  We explore the geometric interpretation of a diabatic, two-photon Raman
process as a rotation on the Bloch sphere for a pseudo-spin-1/2 system. The
spin state of a spin-1/2 quantum system can be described by a point on the
surface of the Bloch sphere, and its evolution during a Raman pulse is a
trajectory on the sphere determined by properties of the optical beams: the
pulse area, the relative intensities and phases, and the relative frequencies.
We experimentally demonstrate key features of this model with a $^{87}$Rb
spinor Bose-Einstein condensate, which allows us to examine spatially dependent
signatures of the Raman beams. The two-photon detuning allows us to precisely
control the spin density and imprinted relative phase profiles, as we show with
a coreless vortex. With this comprehensive understanding and intuitive
geometric interpretation, we use the Raman process to create and tailor as well
as study and characterize exotic topological spin textures in spinor BECs.
