Quantum description of nuclear spin cooling in a quantum dot

We study theoretically the cooling of an ensemble of nuclear spins coupled to the spin of a localized electron in a quantum dot. We obtain a master equation for the state of the nuclear spins interacting with a sequence of polarized electrons that allows us to study quantitatively the cooling process including the effect of nuclear spin coherences, which can lead to ``dark states'' of the nuclear system in which further cooling is inhibited. We show that the inhomogeneous Knight field mitigates this effect strongly and that the remaining dark-state limitations can be overcome by very few shifts of the electron wave function, allowing for cooling far beyond the dark-state limit. Numerical integration of the master equation indicates that polarizations larger than 90% can be achieved within a millisecond time scale.