Memories of Initial States in the Dynamics of Disordered Systems*

We consider the dynamics of a one dimensional closed disordered system of non-interacting bosons/fermions which is initialized to a Fock state with a pattern of 0 and 1 particles on alternating sites. We show that in the long time limit, the imbalance between the densities in the two sublattices reach a finite value in the localized phase given by, I(∞) = Tanh ( 1/ 2 ξ), where ξ is the localization length. For a chain with random potential disorder, the imbalance is finite for any disorder, whereas for the Aubrey Andre model, it shows a localization-delocalization transition. For a modified Aubrey Andre model, the imbalance as a function of the disorder shows a kink when the mobility edge first appears. We find that in this case, I(∞) = ∑_i Tanh [ 1/ 2 ξⁱ], where ξⁱ is the localization length corresponding to each of the three bands in the model. Our work relates experimentally measureable non-equilibrium quantities in these systems to the localization length and hence shows a new method for extracting the localization length in these systems.