Equation-of-motion treatment of hyperfine interaction in a quantum dot

Isolated electron spins in semiconductor nanostructures are promising qubit candidates for a solid state quantum computer, There have seen truly impressive experimental progresses in the study of single spins in the past two years. In this paper we analytically solve the {\it Non-Markovian} single electron spin dynamics due to inhomogeneous hyperfine couplings with surrounding nuclei in a quantum dot. We use the equation-of-motion method assisted with a large field expansion in a full quantum mechanical treatment. We recover the exact solution for fully polarized nuclei. By considering virtual nuclear spin flip-flops mediated by the electron, which generate fluctuations in the Overhauser field (the nuclear field) for the electron spin, we find that the decay amplitude of the transverse electron spin correlation function for partially polarized nuclear spin configurations is of the order unity instead of $\text{O}(1/N)$ ($N$ being the number of nuclei in the dot) obtained in previous studies. We show that the complete amplitude decay can be understood with the spectrum broadening of the correlation function near the electron spin Rabi frequency induced by nuclear spin flip-flops. Our results show that a 90% nuclear polarization can enhance the electron spin $T_2$ time by more than one order of magnitude in some parameter regime. In the long time limit, the envelope of the transverse electron spin correlation function has a non-exponential $1/t^2$ decay in the presence of both polarized and unpolarized nuclei.