Spin 1 condensates at thermal equilibrium : a $SU(3)$ coherent state approach

TL;DR

This paper develops an $SU(3)$ coherent state framework to analyze spin 1 Bose-Einstein condensates at thermal equilibrium, accounting for magnetization conservation and finite particle number effects.

Contribution

It introduces a novel $SU(3)$ coherent state approach to describe thermal equilibrium states of spin 1 condensates, including magnetization conservation and finite-size fluctuations.

Findings

Partition function constructed for large N

Probability distribution of Zeeman populations derived

Finite N fluctuations of collective spin analyzed

Abstract

We propose a theoretical framework based on $SU(3)$ coherent states as a convenient tool to describe the collective state of a Bose-Einstein condensate of spin 1 atoms at thermal equilibrium. We work within the single-mode approximation, which assumes that all atoms condense in the same spatial mode. In this system, the magnetization $m_z$ is conserved to a very good approximation. This conservation law is included by introducing a prior distribution for $m_z$ and constructing a generalized statistical ensemble that preserves its first moments. In the limit of large particle numbers, we construct the partition function at thermal equilibrium and use it to compute various quantities of experimental interest, such as the probability distribution function and moments of the population in each Zeeman state. When $N$ is large but finite (as in typical experiments, where $N\sim 10^3-10^5$), we find that fluctuations of the collective spin can be important.