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arXiv · 1803.11026

Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons

Lea Boßmann·Stefan Teufel
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Abstract

We consider the dynamics of $N$ interacting bosons initially forming a Bose-Einstein condensate. Due to an external trapping potential, the bosons are strongly confined in two dimensions, where the transverse extension of the trap is of order $\varepsilon$. The non-negative interaction potential is scaled such that its range and its scattering length are both of order $(N/\varepsilon^2)^{-1}$, corresponding to the Gross-Pitaevskii scaling of a dilute Bose gas. We show that in the simultaneous limit $N\rightarrow\infty$ and $\varepsilon\rightarrow 0$, the dynamics preserve condensation and the time evolution is asymptotically described by a Gross-Pitaevskii equation in one dimension. The strength of the nonlinearity is given by the scattering length of the unscaled interaction, multiplied with a factor depending on the shape of the confining potential. For our analysis, we adapt a method by Pickl to the problem with dimensional reduction and rely on the derivation of the one-dimensional NLS equation for interactions with softer scaling behaviour.

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Lea Boßmann, Stefan Teufel. 2018-03-29. Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons. https://doi.org/10.1007/s00023-018-0738-7

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