  An electron confined in a quantum dot interacts with its local nuclear spin
environment through the hyperfine contact interaction. This interaction
combined with external control and relaxation or measurement of the electron
spin allows for the generation of dynamic nuclear polarization. The quantum
nature of the nuclear bath, along with the interplay of coherent external
fields and incoherent dynamics in these systems renders a wealth of intriguing
phenomena seen in recent experiments such as electron Zeeman frequency
focusing, hysteresis, and line dragging. We develop in detail a fully quantum,
self-consistent theory that can be applied to such experiments and that
moreover has predictive power. Our theory uses the operator sum representation
formalism in order to incorporate the incoherent dynamics caused by the
additional, Markovian bath, which in self-assembled dots is the vacuum field
responsible for electron-hole optical recombination. The beauty of this
formalism is that it reduces the complexity of the problem by encoding the
joint dynamics of the external coherent and incoherent driving in an effective
dynamical map that only acts on the electron spin subspace. This together with
the separation of timescales in the problem allows for a tractable and
analytically solvable formalism. The key role of entanglement between the
electron spin and the nuclear spins in the formation of dynamic nuclear
polarization naturally follows from our solution. We demonstrate the theory in
detail for an optical pulsed experiment and present an in-depth discussion and
physical explanation of our results.
