Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems

Ofir E. Alon, Alexej I. Streltsov, and Lorenz S. Cederbaum
Phys. Rev. A 77, 033613 – Published 14 March 2008

Abstract

The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schrödinger equation of N interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or MCTDHB(M), where M specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number M of orbitals is demonstrated in a specific example of four interacting bosons in a double well.

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  • Received 8 March 2007

DOI:https://doi.org/10.1103/PhysRevA.77.033613

©2008 American Physical Society

Authors & Affiliations

Ofir E. Alon*, Alexej I. Streltsov, and Lorenz S. Cederbaum

  • Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany

  • *ofir@pci.uni-heidelberg.de
  • alexej@pci.uni-heidelberg.de
  • lorenz.cederbaum@pci.uni-heidelberg.de

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Issue

Vol. 77, Iss. 3 — March 2008

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