2007/03/09 by Ofir E. Alon, Alexej I. Streltsov, Lorenz S. Cederbaum · 1 citation
Physics and Astronomy · #cond-mat.other
paper · pdf · doi:10.1103/physreva.77.033613
published as Phys. Rev. A 77, 033613 (2008) · 30 pages, 2 figures
arxiv created 2007/03/09 · arxiv updated 2009/12/01
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 wavefunction is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where \it both the expansion coefficients \it and the permanents (orbitals) themselves are \it 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 non-linear integro-differential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multi-configurational 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.