Zero temperature geometric spin dephasing on a ring in presence of an Ohmic environment

We study zero temperature spin dynamics of a particle confined to a ring in presence of spin orbit coupling and Ohmic electromagnetic fluctuations. We show that the dynamics of the angular position $\theta(t)$ are decoupled from the spin dynamics and that the latter is mapped to certain correlations of a spinless particle. We find that the spin correlations in the $z$ direction (perpendicular to the ring) are finite at long times, i.e. do not dephase. The parallel (in plane) components for spin $\half$ do not dephase at weak dissipation but they probably decay as a power law with time at strong dissipation.