Quasi-one-dimensional approximation for Bose-Einstein condensates transversely trapped by a funnel potential

TL;DR

This paper derives an effective one-dimensional equation for Bose-Einstein condensates in a funnel-shaped trap, accurately capturing their static and dynamic behavior despite the potential's singularity.

Contribution

It introduces a novel 1D nonpolynomial Schrödinger equation for BECs in a funnel potential, validated against full 3D simulations.

Findings

The 1D model accurately predicts static properties of BECs.

The 1D model reliably describes dynamical evolution of BECs.

Wave functions remain regular despite the singular potential.

Abstract

Starting from the standard three-dimensional (3D) Gross-Pitaevskii equation (GPE) and using a variational approximation, we derive an effective one-dimensional nonpolynomial Schr\"odinger equation (1D-NPSE) governing the axial dynamics of atomic Bose-Einstein condensates (BECs) under the action of a singular but physically relevant funnel-shaped transverse trap, i.e., an attractive 2D potential $\sim-1/r$ (where $r$ is the radial coordinate in the transverse plane), in combination with the repulsive self-interaction. Wave functions of the trapped BEC are regular, in spite of the potential's singularity. The model applies to a condensate of particles (small molecules) carrying a permanent electric dipole moment in the field of a uniformly charged axial thread, as well as to a quantum gas of magnetic atoms pulled by an axial electric current. By means of numerical simulations, we verify that the effective 1D-NPSE provides accurate static and dynamical results, in comparison to the full 3D GPE, for both repulsive and attractive signs of the intrinsic nonlinearity.