  We theoretically investigate the controlled dynamic polarization of lattice
nuclear spins in GaAs double quantum dots containing two electrons. Three
regimes of long-term dynamics are identified, including the build up of a large
difference in the Overhauser fields across the dots, the saturation of the
nuclear polarization process associated with formation of so-called "dark
states," and the elimination of the difference field. We show that in the case
of unequal dots, build up of difference fields generally accompanies the
nuclear polarization process, whereas for nearly identical dots, build up of
difference fields competes with polarization saturation in dark states. The
elimination of the difference field does not, in general, correspond to a
stable steady state of the polarization process.
