Mapping the Extended Lyman-Alpha Emission within the Circumgalactic Medium of Quasars Hosted by Dusty Starbursts with CubeCarve

We present a study of extended Ly$α$ emission around four quasars hosted by dusty starbursts, which are composite systems thought to represent a transitional stage in quasar evolution. To extract faint CGM emission in the presence of bright point sources, we introduce {\it CubeCarve}, a dual-channel deconvolution algorithm that separates unresolved quasar emission from spatially extended structure. This approach enables reliable recovery of \Lya\ emission projected onto the quasar position without introducing subtraction artifacts. Using {\it CubeCarve}, we find that the \Lya\ surface brightness profiles of these systems are, on average, fainter and shallower than those of quasars of similar bolometric luminosities. We also find that the total integrated \Lya\ luminosities of the nebulae are lower in systems whose host galaxies exhibit brighter far-infrared emission. These results suggest that the CGM conditions in composite systems differ from those in the broader quasar population. Our study highlights both the physical diversity of quasar CGM environments and the effectiveness of {\it CubeCarve} for recovering diffuse emission in modern IFU datasets.