  A recent experiment [E. A. Chekhovich et al., Phys. Rev. Lett. 104, 066804
(2010)] has demonstrated that high nuclear spin polarization can be achieved in
self-assembled quantum dots by exploiting an optically forbidden transition
between a heavy hole and a trion state. However, a fully polarized state is not
achieved as expected from a classical rate equation. Here, we theoretically
investigate this problem with the help of a quantum master equation and we
demonstrate that a fully polarized state cannot be achieved due to formation of
a nuclear dark state. Moreover, we show that the maximal degree of polarization
depends on structural properties of the quantum dot.
