Adaptive grid refinement for a model of two confined and interacting atoms

Authors:
William F. Mitchell;Eite Tiesinga
Affiliations:
Mathematical and Computational Sciences Division, 100 Bureau Drive Stop 8910, National Institute of Standards and Technology, Gaithersburg, MD 20899-8910, USA;Atomic Physics Division, 100 Bureau Drive Stop 8423, National Institute of Standards and Technology, Gaithersburg, MD 20899-8423, USA
Venue:
Applied Numerical Mathematics - Adaptive methods for partial differential equations and large-scale computation
Year:
2005

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Abstract

We have applied adaptive grid refinement to solve a two-dimensional Schrodinger equation in order to study the feasibility of a quantum computer based on extremely-cold neutral alkali-metal atoms. Qubits are implemented as motional states of an atom trapped in a single well of an optical lattice of counter-propagating laser beams. Quantum gates are constructed by bringing two atoms together in a single well leaving the interaction between the atoms to cause entanglement. For special geometries of the optical lattices and thus shape of the wells, quantifying the entanglement reduces to solving for selected eigenfunctions of a Schrodinger equation that contains a two-dimensional Laplacian, a trapping potential that describes the optical well, and a short-ranged interaction potential. The desired eigenfunctions correspond to eigenvalues that are deep in the interior of the spectrum where the trapping potential becomes significant. The spatial range of the interaction potential is three orders of magnitude smaller than the spatial range of the trapping potential, necessitating the use of adaptive grid refinement.