Quantum stirring of particles in closed devices
  We study the quantum analog of stirring of water inside a cup using a spoon. This can be regarded as a prototype example for quantum pumping in closed devices. The current in the device is induced by translating a scatterer. Its calculation is done using the Kubo formula approach. The transported charge is expressed as a line integral that encircles chains of Dirac monopoles. For simple systems the results turn out to be counter intuitive: e.g. as we move a small scatterer "forward" the current is induced "backwards". One should realize that the route towards quantum-classical correspondence has to do with "quantum chaos" considerations, and hence assumes greater complexity of the device. We also point out the relation to the familiar $S$ matrix formalism which is used to analyze quantum pumping in open geometries. 
Gilad Rosenberg and Doron Cohen