  We consider a trapped atomic system in the presence of spatially varying
laser fields. The laser-atom interaction generates a pseudospin degree of
freedom (referred to simply as spin) and leads to an effective spin-orbit
coupling for the fermions in the trap. Reflections of the fermions from the
trap boundaries provide a physical mechanism for effective momentum relaxation
and non-trivial spin dynamics due to the emergent spin-orbit coupling. We
explicitly consider evolution of an initially spin-polarized Fermi gas in a
two-dimensional harmonic trap and derive non-equilibrium behavior of the spin
polarization. It shows periodic echoes with a frequency equal to the harmonic
trapping frequency. Perturbations, such as an asymmetry of the trap, lead to
the suppression of the spin echo amplitudes. We discuss a possible experimental
setup to observe spin dynamics and provide numerical estimates of relevant
parameters.
