Electron spin dephasing by hyperfine interaction with nuclei in quantum dots

The problem of the dynamics of an electron spin coupled by hyperfine (hf) interaction to nuclear spins has been a focus of large theoretical attention, since the interaction with the nuclear bath is the most limiting decoherence mechanism in spin qubits based on quantum dots made of III-V materials. I will present a theory of pure dephasing decoherence which gives predictions for electron dynamics in narrowed state free induction decay, spin echo, and under higher order dynamical decoupling sequences [1,2]. In this theory we take advantage of the long-range character of hf-mediated interactions (which couple remote nuclei via virtual flip-flops with the electron spin), and we resum the leading terms in 1/N expansion of the decoherence time-evolution function (N being the large number of nuclei interacting appreciably with the electron spin). For the case of a thermal uncorrelated bath this approach is applicable as long as the electron Zeeman splitting is larger than the typical Overhauser shift of the electron energy (i.e.~magnetic field must be larger than a few mT in a large GaAs dot). For the spin echo evolution we show that the dominant decoherence process at low fields is due to interactions between nuclei having different Zeeman energies (i.e.~nuclei of As and two isotopes of Ga in GaAs). The robustness of this theory is verified by comparison with a numerical simulation of spin echo in a system with $N = 20$ nuclei. I will also discuss the application of our approach to singlet-triplet qubit in a double quantum dot. \\[4pt] [1] L. Cywinski, W.M. Witzel, and S. Das Sarma, Phys. Rev. Lett. {\bf 102}, 057601 (2009). \\[0pt] [2] L. Cywinski, W.M. Witzel, and S. Das Sarma, Phys. Rev. B {\bf 79}, 245314 (2009).