  Effective nonlinear optical interactions are essential for many applications
in modern photonics. In this paper, we investigate the role of the nonlinear
response of a material to improve quantum metrology. In particular, the
collective optical behavior of an atomic ensemble is applied to enhance
frequency estimation through one of the atoms. Moreover, we introduce
Hilbert-Schmidt speed, an easily computable theoretical tool, to monitor the
variations of linear as well as nonlinear refractive indices and evaluate the
strength of the nonlinear response of optical materials. Furthermore, we
illustrate that quantum Fisher information and Hilbert-Schmidt speed can
efficiently detect negative permittivity and refractive index, which is of
great importance from a practical point of view.
