Generalized Gross-Pitaevskii equation adapted to the $U(5)\supset SO(5)\supset SO(3)$ symmetry for spin-2 condensates

TL;DR

This paper derives and solves a generalized Gross-Pitaevskii equation for spin-2 condensates with specific symmetries, analyzing ground state properties, stability factors, and relating observable quantities to experimental parameters.

Contribution

It introduces a generalized equation tailored to $U(5)\supset SO(5)\supset SO(3)$ symmetry and provides a detailed analysis of ground state degeneracy, stability, and observable relations for spin-2 condensates.

Findings

Ground state degeneracy and stability depend on interaction parameters and magnetization.

Different phases (ferro-, polar, cyclic) can coexist or mix depending on conditions.

Derived formulas relate root mean square radius to particle number, trap frequency, and interaction features.

Abstract

A generalized Gross-Pitaevskii equation adapted to the $U(5)\supset SO(5)\supset SO(3)$ symmetry has been derived and solved for the spin-2 condensates. The spin-textile and the degeneracy of the ground state (g.s.) together with the factors affecting the stability of the g.s., such as the gap and the level density in the neighborhood of the g.s., have been studied. Based on a rigorous treatment of the spin-degrees of freedom, the spin-textiles can be understood in a $N$-body language. In addition to the ferro-, polar, and cyclic phases, the g,s, might in a mixture of them when $0< M< 2N$ ($M$ is the total magnetization). The great difference in the stability and degeneracy of the g.s. caused by varying $\varphi $ (which marks the features of the interaction) and $M$ is notable. Since the root mean square radius $R_{rms}$ is an observable, efforts have been made to derive a set of formulae to relate $R_{rms}$ and $% N$, $\omega $(frequency of the trap), and $\varphi $. These formulae provide a way to check the theories with experimental data.