In this article, the fundamental physics of multi-exciton states in semiconductor nanocrystals is reviewed focusing on the mesoscopic enhancement of the excitonic radiative decay rate and the excitonic optical nonlinearity and the mechanism of their saturation with increase of the nanocrystal size. In the case of the radiative decay rate the thermal excitation of excited exciton states having small oscillator strength within the homogeneous linewidth of the exciton ground state is essential in determining the saturation behavior. The weakly correlated exciton pair states are found to cause a cancellation effect in the third-order nonlinear optical susceptibility at the exciton resonance, providing the first consistent understanding of the experimentally observed saturation of the mesoscopic enhancement of the excitonic 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. Furthermore, ultrafast relaxation processes of biexcitons are discussed in conjunction with the observed very fast rise of the biexciton gain in nanocrystals. In prospect of future progress in research, the theoretical formulation to calculate the triexciton states as one of the multi-exciton states beyond the biexciton is presented for the first time including the electron-hole exchange interaction.
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