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arXiv:0901.1852 · Physica E Low-dimensional Systems and Nanostructures

Interacting bosons beyond the Gross–Pitaevskii mean field

Martín Heimsoth, M. Bönitz

Published Jun 25, 2009• 12 citations
arXivHTMLPDF
DOI
10.1016/j.physe.2009.06.040

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Fragmentation of a two-boson system in an external potential with contact interaction (i.e., $\ensuremath{\delta}$-function interaction) is studied using a full variational treatment. The results are compared to those obtained by the standard mean-field method (Gross-Pitaevskii equation) and the more recently developed best mean-field method. The numerically exact calculations show a transition to fragmentation as a function of the boson-boson interaction strength. The Gross-Pitaevskii solution cannot describe fragmentation. The respective energy exhibits a bifurcation which can be viewed as a hint that fragmentation may take place in reality. The best mean-field approach, on the other hand, can predict the fragmentation of the system and give the correct physical description of the fragmented state. The Gross-Pitaevskii energy diverges as the boson-boson interaction goes to infinity, while the exact and the best mean-field energies saturate.

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Strongly-interacting bosons in two-dimensional harmonic traps are described through breaking of rotational symmetry at the Hartree-Fock level and subsequent symmetry restoration via projection techniques, thus incorporating correlations beyond the Gross-Pitaevskii (GP) solution. The bosons localize and form polygonal-ringlike crystalline patterns, both for a repulsive contact potential and a Coulomb interaction, as revealed via conditional-probability-distribution analysis. For neutral bosons, the total energy of the crystalline phase saturates in contrast to the GP solution, and its spatial extent becomes smaller than that of the GP condensate. For charged bosons, the total energy and dimensions approach the values of classical pointlike charges in their equilibrium configuration.
In this chapter we consider the quantum statistical mechanics of identical nonrelativistic bosons by introducing the formalism of functional integration for the bosonic field. We then derive the equation of state of non-interacting and weakly-interacting massive bosons taking into account beyond-mean-field regularized Gaussian fluctuations. We also analyze various properties of the mean-field time-dependent Gross-Pitaevskii equation. In particular, we show that this equation is equivalent to the zero-temperature equations of superfluid hydrodynamics proposed in 1949 by Landau. Finally, we discuss topological and solitonic solutions of the Gross-Pitaevskii equation.
Fragmentation of a two-boson system in an external potential with contact interaction (i.e., $\ensuremath{\delta}$-function interaction) is studied using a full variational treatment. The results are compared to those obtained by the standard mean-field method (Gross-Pitaevskii equation) and the more recently developed best mean-field method. The numerically exact calculations show a transition to fragmentation as a function of the boson-boson interaction strength. The Gross-Pitaevskii solution cannot describe fragmentation. The respective energy exhibits a bifurcation which can be viewed as a hint that fragmentation may take place in reality. The best mean-field approach, on the other hand, can predict the fragmentation of the system and give the correct physical description of the fragmented state. The Gross-Pitaevskii energy diverges as the boson-boson interaction goes to infinity, while the exact and the best mean-field energies saturate.