Rings and boxes in dissipative environments

We study a particle on a ring in presence of a dissipative Caldeira-Leggett environment and derive its response to a DC field [1]. We find, through a 2-loop renormalization group analysis, that a large dissipation parameter $\eta$ flows to a fixed point $\eta^R=\hbar/2\pi$. We also reexamine the mapping of this problem to that of the Coulomb box and show that the relaxation resistance, of recent interest, has a certain average that is quantized for $\eta>\eta^R$, leading to quantized noise. We propose a box experiment to detect this noise. When the particle carries a spin with spin-orbit interactions [2] we find that the spin correlations in the direction perpendicular to the ring are finite at long times, i.e. do not dephase, while the parallel components may decay as a power law at strong dissipation. \\[4pt] [1] Y. Etzioni, B. Horovitz and P. Le Doussal, Phys. Rev. Lett. {\bf 106}, 166803 (2011). \\[0pt] [2] G. Zar\'{a}nd, G. T. Zim\'{a}nyi and F. Wilhelm, Phys. Rev. B{\bf 62}, 8137 (2000).