Electron spin dephasing by hyperfine-mediated interactions in a nuclear spin bath

We investigate pure dephasing decoherence (free induction decay and spin echo) of a quantum dot spin qubit interacting with a nuclear spin bath. While for infinite magnetic field $B$ the only decoherence mechanism is spectral diffusion due to dipolar flip-flops of nuclear spins, with decreasing $B$ the hyperfine-mediated interactions between the nuclear spins become important. We give a theory [1] of decoherence due to these interactions which takes advantage of their long range nature. For a thermal uncorrelated bath we show that our theory is applicable down to $B\sim$10 mT, allowing for comparison with recent experiments on spin echo in GaAs quantum dots [2].\\ {[1]} L. Cywinski, W.M. Witzel, and S. Das Sarma, preprint arXiv:0809:0003 (2008).\\ {[2]} F.H.L. Koppens, K.C. Nowack, and L.M.K. Vandersypen, Phys. Rev. Lett. \textbf{100}, 236802 (2008).