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Physical Review A

Equivalent linear two-body method for Bose-Einstein condensates in time-dependent harmonic traps

Yeong E. Kim, Alexander L. Zubarev

Published Nov 11, 2002• 14 citations
Publisher

Abstract

The recently developed time-independent effective linear two-body method [J. Phys. B: At., Mol. Opt. Phys. 33, 55 (2000)] has been generalized for time-dependent traps. The method is used to describe the dynamics of trapped Bose-Einstein condensates beyond the Thomas-Fermi regime. The calculated aspect ratios after ballistic expansion are found to be in good agreement with experimental data obtained recently by G\"orlitz et al. [Phys. Rev. Lett. 87, 130402 (2001)].

DOI
10.1103/physreva.66.053602

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The recently developed time-independent effective linear two-body method [J. Phys. B: At., Mol. Opt. Phys. 33, 55 (2000)] has been generalized for time-dependent traps. The method is used to describe the dynamics of trapped Bose-Einstein condensates beyond the Thomas-Fermi regime. The calculated aspect ratios after ballistic expansion are found to be in good agreement with experimental data obtained recently by G"orlitz et al. [Phys. Rev. Lett. 87, 130402 (2001)].
We present an analytical approximation for nonlinear dynamics of trapped Bose-Einstein condensates. The approximation is a substantial improvement over the Thomas-Fermi approximation and is shown to be applicable for systems with a rather small number of atoms N. The calculated aspect ratios after ballistic expansion are found to be in good agreement with those observed in recent experiments. The approximation is useful in providing a simple and easy quantitative computational tool for the analysis of experimental data.
We present an analytical approximation for nonlinear dynamics of trapped Bose-co ndensed gases. The new approximation is a substantial improvement over the Thomas-Fermi approximation and is shown to be applicable for systems with a rather small number of atoms NNN. The calculated aspect ratios after ballistic expansion are found to be in good agreement with those observed in recent experiments.
We derive a unified set of scaling equations for Bose-Einstein condensates in time-dependent harmonic traps, connecting the weakly interacting Gaussian regime to the strongly interacting Thomas-Fermi regime. The exact Gross-Pitaevskii ground-state density is taken as a fixed profile in rescaled coordinates, with its dynamics described by three scaling factors corresponding to compression or expansion along the three spatial directions. The resulting equations contain no adjustable parameter, since their coefficients are determined once from the initial state. They recover analytically the Gaussian variational and Thomas-Fermi scaling equations in their respective limits and satisfy an axis-resolved virial theorem. The approach also yields the global phase and the low-lying collective-mode frequencies of the condensate. We benchmark the model against three-dimensional Gross-Pitaevskii simulations and against measured expansion energies. The model remains accurate across all interaction regimes, in strongly anisotropic traps, and along time-dependent sequences including a relatively fast quench, up to the point where the underlying frozen-profile hypothesis, shared by all three scaling approaches, breaks down.
Linear limit continuation was recently developed as a systematic and effective method for constructing numerically exact solitary waves from their respective linear limits. In this work, we apply the technique to two typical anisotropic harmonic traps in two-dimensional Bose-Einstein condensates to further establish the method and also to find more solitary waves. Many wave patterns are identified in the near-linear regime and they are subsequently continued into the Thomas-Fermi regime, and then they are further continued into the isotropic trap if possible. Finally, the parametric connectivity of the pertinent solitary waves is also discussed.
A method is proposed to design the time dependence of the trap frequency and achieve in a short time an adiabatic-like (frictionless) evolution of Bose-Einstein condensates governed by the Gross-Pitaevskii equation. Different cases depending on the effective dimension of the trap and the interaction regimes are considered. 2D traps are particularly suitable as the method can be applied without the need to impose any additional time-dependent change in the strength of the interatomic interaction or a Thomas-Fermi regime as it occurs for 1D and 3D traps.
A method is proposed to design the time dependence of the trap frequency and achieve in a short time an adiabatic-like (frictionless) evolution of Bose-Einstein condensates governed by the Gross-Pitaevskii equation.Different cases depending on the effective dimension of the trap and the interaction regimes are considered.2D traps are particularly suitable as the method can be applied without the need to impose any additional time-dependent change in the strength of the interatomic interaction or a Thomas-Fermi regime as it occurs for 1D and 3D traps.