Dimensional Crossover of the Dephasing Time in Disordered Mesoscopic Rings: From Diffusive through Ergodic to 0D Behavior

We analyze dephasing by electron interactions in a small disordered quasi-one dimensional (1D) ring weakly coupled to leads, where we recently predicted a crossover for the dephasing time $tPh(T)$ from diffusive or ergodic 1D ($tPh^{-1} propto T^{2/3}, T^{1}$) to $0D$ behavior ($tPh^{-1} propto T^{2}$) as $T$ drops below the Thouless energy $ETh$. We provide a detailed derivation of our results, based on an influence functional for quantum Nyquist noise, and calculate all leading and subleading terms of the dephasing time in the three regimes. Explicitly taking into account the Pauli blocking of the Fermi sea in the metal allows us to describe the $0D$ regime on equal footing as the others. The crossover to $0D$, predicted by Sivan, Imry and Aronov for 3D systems, has so far eluded experimental observation. We will show that for $T ll ETh$, $0D$ dephasing governs not only the $T$-dependence for the smooth part of the magnetoconductivity but also for the amplitude of the Altshuler-Aronov-Spivak oscillations, which result only from electron paths winding around the ring. This observation can be exploited to filter out and eliminate contributions to dephasing from trajectories which do not wind around the ring, which may tend to mask the $T^{2}$ behavior. Thus, the ring geometry holds promise of finally observing the crossover to $0D$ experimentally.