High-order commutator-free exponential time-propagation of driven quantum systems

Details High-order commutator-free exponential time-propagation of driven quantum systems High-order commutator-free exponential time-propagation of driven quantum systems

2011-02-24

arxiv.org/abs/1102.5071v2

Journal of Computational Physics 230, 5930 (2011)

Mathematics

Numerical Analysis

32 pages, 12 figures. Final version as published in J. Comp. Phys

Scientific paper

10.1016/j.jcp.2011.04.006

We discuss the numerical solution of the Schr\"odinger equation with a time-dependent Hamilton operator using commutator-free time-propagators. These propagators are constructed as products of exponentials of simple weighted sums of the Hamilton operator. Owing to their exponential form they strictly preserve the unitarity of time-propagation. The absence of commutators or other computationally involved operations allows for straightforward implementation and application also to large scale and sparse matrix problems. We explain the derivation of commutator-free exponential time-propagators in the context of the Magnus expansion, and provide optimized propagators up to order eight. An extensive theoretical error analysis is presented together with practical efficiency tests for different problems. Issues of practical implementation, in particular the use of the Krylov technique for the calculation of exponentials, are discussed. We demonstrate for two advanced examples, the hydrogen atom in an electric field and pumped systems of multiple interacting two-level systems or spins that this approach enables fast and accurate computations.

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