Title: Controllable generation and spatial phase-modulation of vortex beams in cascade-type atomic ensembles

Abstract: We propose a four-level cascade-type atomic system in which two reverse four-wave mixing (FWM) processes can coexist via optical transitions in distinct pathways. This enables the generation of high- or low-frequency vortex beams on demand and facilitates spatial phase modulation of the vortex beam intensity distribution. The two reverse FWM processes can yield the sum-frequency vortex beam with high frequency and the difference-frequency vortex beam with low frequency, corresponding to frequency up-conversion and down-conversion, respectively. The closed-loop configuration's phase sensitivity allows for asymmetric phase modulation of the spatial distributions of the two output vortex beams. Additionally, increasing the phase enables spatial rotation of the output beams. Our proposed scheme holds promise for potential applications in quantum nonlinear optics and vortex exchange.