Explicit schemes for time propagating many-body wavefunctions

Accurate theoretical data on many time-dependent processes in atomic and molecular physics and in chemistry require the direct numerical solution of the time-dependent Schr\"odinger equation, thereby motivating the development of very efficient time propagators. These usually involve the solution of very large systems of first order differential equations that are characterized by a high degree of stiffness. We analyze and compare the performance of the explicit one-step algorithms of Fatunla and Arnoldi. Both algorithms have exactly the same stability function, therefore sharing the same stability properties that turn out to be optimum. Their respective accuracy however differs significantly and depends on the physical situation involved. In order to test this accuracy, we use a predictor-corrector scheme in which the predictor is either Fatunla's or Arnoldi's algorithm and the corrector, a fully implicit four-stage Radau IIA method of order 7. We consider two physical processes. The first one is the ionization of an atomic system by a short and intense electromagnetic pulse; the atomic systems include a one-dimensional Gaussian model potential as well as atomic hydrogen and helium, both in full dimensionality. The second process is the decoherence of two-electron quantum states when a time independent perturbation is applied to a planar two-electron quantum dot where both electrons are confined in an anharmonic potential. Even though the Hamiltonian of this system is time independent the corresponding differential equation shows a striking stiffness. For the one-dimensional Gaussian potential we discuss in detail the possibility of monitoring the time step for both explicit algorithms. In the other physical situations that are much more demanding in term of computations, we show that the accuracy of both algorithms depends strongly on the degree of stiffness of the problem.Comment: 24 pages, 14 Figure

Multiresolution schemes for time-scaled propagation of wave packets

We present a detailed analysis of the time scaled coordinate approach and its implementation for solving the time-dependent Schr\"odinger equation describing the interaction of atoms or molecules with radiation pulses. We investigate and discuss the performance of multi-resolution schemes for the treatment of the squeezing around the origin of the bound part of the scaled wave packet. When the wave packet is expressed in terms of B-splines, we consider two different types of breakpoint sequences: an exponential sequence with a constant density and an initially uniform sequence with a density of points around the origin that increases with time. These two multi-resolution schemes are tested in the case of a one-dimensional gaussian potential and for atomic hydrogen. In the latter case, we also use Sturmian functions to describe the scaled wave packet and discuss a multi-resolution scheme which consists in working in a sturmian basis characterized by a set of non-linear parameters. Regarding the continuum part of the scaled wave packet, we show explicitly that, for large times, the group velocity of each ionized wave packet goes to zero while its dispersion is suppressed thereby explaining why, eventually, the scaled wave packet associated to the ejected electrons becomes stationary. Finally, we show that only the lowest scaled bound states can be removed from the total scaled wave packet once the interaction with the pulse has ceased

Interaction laser-atome : nouvelles approches théoriques dépendantes du temps

Ce travail porte sur le développement de nouvelles méthodes dépendantes du temps pour l'étude de l'interaction laser-atome; il vise deux objectifs. Le premier fait suite au désaccord significatif existant entre les résultats de différents calculs de section efficace de double ionisation par absorption de deux photons de l'atome d'hélium. Il s'agit de déterminer si ce désaccord est lié à la prise en compte ou non de la corrélation électronique dans la voie de sortie. Pour répondre à cette question, nous développons une méthode hybride combinant la résolution numérique de l'équation de Schrödinger (ES) avec une méthode de type R-matrice. L’analyse des résultats pour l'ionisation de l'hélium par un et deux photons montre que le désaccord n’est pas lié à la corrélation dans la voie de sortie. Le second objectif, lié à l'inexistence de fonctions d'onde analytiques décrivant les continua multiples, est de développer une méthode permettant d'éviter la projection des paquets d'ondes (PO) sur ces fonctions pour en extraire l'information pertinente. Cet objectif est atteint en introduisant dans l’ES un facteur d’échelle dépendant du temps qui conduit à un confinement du PO. Cette méthode est analysée en l'appliquant à l'étude de l’interaction d’atomes à un électron actif avec une impulsion laser. Sa mise en oeuvre nécessite par ailleurs le développement d'algorithmes performants de propagation temporelle des PO. Différents algorithmes sont étudiés en détail. Pour tester ces derniers, nous étudions en profondeur l'ionisation de l'hydrogène atomique en régime basse fréquence pour expliquer l'inattendue présence des structures observées à basse énergie dans les spectres de photo-ionisation d’atomes et molécules, mesurés dans la direction de polarisation du champ.(SC - Sciences) -- UCL, 201

