Explicit schemes for time propagating many-body wave functions

Abstract

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.

Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit

“…It has been a crucial tool in solving the soft Coulomb problem because, as β gets small, the function is very stiff near the origin demanding a very small step size, too small to be used for all elements. Although, the formal 053319-2 definition of the stiffness of a DE is not clearly established, it is generally known to be based on a first-order Taylor series expansion [10,12,13]. Our algorithm essentially extends this idea to higher order and addresses a similar concept in a more practical setting to estimate element size.…”

Dissipative dynamics of a driven qubit: Interplay between nonadiabatic dynamics and noise effects from the weak to strong coupling regime

“…This task can be performed by employing a discretization of the bath modes entering in Eq. (3), a suitable truncation scheme of the bath Hilbert space, followed by the application of SIL method [40][41][42]. The discretization of the bath modes can be performed by choosing a density of states ρ(ω) and by fixing the total number of bosonic modes M in the range [0, 2ω c ]; here we adopt an exponentially decreasing density of states…”

“…We employ an exact numerical approach based on a truncation scheme of the bath Hilbert space and on the Short Iterative Lanczos (SIL) diagonalization [35][36][37][38][39][40][41], which allows us to follow the dynamics of the observables of both the qubit system and the environment, without the need of tracing out the bath degrees of freedom. This method has proved useful in reproducing the correct physical behavior of the SBM in the weak coupling regime [42], and we show how the inclusion of higher order excitation processes in the physical description can noticeably widen the range of coupling strengths to be investigated, allowing us to describe the physics from intermediate to strong coupling regime where no analytical scheme is known to hold.…”

The Explicit Fatunla’s Method for First-Order Stiff Systems of Scalar Ordinary Differential Equations: Application to Robertson Problem

“…The Explicit Fatunla's method (EFM) has been used by Frapiccini et al [20] to solve numerically the time-dependent Schrödinger equation describing physical processes whose complexity requires the use of state-of-the-art methods.…”