Effective Frenkel Hamiltonian for optical nonlinearities in semiconductors: Application to magnetoexcitons

Closed Green-function expressions for the third-order response of semiconductors are derived by mapping the two-band model onto the much simpler molecular (Frenkel) Hamiltonian. The signatures of two-exciton resonances are incorporated through the exciton scattering matrix, totally avoiding the explicit calculation of two-exciton states. Exact expressions for the nonlinear optical response of two-dimensional semiconductor nanostructures in a strong perpendicular magnetic field are derived by truncating at the $n=1\mathrm{}$ Landau level, and using the symmetry of the system and a group-theoretical analysis. We find that the nonlinear optical response depends crucially on asymmetries between particle-particle and particle-hole Coulomb interactions.