A Parabolic Model for the Dimple Potentials

TL;DR

This paper introduces a truncated parabolic potential as a more accurate and practical model for dimple potentials in Bose-Einstein condensates, demonstrating its equivalence to the Dirac delta potential in key quantum properties.

Contribution

It proposes the truncated parabolic potential as a superior alternative to the Dirac delta for modeling dimple potentials, with theoretical validation of its spectral and scattering similarities.

Findings

Truncated parabolic function approximates the Dirac delta function.

The potential yields identical bound states, tunneling, and reflection amplitudes as the delta potential.

It offers a more realistic model for dimple potentials in Bose-Einstein condensates.

Abstract

We study truncated parabolic function and demonstrate that it is a representation of the Dirac delta function. We also show that the truncated parabolic function, used as a potential in the Schr\"{o}dinger equation, has the same bound state spectrum, tunneling and reflection amplitudes with the Dirac delta potential as the width of the parabola approximates to zero. Dirac delta potential is used to model dimple potentials which are utilized to increase the phase-space density of a Bose-Einstein condensate in a harmonic trap. We show that harmonic trap with a delta function at the origin is a limit case of the harmonic trap with a symmetric truncated parabolic potential around the origin. Therefore, we propose that the truncated parabolic potential is a better candidate for modeling the dimple potentials.