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- ItemPopulation transfer to high angular momentum states in infrared-assisted XUV photoionization of helium(Bristol : IOP Publ., 2020) Mayer, Nicola; Peng, Peng; Villeneuve, David M.; Patchkovskii, Serguei; Ivanov, Misha; Kornilov, Oleg; Vrakking, Marc J.J.; Niikura, HiromichiAn extreme-ultraviolet (XUV) laser pulse consisting of harmonics of a fundamental near-infrared (NIR) laser frequency is combined with the NIR pulse to systematically study two-color photoionization of helium atoms. A time-resolved photoelectron spectroscopy experiment is carried out where energy- A nd angle-resolved photoelectron distributions are obtained as a function of the NIR intensity and wavelength. Time-dependent Schrödinger equation calculations are performed for the conditions corresponding to the experiment and used to extract residual populations of Rydberg states resulting from excitation by the XUV + NIR pulse pair. The residual populations are studied as a function of the NIR intensity (3.5 × 1010-8 × 1012 W cm-2) and wavelength (760-820 nm). The evolution of the photoelectron distribution and the residual populations are interpreted using an effective restricted basis model, which includes the minimum set of states relevant to the features observed in the experiments. As a result, a comprehensive and intuitive picture of the laser-induced dynamics in helium atoms exposed to a two-color XUV-NIR light field is obtained. © 2020 The Author(s). Published by IOP Publishing Ltd.
- ItemImplementing exact absorbing boundary condition for the linear one-dimensional Schrödinger problem with variable potential by Titchmarsh-Weyl theory(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Ehrhardt, Matthias; Zheng, ChunxiongA new approach for simulating the solution of the time-dependent Schrödinger equation with a general variable potential will be proposed. The key idea is to approximate the Titchmarsh-Weyl m-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to a suitable spectral parameter. With the proposed method we can overcome the usual high-frequency restriction for absorbing boundary conditions of general variable potential problems. We end up with a fast computational algorithm for absorbing boundary conditions that are accurate for the full frequency band
- ItemNumerical simulation of waves in periodic structures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Ehrhardt, Matthias; Han, Houde; Zheng, ChunxiongIn this work we present a new numerical technique for solving periodic structure problems. This new approach possesses several advantages. First, it allows for a fast evaluation of the Robin-to-Robin operator for periodic array problems. Secondly, this computational method can also be used for bi-periodic structure problems with local defects. Our strategy is an improvement of the recently developed recursive doubling process by Yuan and Lu. In this paper we consider several problems, such as the exterior elliptic problems with strong coercivity, the time-dependent Schrödinger equation and finally the Helmholtz equation with damping.
- ItemA class of probabilistic models for the Schrödinger equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Wagner, WolfgangA class of stochastic particle models for the spatially discretized time-dependent Schrödinger equation is constructed. Each particle is characterized by a complex-valued weight and a position. The particle weights change according to some deterministic rules between the jumps. The jumps are determined by the creation of offspring. The main result is that certain functionals of the particle systems satisfy the Schrödinger equation. The proofs are based on the theory of piecewise deterministic Markov processes.
- ItemA random walk model for the Schrödinger equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Wagner, WolfgangA random walk model for the spatially discretized time-dependent Schrödinger equation is constructed. The model consists of a class of piecewise deterministic Markov processes. The states of the processes are characterized by a position and a complex-valued weight. Jumps occur both on the spatial grid and in the space of weights. Between the jumps, the weights change according to deterministic rules. The main result is that certain functionals of the processes satisfy the Schrödinger equation.
- ItemNumerical simulation of quantum waveguides(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Arnold, Anton; Ehrhardt, Matthias; Schulte, MaikeThis chapter is a review of the research of the authors from the last decade and focuses on the mathematical analysis of the Schrödinger model for nano-scale semiconductor devices. We discuss transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a two dimensional domain. First we derive the two dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson-type finite difference scheme and a compact nine-point scheme. For this difference equations we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. The presented discrete boundary-valued problem is unconditionally stable and completely reflection-free at the boundary. Then, since the DTBCs for the Schrödinger equation include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate DTBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In several numerical tests we illustrate the perfect absorption of outgoing waves independent of their impact angle at the boundary, the stability, and efficiency of the proposed method. Finally, we apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow.
- ItemA fast solution method for time dependent multidimensional Schrödinger equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Lanzara, Flavia; Mazya, Vladimir; Schmidt, GuntherIn this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schrödinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200.
- ItemA random cloud model for the Schrödinger equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Wagner, WolfgangThe paper is concerned with the construction of a stochastic model for the spatially discretized time-dependent Schrödinger equation. The model is based on a particle system with a Markov jump evolution. The particles are characterized by a sign (plus or minus), a position (discrete grid) and a type (real or imaginary). The jumps are determined by the creation of offsprings. The main result is the construction of a family of complex-valued random variables such that their expected values coincide with the solution of the Schrödinger equation.