# Finite-size effects in the dynamics of few bosons in a ring potential

## TL;DR

This paper investigates how finite-size effects influence the dynamics of a small number of ultra-cold bosons in a ring potential, revealing multiple characteristic timescales and comparing mean-field and exact solutions.

## Contribution

It identifies three key timescales in the evolution of few-boson systems and compares mean-field predictions with exact solutions, highlighting finite-size effects.

## Key findings

- Three characteristic timescales are identified: rotation, decay, and collapse-revival.
- Finite-size effects cause deviations from mean-field dynamics in small systems.
- Exact solutions show similar dynamical features to mean-field predictions for stirred systems.

## Abstract

We study the temporal evolution of a small number $N$ of ultra-cold bosonic atoms confined in a ring potential. Assuming that initially the system is in a solitary-wave solution of the corresponding mean-field problem, we identify significant differences in the time evolution of the density distribution of the atoms when it instead is evaluated with the many-body Schr\"odinger equation. Three characteristic timescales are derived: the first is the period of rotation of the wave around the ring, the second is associated with a "decay" of the density variation, and the third is associated with periodic "collapses" and "revivals" of the density variations, with a factor of $\sqrt N$ separating each of them. The last two timescales tend to infinity in the appropriate limit of large $N$, in agreement with the mean-field approximation. These findings are based on the assumption of the initial state being a mean-field state. We confirm this behavior by comparison to the exact solutions for a few-body system stirred by an external potential. We find that the exact solutions of the driven system exhibit similar dynamical features.