14 - Brownian Motion: Classical and Semiclassical Entanglement

Summary

We show that for two classical Brownian particles there exists an analog ofcontinuous-variable quantum entanglement: The common probability distributionof the two coordinates and the corresponding coarse-grained velocitiescannot be prepared via mixing of any factorized distributions referring tothe two particles in separate. This is possible for particles which interactedin the past, but do not interact in the present. Three factors are crucial forthe effect: (1) separation of time-scales of coordinate and momentum whichmotivates the definition of coarse-grained velocities; (2) the resulting uncertaintyrelations between the coordinate of the Brownian particle and thechange of its coarse-grained velocity; (3) the fact that the coarse-grained velocity,though pertaining to a single Brownian particle, is defined on a commoncontext of two particles. The Brownian entanglement is a consequenceof a coarse-grained description and disappears for a finer resolution of theBrownian motion. We discuss possibilities of its experimental realizations inexamples of macroscopic Brownian motion.