Spectrum of the Nuclear Environment for GaAs Spin Qubits

Using a singlet-triplet spin qubit as a sensitive spectrometer of the GaAs nuclear spin bath, we demonstrate that the spectrum of Overhauser noise agrees with a classical spin diffusion model over 6 orders of magnitude in frequency, from 1 mHz to 1 kHz, is flat below 10 mHz, and falls as 1/f^{2} for frequency f≳1  Hz. Increasing the applied magnetic field from 0.1 to 0.75 T suppresses electron-mediated spin diffusion, which decreases the spectral content in the 1/f^{2} region and lowers the saturation frequency, each by an order of magnitude, consistent with a numerical model. Spectral content at megahertz frequencies is accessed using dynamical decoupling, which shows a crossover from the few-pulse regime (≲16π pulses), where transverse Overhauser fluctuations dominate dephasing, to the many-pulse regime (≳32 π pulses), where longitudinal Overhauser fluctuations with a 1/f spectrum dominate.