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.