Analysing a two-electron wavepacket by semiclassically propagating its Fourier components in space

In the last few years, the development of high order harmonic generation sources and free electron lasers delivering ultra-intense and ultra-short VUV-XUV pulses has made it possible to study nonlinear processes in atoms and molecules on the electronic time scale. The theoretical support required by the ongoing experiments comes notably in the form of numerical tools intended to solve the time-dependent Schrodinger equation. The wavepacket produced in these approaches has a multichannel character and its analysis in terms of the observed physical channels is a problem in itself. Various solutions have been proposed so far, which all suffer from one or another inconvenience, ranging from very heavy computational costs to the inability to characterize differential cross sections. The purpose of this paper is to propose a new, low-cost and complete method of analysis. It consists in propagating the Fourier components of the wavepacket with respect to the hyper-radius all the way to the genuine asymptotic region where the various channels disentangle from each other based on their kinematics. We demonstrate the feasibility and versatility of this proposal by applying it to two different time-propagation codes in the case of one-photon double ionization of helium using short pulses

Explicit schemes for time propagating many-body wave functions

Accurate theoretical data on many time-dependent processes in atomic and molecular physics and in chemistry require the direct numerical ab initio solution of the time-dependent Schrödinger equation, thereby motivating the development of very efficient time propagators. These usually involve the solution of very large systems of first-order differential equations that are characterized by a high degree of stiffness. In this contribution, we analyze and compare the performance of the explicit one-step algorithms of Fatunla and Arnoldi. Both algorithms have exactly the same stability function, therefore sharing the same stability properties that turn out to be optimum. Their respective accuracy, however, differs significantly and depends on the physical situation involved. In order to test this accuracy, we use a predictor-corrector scheme in which the predictor is either Fatunla's or Arnoldi's algorithm and the corrector, a fully implicit four-stage Radau IIA method of order 7. In this contribution, we consider two physical processes. The first one is the ionization of an atomic system by a short and intense electromagnetic pulse; the atomic systems include a one-dimensional Gaussian model potential as well as atomic hydrogen and helium, both in full dimensionality. The second process is the decoherence of two-electron quantum states when a time-independent perturbation is applied to a planar two-electron quantum dot where both electrons are confined in an anharmonic potential. Even though the Hamiltonian of this system is time independent the corresponding differential equation shows a striking stiffness which makes the time integration extremely difficult. In the case of the one-dimensional Gaussian potential we discuss in detail the possibility of monitoring the time step for both explicit algorithms. In the other physical situations that are much more demanding in term of computations, we show that the accuracy of both algorithms depends strongly on the degree of stiffness of the problem

Interaction of a model atom exposed to strong laser pulses: Role of the Coulomb potential

With the help of the solution of the time-dependent Schrödinger equation in momentum space, we study the above-threshold ionization spectrum resulting from the interaction of atomic hydrogen with an infrared and XUV short laser pulses. Our calculations are based on a model where the kernel of the nonlocal Coulomb potential is replaced by a finite sum of N symmetric separable potentials, each of them supporting one bound state of atomic hydrogen. Here, we consider only the case of 1s, 2s, and 2p states. Thus, the theory fully accounting for the important 1s-2p transition, explains the photoelectron spectrum as well as the total ionization probability for the resonant case. We compared the results given by our theory with the numerical solutions of the time-dependent Schrödinger equation. © 2013 American Physical Society

Time scaling with efficient time-propagation techniques for atoms and molecules in pulsed radiation fields

We present an ab initio approach to solve the time-dependent Schr\"odinger equation to treat electron and photon impact multiple ionization of atoms or molecules. It combines the already known time scaled coordinate method with a new high order time propagator based on a predictor-corrector scheme. In order to exploit in an optimal way the main advantage of the time scaled coordinate method namely that the scaled wave packet stays confined and evolves smoothly towards a stationary state the modulus square of which being directly proportional to the electron energy spectra in each ionization channel, we show that the scaled bound states should be subtracted from the total scaled wave packet. In addition, our detailed investigations suggest that multi-resolution techniques like for instance, wavelets are the most appropriate ones to represent spatially the scaled wave packet. The approach is illustrated in the case of the interaction of an one-dimensional model atom as well as atomic hydrogen with a strong oscillating field.Comment: 26 pages, 9 figures, to be published in Physical Review