions and integer and fractional implementations of the implicit trapezoidal rule. While the Hermite functions and Malmquist-Takenaka functions are suitable for solitons, the Malmquist-Takenaka functions are selected as these are superior for approximating wave packets of increasing frequency (Iserles, Luong and Webb, 2023). Furthermore, they possess properties that are favourable from a computational viewpoint.FindingsResults will show that the proposed approach is effective for simulating NLTLs with both integer and fractional time derivatives. In particular, fractional derivatives are shown to have a significant effect on the evolution of wave packets on a nonlinear transmission line.Originality/valueThe method is novel in using Malmquist-Takenaka basis functions for a spectral solution of a nonlinear transmission line equation with both integer and fractional time derivatives and for examining the effect of fractional derivatives on wave packets.

The de Broglie--Bohm causual (hydrodynamic) formulation of quantum mechanics is computationally implemented in the Lagrangian (moving with the fluid) viewpoint. The quantum potential and force are accurately evaluated with a moving weighted least squares algorithm. The quantum trajectory method is then applied to barrier tunneling on smooth potential surfaces. Analysis of the tunneling mechanism leads to a novel and accurate approximation: shortly after the wave packet is launched, completely neglect all quantum terms in the dynamical equations for motion along the tunneling coordinate.

We present results of real-time tracking of atomic two-electron dynamics in an autoionizing transient wave packet in krypton. A coherent superposition of two Fano resonances is excited with a femtosecond extreme-ultraviolet pulse. The evolution of the corresponding wave packet is subsequently probed with a delayed infrared pulse. In our specific case, we get access to the interference between one- and two-electron excitation channels in the launched wave packet, which is superimposed on its decay through autoionization. A simple model is able to account for the observed dynamical evolution of this wave packet.

Using an analytical model potential which contains resonant and bound states, we show that the decay of the resonances can be simulated by Lindblad operators. For that purpose, the unitary time evolution of an initial Gaussian wave packet in the model potential is compared with the non-unitary time evolution, obtained by solving the Lindblad equation, of the same wave packet in a potential which coincides with the model potential in the region of interest but does not contain resonances. In the latter case, dissipative effects are accounted for by Lindblad operators which lead to phenomenological friction and diffusion constants in the equations of motion. We suggest how those constants can be determined in a non-heuristic way, being directly connected to the width of the resonance in the model potential which we calculate using the complex rotation method.

Transport via the Liouville equation and moments of quantum distribution functions

Recent TOPEX/Poseidon observations show an enhanced (weakened) westward propagation of long planetary Rossby waves at extratropics (Tropics) and they usually propagate faster in the western basin than in the eastern basin in all major oceans. The evolution of a long planetary wave packet in a continuously stratified ocean in response to the various forcing functions is analytically investigated using the wave packet theory. For a wave packet with a large vertical scale, the stratification variation and the vertical shear of the mean zonal current act in concert, causing the wave packet to propagate directly against the mean zonal current—called the counter-Doppler-shift (CDS) effect. It is found that the speed ratio between the zonal baroclinic zonal current and the classic theory increases with the latitude, the eastward zonal current, and the local vertical scale. The vertical scale of a wave packet plays a critical role in the propagation, the structure, and spatial-scale development of a wave...

Floquet states represent intrinsic modes of ionization of an atom in a monochromatic field of constant intensity. To describe atomic wave packets evolving in realistic laser pulses, linear superpositions of Floquet states are required (“multistate Floquet theory”). This gives the possibility of following the evolution of wave packets in terms of the Floquet states that are populated during the pulse. We study here the way in which the Floquet states present in the representation of the wave packet manifest themselves in the excess-photon ionization spectra (EPI/ATI). For the purpose of illustration we choose a 1D atomic model with a soft-core Coulomb potential. We calculate the totality of the Floquet states, at all intensities needed, and generate the corresponding “Floquet map”. We then calculate the EPI spectra for wave packets evolving from the ground state under different types of pulses. By analyzing the location of the lines in the spectrum, and their shapes, we show that they can be associated, in a clear cut and predictable way, to Floquet states responsible for the emission. The understanding of the underlying physics can lead to tailoring laser pulses, such as to obtain EPI signals in a controlled way. Whereas our analysis is applied to theoretical spectra, it would apply, just as well, to experimental ones.

The wave packet perturbation energy propagation and accumulation signal during continuing heavy rain process in west of Sichuan Basin from September 22, 2008 to September 27, 2008 is analyzed and predicted in wave-packet propagation diagnosis method by making use of NCEP/NCAR reanalysis materials. According to the analysis, the distribution and propagation of wave packet can obviously reflect features of generation, lasting and ending of the rain. The high value area of the wave packet is basically consistent with heavy rain areas. The heavy rain process is generally generated in high value period or high phase of the wave packet perturbation energy accumulation. According to the meridional and zonal propagation features of the wave packet value, the continuing heavy rain process in west of Sichuan Basin from September 22, 2008 to September 27, 2008 was mainly affected by West Pacific subtropical high, cold air in northern regions and plateau regions, two water vapor channels from Bay of Bengal and South Sea to the basin as well as Typhoon Hagupit. The evolution of wave packet is meaningful for prediction.

