Nuclear spin dynamics and spin orbit effects in Landau-Zener sweep correlations at the S-$\mathrm{T_+}$ Transition

In GaAs-based double quantum dot spin qubits, nuclear spins have been used for qubit control, but are also an important source of decoherence. The S and $\mathrm{T_+}$ levels exhibit a small avoided crossing as a function of detuning. It has been used for S-$\mathrm{T_+}$ qubit control and for dynamic nuclear polarization (DNP). The transition matrix element contains the nuclear Overhauser fields perpendicular to the external B-field and spin-orbit coupling. We show, both theoretically and experimentally, that nuclear spin dynamics can be seen in the temporal correlation of single-shot measurements after Landau-Zener sweeps across this transition. A semi-classical model of the nuclear spins is sufficient. The dynamics consist of the relative Larmor precession of the three GaAs nuclear spin species in the external B-field and dephasing of the oscillations due to local field fluctuations. Theoretically, it is expected that the absolute Larmor precessions also become visible in the presence of spin-orbit coupling. This can be used to qualitatively and quantitatively observe spin-orbit coupling and to distinguish it from the nuclear spin contribution. Understanding these dynamics is relevant for the fidelity of S-$\mathrm{T_+}$ qubit operations and the effectiveness of DNP.