Spin separation and exchange for quantum dots in the Overhauser field

We describe the spin and charge dynamics of the system of two electrons confined within a double quantum dot defined in a quantum wire. The spin dynamics is driven by the electron motion in presence of the spin-orbit interaction and the randomly varying local Overhauser field due to the nuclear spins. The Schroedinger equation is solved with the time-dependent configuration interaction method that allows for an exact description of the system dynamics. The procedures of the spin separation, exchange and read-out by the spin to charge conversion all induced by the detuning variation are simulated. The rates of the potential variation that are necessary for the spin separation and spin to charge conversion in the context of the Landau-Zener transitions are determined. The average over random configurations of the hyperfine field produce spin exchange results which qualitatively agree with the experimental data.