Crossover from strong to weak exciton confinement and third-harmonic generation on one-dimensional quantum dots

We study the effects of exciton confinement on the nonlinear optical susceptibility of one-dimensional quantum dots. We use a direct numerical diagonalization to obtain the eigenenergies and eigenstates of the discretized Hamiltonian representing an electron–hole pair confined by a semiparabolic potential and interacting with each other via a Coulomb potential. Density matrix perturbation theory is used to compute the nonlinear optical susceptibilities due to third-harmonic generation and the corresponding nonlinear corrections to the refractive index and absorption coefficient. These quantities are analyzed as a function of ratio between the confinement length L and the exciton Bohr radius a₀. The Coulomb potential degrades the uniformity of the level separation. We show that this effect promotes the emergence of multiple resonance peaks in the third-harmonic generation spectrum. In the weak confinement regime β = L/a₀ ? 1, the third-order susceptibility is shown to decay as 1/β⁸ due to the prevalence of the hydrogenoid character of the exciton eigenstates.