  We propose a new method for atomic-scale imaging of spatial current patterns
in nanoscopic quantum networks by using scanning tunneling microscopy (STM). By
measuring the current flowing from the STM tip into one of the leads attached
to the network as a function of tip position, one obtains an atomically
resolved spatial image of "current riverbeds" whose spatial structure reflects
the coherent flow of electrons out of equilibrium. We show that this method can
be successfully applied in variety of network topologies, and is robust against
dephasing effects.
