Dynamics of the Overhauser field under nuclear spin diffusion in a quantum dot

The coherence of electron spins can be enhanced significantly by preparing the nuclear spin polarizations to generate an Overhauser field with small fluctuations. We propose a theoretical model for calculating the long time dynamics of the prepared Overhauser field under nuclear spin diffusion in a quantum dot. We obtained a simplified diffusion equation that can be numerically solved, and we show quantitatively how the Knight shift and the electron-mediated nuclear spin flip-flops affect the nuclear spin diffusion. The results explain several recent experimental observations, where the measured decay time of the Overhauser field is dependent on the external magnetic field, electron spin state in double quantum dots and initial nuclear spin polarization rate.