  We propose and analyze a driven-dissipative quantum sensor that is
continuously monitored close to a dissipative critical point. The sensor relies
on the critical open Rabi model with the spin and phonon degrees of freedom of
a single trapped ion to achieve criticality-enhanced sensitivity. Effective
continuous monitoring of the sensor is realized via a co-trapped ancilla ion
that switches between dark and bright internal states conditioned on a `jump'
of the phonon population which, remarkably, achieves nearly perfect phonon
counting despite a low photon collection efficiency. By exploiting both
dissipative criticality and efficient continuous readout, the sensor device
achieves highly precise sensing of oscillating electric field gradients at a
criticality-enhanced precision scaling beyond the standard quantum limit, which
we demonstrate is robust to the experimental imperfections in real-world
applications.
