Ballistic spin dynamics in Rashba spin-orbit coupled systems

We theoretically study a time evolution of a transient spin grating in a two-dimensional electron gas with a Rashba spin- orbit coupling in a ballistic regime. We calculate the out-of- plane spin Fourier components, which decays with a rapid oscillation. We investigate their lifetime and frequency and analyze their dependence on the grating vector $q$ and the spin- orbit coupling constant $\alpha$. The frequency is proportional to $q$ and $\alpha$, and we find that the lifetime of the spin polarization is proportional to the inverse square of $\alpha$ for the small $q$ limit. Finally we compare our calculation with the case for a diffusive regime. We show that the lifetime of the spin polarization is longer when the grating vector becomes smaller, which is in contrast with the diffusive case.