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arXiv:cond-mat/0211434 · Journal of Physics B Atomic Molecular and Optical Physics

Solutions of the Gross-Pitaevskii equation in two dimensions

M D Lee, S. A. Morgan

Published Jun 25, 2002• 5 citations
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Abstract

In two dimensions the Gross-Pitaevskii equation for a cold, dilute gas of bosons has an energy-dependent coupling parameter describing particle interactions. We present numerical solutions of this equation for bosons in harmonic traps and show that the results can be quite sensitive to the precise form of the coupling parameter that is used.

DOI
10.1088/0953-4075/35/13/313

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In two dimensions the Gross-Pitaevskii equation for a cold, dilute gas of bosons has an energy-dependent coupling parameter describing particle interactions. We present numerical solutions of this equation for bosons in harmonic traps and show that the results can be quite sensitive to the precise form of the coupling parameter that is used.
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We consider many-body effects on particle scattering in one-, two-, and three-dimensional (3D) Bose gases. We show that at T=0T=0T=0 these effects can be modeled by the simpler two-body T matrix evaluated off the energy shell. This is important in 1D and 2D because the two-body T matrix vanishes at zero energy and so mean-field effects on particle energies must be taken into account to obtain a self-consistent treatment of low-energy collisions. Using the off-shell two-body T matrix we obtain the energy and density dependence of the effective interaction in 1D and 2D and the appropriate Gross-Pitaevskii equations for these dimensions. Our results provide an alternative derivation of those of Kolomeisky and co-workers. We present numerical solutions of the Gross-Pitaevskii equation for a 2D condensate of hard-sphere bosons in a trap. We find that the interaction strength is much greater in 2D than for a 3D gas with the same hard-sphere radius. The Thomas-Fermi regime is, therefore, approached at lower condensate populations and the energy required to create vortices is lowered compared to the 3D case.
Through a suitable expansion of the Gross-Pitaevskii equation near the classical turning point, we obtain an explicit solution for the order parameter at the boundary of a trapped Bose gas interacting with repulsive forces. The kinetic energy of the system, in terms of the classical radius R and of the harmonic oscillator length aHO{\mathit{a}}_{\mathrm{HO}}aHO​, follows the law Ekin{\mathit{E}}_{\mathrm{kin}}Ekin​/N\ensuremath{\propto}${\mathit{R}}^{\mathrm{\ensuremath{-}}2}$[ln(R/aHO{\mathit{a}}_{\mathrm{HO}}aHO​)+ const], approaching, for large R, the results obtained by solving numerically the Gross-Pitaevskii equation. The occurrence of a Josephson-type current in the presence of a double trap potential is finally discussed. \textcopyright{} 1996 The American Physical Society.
A method is proposed for a self-consistent evaluation of the coupling constant in the Gross-Pitaevskii equation without involving a pseudopotential replacement. A renormalization of the coupling constant occurs due to medium effects and the trapping potential, e.g., in quasi-1D or quasi-2D systems. It is shown that a simplified version of the Hartree-Fock-Bogoliubov approximation leads to a variational problem for both the condensate and a two-body wave function describing the behavior of a pair of bosons in the Bose-Einstein condensate. The resulting coupled equations are free of unphysical divergences. Particular cases of this scheme that admit analytical estimations are considered and compared to the literature. In addition to the well-known cases of low-dimensional trapping, crossover regimes can be studied. The values of the kinetic, interaction, external, and release energies in low dimensions are also evaluated and contributions due to short-range correlations are found to be substantial.