Memories of initial states and density imbalance in dynamics of interacting disordered systems

We study the dynamics of one and two dimensional disordered lattice bosons/fermions initialized to a Fock state with a pattern of $1$ and $0$ particles on $A$ and ${bar A}$ sites. For non-interacting systems we establish a universal relation between the long time density imbalance between $A$ and ${bar A}$ site, $I(infty)$, the localization length $xi_l$, and the geometry of the initial pattern. For alternating initial pattern of $1$ and $0$ particles in 1 dimension, $I(infty)=tanh[a/xi_l]$, where $a$ is the lattice spacing. For systems with mobility edge, we find analytic relations between $I(infty)$, the effective localization length $tilde{xi}_l$ and the fraction of localized states $f_l$. The imbalance as a function of disorder shows non-analytic behaviour when the mobility edge passes through a band edge. For interacting bosonic systems, we show that dissipative processes lead to a decay of the memory of initial conditions. However, the excitations created in the process act as a bath, whose noise correlators retain information of the initial pattern. This sustains a finite imbalance at long times in strongly disordered interacting systems.