Blurred path-spin entanglement in Stern-Gerlach apparatus: interplay between magnetic inhomogeneity and Larmor precession

We argue that the non-adiabatic evolution of spin states in Stern-Gerlach apparatus can blur the manifestation of path spin entanglement. This fact introduces a contradiction to the usual perception of spin measurement even in a formally ideal situation. Through quantitative calculation, we have pointed out the specific reason behind the breakdown of adiabatic evolution to be the spatial inhomogeneity of applied magnetic field. The angle $\theta$ between the $z$ component of magnetic moment of the particle and applied magnetic field is also a determining factor for the category of evolution of spin states. Adiabaticity always can be restored by choosing a sufficiently small value of $\theta$. Considering azimuthal inhomogeneity, we have examined the nature of path spin entanglement in the context of Stern-Gerlach experiment.