Drastic effects of damping mechanisms on the third-order optical nonlinearity

We have investigated the optical response of superradiant atoms, which undergoes three different damping mechanisms: radiative dissipation ($\gamma_r$), dephasing ($\gamma_d$), and nonradiative dissipation ($\gamma_n$). Whereas the roles of $\gamma_d$ and $\gamma_n$ are equivalent in the linear susceptibility, the third-order nonlinear susceptibility drastically depends on the ratio of $\gamma_d$ and $\gamma_n$: When $\gamma_d \ll \gamma_n$, the third-order susceptibility is essentially that of a single atom. Contrarily, in the opposite case of $\gamma_d \gg \gamma_n$, the third-order susceptibility suffers the size-enhancement effect and becomes proportional to the system size.