Nuclear Dynamics During Landau-Zener Singlet-Triplet Transitions in Double Quantum Dots

We consider nuclear spin dynamics in a two-electron double dot system near the intersection of the electron spin singlet $S$ and the lower energy component $T_{+}$ of the spin triplet. The electron spin interacts with nuclear spins and is influenced by the spin-orbit coupling. Our approach is based on a quantum description of the electron spin in combination with the coherent semiclassical dynamics of nuclear spins. We consider single and double Landau-Zener passages across the $S$-$T_{+}$ anticrossings. For linear sweeps, the electron dynamics is expressed in terms of parabolic cylinder functions. The dynamical nuclear polarization is described by two complex conjugate functions $\Lambda ^{\pm}$ related to the integrals of the products of the singlet and triplet amplitudes ${\tilde{c}}_{S}^{\ast}{\tilde{c}}_{T_{+}}$ along the sweep. The real part $P$ of $\Lambda ^{\pm}$ is related to the $S$-$T_{+}$ spin-transition probability, accumulates in the vicinity of the anticrossing, and for long linear passages coincides with the Landau-Zener probability $P_{LZ}=1-e^{-2\pi \gamma}$, where $\gamma $ is the Landau-Zener parameter. The imaginary part $Q$ of $\Lambda^{+}$ is specific for the nuclear spin dynamics, accumulates during the whole sweep, and for $\gamma \gtrsim 1$ is typically an order of magnitude larger than $P$. $Q$ has a profound effect on the nuclear spin dynamics, by (i) causing intensive shake-up processes among the nuclear spins and (ii) producing a high nuclear spin generation rate when the hyperfine and spin-orbit interactions are comparable in magnitude. We find analytical expressions for the back-action of the nuclear reservoir represented via the change in the Overhauser fields the electron subsystem experiences.