A bloch decomposition-based split-step pseudospectral method for quantum dynamics with periodic potentials

Zhongyi Huang*, Shi Jin, Peter Markowich, Christof Sparber

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

We present a new numerical method for accurate computations of solutions to (linear) one-dimensional Schrödinger equations with periodic potentials. This is a prominent model in solid state physics where we also allow for perturbations by nonperiodic potentials describing external electric fields. Our approach is based on the classical Bloch decomposition method, which allows us to diagonalize the periodic part of the Hamiltonian operator. Hence, the dominant effects from dispersion and periodic lattice potential are computed together, while the nonperiodic potential acts only as a perturbation. Because the split-step communicator error between the periodic and nonperiodic parts is relatively small, the step size can be chosen substantially larger than for the traditional splitting of the dispersion and potential operators. Indeed it is shown by the given examples that our method is unconditionally stable and more efficient than the traditional split-step pseudospectral schemes. To this end a particular focus is on the semiclassical regime, where the new algorithm naturally incorporates the adiabatic splitting of slow and fast degrees of freedom.

Original languageEnglish (US)
Pages (from-to)515-538
Number of pages24
JournalSIAM Journal on Scientific Computing
Volume29
Issue number2
DOIs
StatePublished - Dec 1 2007

Keywords

  • Bloch decomposition
  • Lattice potential
  • Schrödinger equation
  • Semiclassical asymptotics
  • Time-splitting spectral method

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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