Semi-global approach for propagation of the time-dependent Schrödinger equation for time-dependent and nonlinear problems

Abstract

A detailed exposition of highly efficient and accurate method for the propagation of the time-dependent Schrödinger equation [50] is presented. The method is readily generalized to solve an arbitrary set of ODE's. The propagation is based on a global approach, in which large time-intervals are treated as a whole, replacing the local considerations of the common propagators. The new method is suitable for various classes of problems, including problems with a time-dependent Hamiltonian, nonlinear problems, non-Hermitian problems and problems with an inhomogeneous source term. In this paper, a thorough presentation of the basic principles of the propagator is given. We give also a special emphasis on the details of the numerical implementation of the method. For the first time, we present the application for a non-Hermitian problem by a numerical example of a one-dimensional atom under the influence of an intense laser field. The efficiency of the method is demonstrated by a comparison with the common Runge-Kutta approach.

Time-dependent Markovian quantum master equation

“…Solution for the NAME For a time-independent problem it is convenient to transform to the Heisenberg picture, and obtain a set of coupled linear differential equations for the operators, [16,48]. For Hilbert space of dimension N one obtains N 2 − 1 equations which can be solved analytically or by standard numeric methods [51]. In contrast, the solution is more complicated when the GKLS equation has an explicit time dependence.…”

Sixth-order schemes for laser–matter interaction in the Schrödinger equation

“…In principle, the procedure extends in a straightforward way to higher dimensions via tensorisation of the periodic grids and we demonstrate the applicability of the approach using two and three-dimensional examples under ε 3 = ε 4 = 10 −2 . Lastly, we consider a Coloumb potential example from Schaeffer et al [32] under ε 5 = 1.…”

“…which are degree eight and degree ten polynomials in two and three dimensions, respectively, and where the wells are defined by the centres, Lastly, we consider the one-dimensional (soft) Coloumb potential example from Schaeffer et al [32] as our fifth example (fig. 6.2 (centre) and (right)), where V 5 (x) = 2 1 − 1 √ x 2 +1 .…”

“…Among these, some of the notable classes are the family of Magnus-Lanczos methods [23,12,20] that achieve high accuracies but require very small time step (or a large number of Lanczos iterations) and the class of Chebychev polynomial based methods [35,27,32] that can allow the use of much larger time steps at the expense of unitarity.…”

“…These exploit the linearity (in space) of the time-dependent component of the potential and avoid discretisation of the integrals appearing in the Magnus expansion. While the sampling of potentials on an equispaced (temporal) grid can also be achieved effectively by following the approach of Schaeffer et al [32], the approach of Iserles et al [21] was demonstrated to handle highly oscillatory potentials effectively very well.…”