The fundamental physics of multi-exciton states in semiconductor nanocrystals (quantum dots) are discussed focusing on the mesoscopic enhancement of the exitonic optical nonlinearity and the mechanism of their sturation with icrease of the naocrystal size. The wealkly correlated excitohn pair states are found to casue significant cancellation effect in the third-order nonlinear optical susceptibility at the exiton resonance, leading to saturation of the mesoscopic enhancement of the exictonic optical nonlinearity. The presence of the weakly correlated exciton pair states is confirmed convincingly from the good correspondence between theory and experiments on the induced absorption spectra from the exciton state in CuCl nanocrystals. Both dephasing and population relaxation of exitions localized in quantum dot-like islands in narrow GaAs quantum wells are investigated by using the three-pulse stimulated phton echo method. A direct comparison of these two closely related decay processes reveals the presence of pure dephasing that dominates excitonic dephasing at elevated tlemperatures. The pure dephasing contribution arises from coupling of excitonic states with a continum of acoustic phonons and is enhanced by 3D quantum confinement. Both the mangitude and the temperature dependence of the dephasing rate can be described satisfactorily by a model that generalizes the Huang-Rhys theory of F-centers. We clarify the origin of qualitative difference in the temperature dependence of the excitionic dephasing rate for QDs with weak and strong quantum confinement.
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