Universality of the energy spectrum for two interacting harmonically trapped ultra-cold atoms in one and two dimensions

TL;DR

This paper demonstrates the universality of the energy spectrum for two ultra-cold bosonic atoms in harmonic traps in one and two dimensions, showing that low-energy physics is independent of potential details when the range is small.

Contribution

It provides an exact analysis of the energy spectrum for two interacting ultra-cold atoms in harmonic traps across one and two dimensions, highlighting universality and regularization methods.

Findings

Energy spectrum is universal in 1D when potential range is much smaller than oscillator length.

Low-energy physics in 2D is also universal, with a simple regularized pseudopotential method.

Regularization is unnecessary in the zero-range limit in 1D, and carefully handled in 2D.

Abstract

Motivated by the recent article of P. Shea {\it et al.} [Am. J. Phys. {\bf 77} (6), 2009] we examine the exactly solvable problem of two harmonically trapped ultra-cold bosonic atoms interacting {\it via} a short range potential in one and two dimensions. A straightforward application in one dimension shows that the energy spectrum is universal, provided that the range of the potential is much smaller than the oscillator length, in addition to clearly illustrating why regularization is not required in the limit of zero range. The two dimensional problem is less trivial, requiring a more careful treatment as compared to the one dimensional case. Our two dimensional analysis likewise reveals that the low-energy physics is also universal, in addition to providing a simple method for obtaining the appropriately regularized two dimensional pseudopotential.