We prepare coherent dark state ⁸⁷Rb atoms using two spatially extended coherent laser fields. Using standard electro-magnetic induced transparency spectroscopy we measure directly the Laplace transform of the survival probability of atoms in the dark state (drunkards) diffusing in the presence of static traps (policemen) created by shaping a supplementary beam. We show that the non-exponential spectrum is a manifestation of Anderson localization in imaginary time. The non-exponential decay is already observed in the presence of a single trap reflecting the well know fact that any kind of disorder is strongly localizing in dimension d<3.

Motivation for this study comes from a well known problem in statistical physics - the calculation of the survival probability of diffusing particles (drunkard's) in the presence of static traps (policeman's) with a specified distribution in space. For example, when the traps are arranged on a one dimensional lattice with a lattice constant a, the survival probability decay exponentially with a time constant τ ~ a²/2D which is the characteristic time to reach a trap (D is the diffusion constant). Interestingly, if the traps are distributed randomly but with the same density the survival probability is non-exponential (it is stretched), since rare trap-free regions control the asymptotic survival probability.