Electron-spin dephasing via hyperfine interaction in a quantum dot: An equation-of-motion calculation of electron-spin correlation functions

In this paper we develop an equation-of-motion (EOM) approach to study the non-Markovian single-electron-spin dynamics due to its inhomogeneous hyperfine coupling to the surrounding nuclei in a quantum dot. In particular, we identify an electron-spin correlation function that fully represents the electron-spin quantum coherence. Using the EOM method, we recover the exact solution of electron-spin decoherence for the case of a fully polarized nuclear reservoir. By considering nuclear-spin flip-flops mediated by virtual electron flips, which generate fluctuations in the Overhauser field (the nuclear field) for the electron spin, we find that the free-induction decay of the electron-spin correlation function for partially polarized and unpolarized nuclear-spin configurations is of the order unity instead of O (1/N) (N being the number of nuclei in the dot) obtained in previous studies. We show that the complete amplitude decay corresponds to the spectral broadening of the correlation function near the electron-spin Rabi frequency induced by nuclear-spin flip-flops. Our results show that a 90% nuclear-spin polarization can enhance the electron-spin coherence time by more than 1 order of magnitude. In the long-time limit, the envelope of the transverse electron-spin correlation function has a nonexponential 1/ t2 decay in the presence of both polarized and unpolarized nuclei.