  Present protocols of criticality-enhanced sensing with open quantum sensors
assume direct measurement of the sensor and omit the radiation quanta emitted
to the environment, thereby potentially missing valuable information. Here we
propose a protocol for criticality-enhanced sensing via continuous observation
of the emitted radiation quanta. Under general assumptions, we establish a
scaling theory for the global quantum Fisher information of the joint system
and environment state at dissipative critical points. We derive universal
scaling laws featuring transient and long-time behavior governed by the
underlying critical exponents. Importantly, such scaling laws exceed the
standard quantum limit and can in principle saturate the Heisenberg limit. To
harness such advantageous scaling, we propose a practical sensing scheme based
on continuous detection of the emitted quanta as realized experimentally in
various quantum-optical setups. In such a scheme a single interrogation
corresponds to a (stochastic) quantum trajectory of the open system evolving
under the nonunitary dynamics dependent on the parameter to be sensed and the
backaction of the continuous measurement. Remarkably, we demonstrate that the
associated precision scaling significantly exceeds that based on direct
measurement of the critical steady state, thereby establishing the metrological
value of the continuous detection of the emitted quanta at dissipative
criticality. We illustrate our protocol via counting the photons emitted by the
open Rabi model, a paradigmatic model for the study of dissipative phase
transition with finite components. Our protocol is applicable to generic
quantum-optical open sensors permitting continuous readout, and may find
applications at the frontier of quantum sensing, such as the human-machine
interface, magnetic diagnosis of heart disease, and zero-field nuclear magnetic
resonance.
