  Recently, we had proposed an optically-pumped five-level Gain EIT (GEIT)
system, which has a transparency dip superimposed on a gain profile and
exhibits a negative dispersion suitable for the white light cavity (WLC)
enhanced interferometric gravitational wave detector [Phys. Rev. D. 92, 082002
(2015)]. Using this system as the negative dispersion medium (NDM) in the
WLC-SR (signal recycling) scheme, we get an enhancement in the quantum noise
(QN) limited sensitivity-bandwidth product by a factor of ~18. We have also
shown how to realize such a system in practice using Zeeman sublevels in
$^{87}$Rb at 795nm [Opt. Commun. 402, 382-388 (2017)]. However, aLIGO operates
at 1064nm 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 an NDM at 1064nm 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 GEIT case, we use the
compound-cavity signal-recycling (CC-SR) scheme, where an auxiliary mirror is
inserted in the dark port of the detector, and show that the enhancement factor
can be as high as ~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 CC-SR system.
