Monitoring variations of refractive index via Hilbert-Schmidt speed and applying this phenomenon to improve quantum metrology

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.