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Theoretical Concepts and Numerical Methods

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Tunneling Dynamics in Open Ultracold Bosonic Systems

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Abstract

The scope of this chapter is to equip the reader with the necessary conventions and tools from the second quantization formalism. Quantities of interest like reduced density matrices and normalized correlation functions are defined. The theoretical framework of the methods used throughout the remainder of the book is outlined. In detail, the derivation of the time-dependent Gross–Pitaevskii equation, the time-dependent multi-orbital mean-field equations of motion as well as the multiconfigurational time-dependent Hartree method for bosons are given. Furthermore, the time-evolved block decimation for the Bose–Hubbard Hamiltonian and the discrete non-linear Schrödinger equation are introduced, as they are the subject of a later comparison.

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Notes

  1. 1.

    For didactical reasons, the time-dependencies are omitted in various places in this section.

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  1. Condensed Matter Theory and Quantum Computing Group, University of Basel, Basel, Switzerland

    Axel U. J. Lode

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Lode, A.U.J. (2015). Theoretical Concepts and Numerical Methods. In: Tunneling Dynamics in Open Ultracold Bosonic Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-07085-8_2

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