Spin-orbit effects on nuclear state preparation at the $S-T_{+}$ anti-crossing in double quantum dots

We explore the interplay of spin-orbit and hyperfine effects on the nuclear preparation schemes in two-electron double quantum dots, e.g. in GaAs. The quantity of utmost interest is the electron spin decoherence time $T_{2}^{*}$ in dependence of the number of sweeps through the electron spin singlet $S$ triplet $T_{+}$ anti-crossing. Decoherence of the electron spin is caused by the difference field induced by the nuclear spins. We study the case where a singlet $S(2,0)$ is initialized, in which both electrons are in the left dot. Subsequently, the system is driven repeatedly through the anti-crossing and back using linear electrical bias sweeps. Our model describes the passage through the anti-crossing with a large number of equally spaced, step-like parameter increments. We develop a numerical method describing the nuclear spins fully quantum mechanically, which allows us to track their dynamics. Both Rashba and Dresselhaus spin-orbit terms do depend on the angle $\theta$ between the $[110]$ crystallographic and the inter-dot axis. Our results show that the suppression of decoherence (and therefore the enhancement of $T_{2}^{*}$) is inversely proportional to the strength of the spin-orbit interaction, which is tuned by varying the angle $\theta$.