Dissipative phase transition and metrology in collectively pumped superradiance

We study a many-atom system exhibiting two competing collective processes: collective decay and collective pumping of excitations, relevant e.g. in cavity QED platforms. We find that the steady state exhibits a sharp transition as a function of the pumping strength, between fully depopulated and fully populated states. We devise a metrological protocol for measuring system parameters by scanning the pumping around the critical point, finding that the sensitivity scales as 1/N , thus beating the standard quantum limit thanks to the buildup of correlations. Crucially, our theoretical analysis, verified numerically, goes beyond the adiabatic regime of an infinite scan time: we study non-equilibrium relaxation dynamics around the transition and their effect on the sensitivity, revealing that the favorable 1/N scaling survives well beyond adiabaticity. Apart from its direct impact on metrology with superradiant atomic systems, our general analysis provides new perspectives for studying the metrological utility of dissipative phase transitions in realistic finite-time protocols.