Figure 11

The maximum and minimum gain spectra. (a) The maximum gain spectra$|g_{{\varphi }_{\max }}(\omega )|$,and (b) the minimum gain spectra $|g_{{\varphi }_{\min }}(\omega )|$,for linearisation about the fixed point at the origin for$ϵ=0$.The gain is dependent upon four variables: the scaled parametric pumping${\kappa }^{\prime }=\frac{\kappa }{\gamma }$;the scaled detuning ${\Delta }^{\prime }=\frac{\Delta }{\gamma }$;and the scaled probed frequency ${\omega }^{\prime }=\frac{\omega }{\gamma }$and phase ϕ. Here, we plot the spectra against only theparametric pumping and the probed frequency. The different coloured sheets showdifferent detunings: the red (innermost) surface shows the gain for no detuning${\Delta }^{\prime }=0$;the green (central) surface shows the gain for $|{\Delta }^{\prime }|=1$; and theblue (outermost) surface shows the gain for $|{\Delta }^{\prime }|=2$. Where thesurface is not plotted (other than where it is truncated around thesingularities at $({\kappa }^{\prime 2})=1+{\Delta }^{\prime 2}$)it is because the origin does not exist as a stable semi-classical fixed pointto be linearised about for those parameter values. We choose the optimal phaseϕ to produce the (a) maximum gain, and (b) minimum gain, at DCfor each pumping power and detuning.