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

TL;DR

This paper derives a one-dimensional Gross-Pitaevskii equation from the three-dimensional quantum dynamics of strongly confined bosons, demonstrating that the condensate's evolution can be effectively described in reduced dimensions.

Contribution

It extends the derivation of the 1D Gross-Pitaevskii equation to strongly confined bosons with a specific scaling of interactions, adapting Pickl's method for dimensional reduction.

Findings

Condensation is preserved in the limit of large N and vanishing confinement width.

The dynamics are asymptotically governed by a 1D Gross-Pitaevskii equation.

The nonlinearity strength depends on the unscaled scattering length and trap shape.

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.