Hysteresis in the complex nonlinear refractive index of a homogeneous and isotropic medium

We calculate the permittivity, $\epsilon(\omega)$, for a medium with a quadratic electro-optic effect, modeling it as a Duffing oscillator. The nonlinear refractive index $n(\omega, E(\omega))$ and the nonlinear absorption coefficient $\alpha(\omega, E(\omega))$ exhibit hysteresis when the light intensity is varied [here $E(\omega)$ is the electric field strength at angular frequency $\omega$], and when the light frequency is varied. $n(\omega, E(\omega))$ can be negative when the resonances in the permittivity and permeability are close to one another.