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Motivated by the recent experiments [Chen et al., Phys. Rev. Lett 121, 113204(2018), Chen et al., Phys. Rev. Lett. 121, 250401 (2018)], we investigate thelow-lying excitation spectrum of the ground-state phases ofspin-orbital-angular-momentum-coupled (SOAM-coupled) spin-1 condensates.Atvanishing detuning, a ferromagnetic SOAM-coupled spin-1 BEC can have twoground-state phases, namely coreless and polar-core vortex states, whereas anantiferromagnetic BEC supports only polar-core vortex solution. The angularmomentum per particle, longitudinal magnetization, and excitation frequenciesdisplay discontinuities across the phase boundary between the coreless vortexand polar-core vortex phases. The low-lying excitation spectrum evaluated bysolving the Bogoliubov-de-Gennes equations is marked by avoided crossings andhence the hybridization of the spin and density channels. The spectrum isfurther confirmed by the dynamical evolution of the ground state subjected to aperturbation suitable to excite a density or a spin mode and a variationalanalysis for the density-breathing mode.

Quantum phases and spectrum of collective modes in a spin-1 BEC with spin-orbital-angular-momentum coupling