  We study the exciton dynamics in an optically excited nanocrystal quantum
dot. Multiple exciton formation is more efficient in nanocrystal quantum dots
compared to bulk semiconductors due to enhanced Coulomb interactions and the
absence of conservation of momentum. The formation of multiple excitons is
dependent on different excitation parameters and the dissipation. We study this
process within a Lindblad quantum rate equation using the full many-particle
states. We optically excite the system by creating a single high energy exciton
$E_{SX}$ in resonance to a double exciton $E_{DX}$. With Coulomb
electron-electron interaction, the population can be transferred from the
single exciton to the double exciton state by impact ionisation (inverse Auger
process). The ratio between the recombination processes and the absorbed
photons provide the yield of the structure. We observe a quantum yield of
comparable value to experiment assuming typical experimental conditions for a
$4$ nm PbS quantum dot.
