Asymptotic expressions for the hyperfine populations in the ground state of spin-1 condensates against a magnetic field

TL;DR

This paper derives and validates asymptotic analytical expressions for hyperfine population variations in spin-1 condensates under magnetic fields, covering different regimes for Rb and Na atoms.

Contribution

It provides the first comprehensive set of asymptotic formulas for hyperfine populations in spin-1 condensates across all magnetic field strengths.

Findings

Analytical expressions match numerical results within specified B ranges.

For Rb, formulas cover all B from 0 to infinity.

For Na, formulas are valid only at very weak or strong B.

Abstract

Based on the perturbation theory up to the second order, analytical asymptotic expressions for the variation of the population of hyperfine component $\mu=0 $ particles in the ground state of spin-1 condensates against a magnetic field $B$ has been derived. The ranges of $B$ in which the asymptotic expressions are applicable have been clarified via a comparison of the numerical results from the analytical expressions and from a diagonalization of the Hamiltonian in a complete spin-space. It was found that, For $^{87}$Rb, the two analytical expressions, one for a weak and the other one for a strong field, together cover the whole range of $B$ from 0 to infinite. For Na, the analytical expressions are valid only if $B$ is very weak or sufficiently strong.