Mesoscopic decoherence in Aharonov-Bohm rings

We study electron decoherence by measuring the temperature dependence of Aharonov-Bohm (AB) oscillations in quasi-one-dimensional rings, etched in a high-mobility $\mathrm{GaAs}/{\mathrm{Ga}}_x{\mathrm{Al}}_{1-x}\mathrm{As}$ heterostructure. The oscillation amplitude is influenced both by phase breaking and by thermal averaging. Thermal averaging is important when the temperature approaches the energy scale on which the AB oscillations shift their phase. For the phase breaking, it is demonstrated that the damping of the oscillation amplitude is proportional to the length of the interfering paths. For temperatures T from 0.3 to 4 K we find the phase-coherence length $L_{\phi }$ $\propto T^{-1},$ close to what has been reported for open quantum dots. This might indicate that the $T^{-1}$ decoherence rate is a general property of open and ballistic mesoscopic systems.