Critical quantum metrology beyond adiabaticity in collectively pumped superradiance

Critical metrology relies on the high sensitivity of systems to parameter changes near a phase transition to extract information with high precision. Such methods usually require working under adiabatic conditions, which implies long preparation times due to critical slowing down. Here instead we demonstrate how favorable sensitivity can be maintained for preparation rates exceeding the adiabatic limit by exploiting knowledge of the non-equilibrium dynamics of the system. Our work focuses on a model of collective dissipation (superradiance) balanced by collective pumping, analyzing the sensitivity around a first order dissipative phase transition. We devise a sensing protocol based on characterizing the hysteresis formed beyond adiabatic conditions, finding the sensing uncertainty $r^{\alpha}/N$ , exhibiting a favorable Heisenberg-like scaling with the particle number $N$ and only a weak power-law dependence $\alpha \approx 0.17$ on the preparation rate $r$. We discuss the implementation of the model and potential applications to metrology within cavity QED setups.