We investigate the steady-state spin diffusion for ultracold spin-orbit-coupled 40K gas by the kinetic spin Bloch equation approach both analytically and numerically. Four configurations, i.e., the spin diffusions along two specific directions with the spin polarization perpendicular (transverse configuration) and parallel (longitudinal configuration) to the effective Zeeman field are studied. It is found that the behaviors of the steady-state spin diffusion for the four configurations are very different, which are determined by three characteristic lengths the mean free path lτ, the Zeeman oscillation length lΩ, and the spin-orbit coupling oscillation length lα. It is analytically revealed and numerically confirmed that by tuning the scattering strength, the system can be divided into five regimes I, weak scattering regime (lτ≳lΩ,lα); II, Zeeman field-dominated moderate scattering regime (lΩ≪lτ≪lα); III, spin-orbit coupling-dominated moderate scattering regime (lα≪lτ≪lΩ); IV, relatively strong scattering regime (lcτ≪lτ≪lΩ,lα); V, strong scattering regime (lτ≪lΩ,lα,lcτ), with lcτ representing the crossover length between the relatively strong and strong scattering regimes. In different regimes, the behaviors of the spacial evolution of the steady-state spin polarization are very rich, showing different dependencies on the scattering strength, Zeeman field and spin-orbit coupling strength. The rich behaviors of the spin diffusions in different regimes are hard to understand in the framework of the simple drift-diffusion model or the direct inhomogeneous broadening picture in the literature. However, almost all these rich behaviors can be well understood by means of our modified drift-diffusion model and/or modified inhomogeneous broadening picture. Specifically, several anomalous features of the spin diffusion are revealed, which are in contrast to those obtained from both the simple drift-diffusion model and the direct inhomogeneous broadening picture.
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