Matched optical vortices of slow light using a tripod coherently prepared scheme

This paper explores the propagation of optical vortices of slow light inside a four-level tripod atomic light-matter coupling system. Initially, the system is prepared in a coherent superposition of two out of the three lower levels by, for example, coherent population trapping, while the third lower state remains unoccupied. The unoccupied state is coupled to a strong control laser field with a constant Rabi frequency, lacking orbital angular momentum (OAM). Simultaneously, one of the remaining lower states interacts with a weak vortex beam. The third lower state has no initial coupling to any field. This arrangement effectively closes the level transitions, resulting in a phase-dependent configuration. By solving the Maxwell-Schr\"odinger equations, we provide analytical evidence that the application of a strong control field can generate an additional optical vortex of slow light. This vortex possesses the same OAM as the incident vortex beam. Furthermore, we explore the matching of optical vortices at different propagation distances, contingent upon the intensity of the control field. Additionally, our analysis extends to a more complex five-level tripod and $\mathrm{\ensuremath{\Lambda}}$ scheme, where we introduce two additional strong control fields to prepare the atoms in a coherent superposition of two lower levels.