  We applied low temperature diffraction limited confocal optical microscopy to
spatially resolve, and spectroscopically study photoluminescence from single
self-assembled semiconductor quantum dots. Using selective wavelength imaging
we unambiguously demonstrated that a single photoexcited quantum dot emits
light in a few very narrow spectral lines. The measured spectrum and its
dependence on the power of either cw or pulsed excitation are explained by
taking carrier correlations into account. We solve numerically a many body
Hamiltonian for a model quantum dot, and we show that the multi line emission
spectrum is due to optical transitions between confined exciton multiplexes. We
furthermore show that the electron-electron and hole-hole exchange interaction
is responsible for the typical appearance of pairs in the photoluminescence
spectra and for the appearance of red shifted new lines as the excitation power
increases. The fact that only a few spectral lines appear in the emission
spectrum strongly indicates fast thermalization. This means that a multiexciton
relaxes to its ground state much faster than its radiative lifetime.
