  The experimental realization of emergent spin-orbit coupling through
laser-induced Raman transitions in ultracold atoms paves the way for exploring
novel superfluid physics and simulating exotic many-body phenomena. A recent
proposal with the use of Laguerre-Gaussian lasers enables another fundamental
type of coupling between spin and orbital angular momentum (SOAM) in ultracold
atoms. We hereby study quantum phases of a realistic Bose-Einstein condensate
(BEC) with this synthetic SOAM coupling in a disk-shaped geometry, respecting
radial inhomogeneity of the Raman coupling. We find that the experimental
system naturally resides in a strongly interacting regime in which the phase
diagram significantly deviates from the single-particle picture. The interplay
between SOAM coupling and interaction leads to rich structures in spin-resolved
position and momentum distributions, including a stripe phase and various types
of immiscible states. Our results would provide a guide for an experimental
investigation of SOAM-coupled BECs.