We use the Korteweg–de Vries (KdV) equation, supplemented with several forcing/friction terms, to describe the evolution of wind-driven water wave packets in shallow water. The forcing/friction terms describe wind-wave growth due to critical level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave stress in the air near the interface induced by a turbulent wind and wave decay due to a turbulent bottom boundary layer. The outcome is a modified KdV–Burgers equation that can be a stable or unstable model depending on the forcing/friction parameters. To analyse the evolution of water wave packets, we adapt the Whitham modulation theory for a slowly varying periodic wave train with an emphasis on the solitary wave train limit. The main outcome is the predicted growth and decay rates due to the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to validate the theory for various cases of initial wave amplitudes and growth and/or decay parameter ranges. The results from the modulation theory agree well with these simulations. In most cases we examined, many solitary waves are generated, suggesting the formation of a soliton gas.

We ionized He, Ne and Ar in the tunnelling regime with a linearly polarized laser pulse and measured the ion-recoil momentum distribution on an axis perpendicular to laser polarization for two different wavelengths (λ = 800 nm and λ = 1800 nm). We observed a significantly narrower distribution for 800 nm than for 1800 nm. Classical simulations of the electron wave packet's evolution under the combined influence of the ion's Coulomb potential and the laser field show that the narrowing is caused by Coulomb focusing of the electron wave packet during recollisions. The narrowing is sensitive on the longitudinal momentum of the wave packet after tunnelling, which suggests a way of measuring it. Measurements at 800 nm in circular polarization–for which no recollision occurs–do not exhibit this narrowing of the perpendicular momentum distribution. Simulations for circular polarization show another aspect of Coulomb focusing: the Coulomb potential also affects the wave packet's transverse momentum as it leaves the saddle point immediately after tunnelling.

We study the space-time evolution of ``flavor'' neutrino wave-packets at finite temperature and density in the early Universe prior to BBN. We implement non-equilibrium field theory methods and linear response to study the space-time evolution directly from the effective Dirac equation in the medium. There is a rich hierarchy of time scales associated with transverse and longitudinal dispersion and coherence. A phenomenon of ``freezing of coherence'' is a result of a competition between longitudinal dispersion and the separation of wave-packets of propagating modes in the medium. Near a resonance the coherence and oscillation time scales are enhanced by a factor $1/\sin2\theta$ compared to the vacuum. Collisional relaxation via charged and neutral currents occurs on time scales much shorter than the coherence time scale and for small vacuum mixing angle, shorter than the oscillation scale. Assuming that the momentum spread of the initial wave packet is determined by the large angle scattering mean free path of charged leptons, we find that the transverse dispersion time scale is the shortest and is responsible for a large suppression in both the survival and transition probabilities on time scales much shorter than the Hubble time. For small mixing angle the oscillation time scale is \emph{longer} than the collisional relaxation scale. The method also yields the evolution of right-handed wave packets. Corrections to the oscillation frequencies emerge from wave-packet structure as well as from the energy dependence of mixing angles in the medium.

Bohm's non-relativistic approach to quantum mechanics is generalized to the relativistic domain. The non-relativistic concept of a «quantum potential» is transcribed into the corresponding «quantum force». This quantum force can be deduced from the recently established theory of relativistic Schrodinger equations in a similar way as the original quantum potential is deduced from the ordinary Schrodinger equation. The leading term of the new quantum force turns out as the well-known classical Lorentz force of electrodynamics, but modified by some quantum corrections. In the non-relativistic limit, Bohm's result is recovered again for a scalar particle.

: Our basic aim is to achieve a better understanding of the turbulent flow of the oceans in terms of the laws that govern the behavior of vortices and waves and their interactions. In our internal-wave project, we are using numerical simulations to study the dynamical evolution of internal wave packets. We plan to study in detail the process of reflection of a packet from regions of weak density gradient. Packets propagating away from the thermocline may be reflected back toward the thermocline. In the process of reflection, high amplitude packets may break causing mixing. This would tend to erode the boundaries of the thermocline. In our topographic work, we wish to understand the role that bottom topography plays in permitting or inhibiting the bifurcations of coastal currents. We are investigating the role of topography in establishing the circulation in the Adriatic. In particular we want to understand the circulation above the Mid and South Adriatic Pits.

The author's general variational method is applied to the case of a particle for which second moments are important but third and higher moments are negligible. Equations of motion are obtained for the angular momentum and for the centre of mass, equations (12·35) and (12·41), with arbitrary external forces X.The angular forces are then calculated for a charged particle with electric and magnetic moments moving in a general electromagnetic field, on the assumption that the effect of a certain part of the energy tensor, Tiii of (15·17), is negligible. This leads to the equations of angular motion, (17·13), from which it is inferred that, in order that the magnitude of the angular momentum may be integrable, the angular momentum, electric and magnetic moments must all be parallel in a frame of reference in which the particle is instantaneously at rest.The linear forces are then calculated for the case of no electric moment, leading to the equations for linear motion (18·10). From these it is inferred that, in order that the mass may be integrable, the ratio of the magnetic moment to the angular momentum must be constant.