  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.
