We analyze possible nonlinear exciton-exciton correlation effects in theoptical response of semiconductors by using a time-dependent density-functionaltheory (TDDFT) approach. For this purpose, we derive the nonlinear(third-order) TDDFT equation for the excitonic polarization. In this equation,the nonlinear time-dependent effects are described by the time-dependent(non-adiabatic) part of the effective exciton-exciton interaction, whichdepends on the exchange-correlation (XC) kernel. We apply the approach to studythe nonlinear optical response of a GaAs quantum well. In particular, wecalculate the 2D Fourier spectra of the system and compare it with experimentaldata. We find that it is necessary to use a non-adiabatic XC kernel to describeexcitonic bound states - biexcitons, which are formed due to the retarded TDDFTexciton-exciton interaction.
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