Lower bound for the population of hyperfine component $\mu =0$ particles in the ground state of spin-1 condensates

TL;DR

This paper derives an analytical lower bound for the number of hyperfine $ ext{μ}=0$ particles in the ground state of spin-1 condensates under magnetic fields, conserving magnetization, and validates its accuracy through numerical examples.

Contribution

It provides the first analytical expression for the lower bound of hyperfine $ ext{μ}=0$ particles in spin-1 condensates with conserved magnetization, applicable across a broad parameter range.

Findings

The analytical lower bound closely matches the actual $ ho_0$ in many parameter regimes.

The derived expression allows simple evaluation of $ ho_0$ without complex calculations.

Numerical examples confirm the applicability of the analytical formula.

Abstract

An analytical expression for the lower bound of the average number of hyperfine component $\mu =0$ particles in the ground state of spin-1 condensates (denoted as $\overset{\_\_}{\rho _{0}}$) under a magnetic field has been derived. In the derivation the total magnetization $M$ is kept rigorously conserved. Numerical examples are given to show the applicability of the analytical expression. It was found that, in a broad domain of parameters specified in the paper, the lower bound is very close to the actual $\overset{\_\_}{\rho _{0}}$. Thereby, in this domain, $\overset{\_\_}{% \rho _{0}}$ can be directly evaluated simply by using the analytical expression.