Unambiguous identification of dipole and breathing modes in the vortex necklace phase

Determine whether the dipole and breathing collective modes can be unambiguously identified in the vortex necklace phase of a spin-orbital-angular-momentum-coupled spin-1 Bose-Einstein condensate with angular momentum transfer l = 4ħ in a quasi-2D harmonic trap and antiferromagnetic interactions, given that the expectation values of standard probing observables (x, y, x^2, y^2, and their spin-weighted counterparts) do not exhibit single dominant oscillation frequencies and show multiple peaks in their Fourier spectra under the perturbative identification approach described.

Background

The paper studies collective excitations in a spin-orbital-angular-momentum (SOAM)-coupled spin-1 Bose-Einstein condensate featuring three ground-state phases: annular stripe (AS), vortex necklace (VN), and zero angular momentum (ZAM). Using Bogoliubov analysis and a perturbative identification method based on the time evolution of expectation values of selected observables, the authors successfully identify low-lying dipole and breathing modes in the AS and ZAM phases.

For the VN phase, however, the time-dependent expectation values of the chosen observables lack single dominant frequencies and exhibit multiple Fourier peaks, which prevents a clear assignment of dipole and breathing modes. This explicit limitation leaves unresolved the question of how to unambiguously identify these modes in the VN phase of the considered SOAM-coupled spin-1 condensate.