Cancellation of size-linear terms in the third-order nonlinear susceptibility: Frenkel excitons in a periodic chain

Abstract

For a system of noninteracting Frenkel excitons in a one-dimensional lattice of size N with periodic boundary conditions, the third-order optical susceptibility ${\mathrm{\ensuremath{\chi}}}^{(3)}$ has been calculated rigorously in a nonlocal form with arbitrary dependence on external-field frequencies. Among the various terms in ${\mathrm{\ensuremath{\chi}}}^{(3)}$ (per unit volume), those explicitly proportional to N in the long-wavelength approximation have been shown to cancel out completely for arbitrary N. The remaining terms, including the effect of nonlocality, reduce to the well-known result of a two-level system in the limit of vanishing transfer energy. There remain N-dependent factors in ${\mathrm{\ensuremath{\chi}}}^{(3)}$, with different functional forms for even and odd N, but they all approach unity in the limit of large N.