  In this manuscript, we propose a new sensing mechanism to enhance the
sensitivity of a quantum system to nonlinearities by homodyning the amplitude
quadrature of the cavity field. The system consists of two dissipatively
coupled cavity modes, one of which is subject to single- and two-photon drives.
In the regime of low two-photon driving strength, the spectrum of the system
acquires a real spectral singularity. We find that this singularity is very
sensitive to the two-photon drive and nonlinearity of the system, and compared
to the previous nonlinearity sensor, the proposed sensor achieves an
unprecedented sensitivity around the singularity point. Moreover, the scheme is
robust against fabrication imperfections. This work would open a new avenue for
quantum sensors, which could find applications in many fields, such as the
precise measurement and quantum metrology.
