Optomechanical resonator as a negative dispersion medium for enhancing the sensitivity bandwidth in a gravitational-wave detector

Recently, we proposed an optically pumped five-level gain with electromagnetically induced transparency system, which has a transparency dip superimposed on a gain profile and exhibits a negative dispersion suitable for the white light cavity enhanced interferometric gravitational-wave detector [Phys. Rev. D 92, 082002 (2015)]. Using this system as the negative dispersion medium in the white light cavity signal-recycling scheme, we get an enhancement in the quantum noise limited sensitivity--bandwidth product by a factor of $\ensuremath{\sim}18$. We have also shown how to realize such a system in practice using Zeeman sublevels in $^{87}\mathrm{Rb}$ at 795 nm [Opt. Commun. 402, 382 (2017)]. However, the Advanced Laser Interferometric Gravitational-Wave Observatory (aLIGO) operates at 1064 nm, and suitable transitions in Rb or other alkali atoms are not available at this wavelength. Therefore, it is necessary to consider a system that is consistent with the operating wavelength of aLIGO. Here, we present the realization of such a negative dispersion medium at 1064 nm with a microresonator, which supports optomechanical interaction. A strong control field is applied at a higher frequency, and, under certain conditions, a probe field at a lower frequency experiences a peak at the center of an absorption profile and a negative dispersion in the transmission. Unlike in the gain with electromagnetically induced transparency case, we use the compound-cavity signal-recycling scheme, in which an auxiliary mirror is inserted in the dark port of the detector, and show that the enhancement factor can be as high as $\ensuremath{\sim}15$. However, using the parameters required for the sensitivity enhancement, the optomechanical system enters an instability region where the control field is depleted. We present an observer-based feedback control process used to stabilize the system.