# Bose-Einstein condensation in spherically symmetric traps

## TL;DR

This paper provides a detailed analysis of Bose-Einstein condensation in spherical traps, comparing numerical and semi-classical methods to determine critical temperatures and exploring the effects of geometry and dimensionality on the condensate.

## Contribution

It introduces a comprehensive calculation approach for critical temperatures in spherical traps and examines the influence of geometry and dimensionality, bridging theory with experimental feasibility.

## Key findings

- Numerical and semi-classical methods yield comparable critical temperatures.
- Geometry of the trap significantly affects the density of states and eigenstates.
- Results are compatible with current cold atom experimental conditions.

## Abstract

We present a pedagogical introduction to Bose-Einstein condensation in traps with spherical symmetry, namely the spherical box and the thick shell, sometimes called bubble trap. In order to obtain the critical temperature for Bose-Einstein condensation, we describe how to calculate the cumulative state number and density of states in these geometries, using numerical and analytical (semi-classical) approaches. The differences in the results of both methods are a manifestation of Weyl's theorem, i.e., they reveal how the geometry of the trap (boundary condition) affects the number of the eigenstates counted. Using the same calculation procedure, we analyzed the impact of going from three-dimensions to two-dimensions, as we move from a thick shell to a two-dimensional shell. The temperature range we obtained, for most commonly used atomic species and reasonable confinement volumes, is compatible with current cold atom experiments, which demonstrates that these trapping potentials may be employed in experiments.