# Immiscible and miscible states in binary condensates in the ring geometry

## TL;DR

This paper investigates the existence and stability of mixed and demixed binary Bose-Einstein condensate states in ring geometries, analyzing their stability through eigenvalue spectra and simulations across 1D and 2D configurations.

## Contribution

It provides a comprehensive analysis of stability for miscible and immiscible states in ring-shaped condensates, including analytical and numerical results for various vorticity configurations.

## Key findings

- Stable radially-demixed states can carry arbitrary vorticity.
- Increasing vorticity can enhance the stability of certain states.
- Demixed states include single, double, and triple-peak structures.

## Abstract

We report detailed investigation of the existence and stability of mixed and demixed modes in binary atomic Bose-Einstein condensates with repulsive interactions in a ring-trap geometry. The stability of such states is examined through eigenvalue spectra for small perturbations, produced by the Bogoliubov-de Gennes equations, and directly verified by simulations based on the coupled Gross-Pitaevskii equations, varying inter- and intra-species scattering lengths so as to probe the entire range of miscibility-immiscibility transitions. In the limit of the one-dimensional (1D) ring, i.e., a very narrow one, stability of mixed states is studied analytically, including hidden-vorticity (HV) modes, i.e., those with opposite vorticities of the two components and zero total angular momentum. The consideration of demixed 1D states reveals, in addition to stable composite single-peak structures, double- and triple-peak ones, above a certain particle-number threshold. In the 2D annular geometry, stable demixed states exist both in radial and azimuthal configurations. We find that stable radially-demixed states can carry arbitrary vorticity and, counter-intuitively, the increase of the vorticity enhances stability of such states, while unstable ones evolve into randomly oscillating angular demixed modes. The consideration of HV states in the 2D geometry expands the stability range of radially-demixed states.