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We carefully examine critical metrology and present an improved criticalquantum metrology protocol which relies on quenching a system exhibiting asuperradiant quantum phase transition beyond its critical point. We show thatthis approach can lead to an exponential increase of the quantum Fisherinformation in time with respect to existing critical quantum metrologyprotocols relying on quenching close to the critical point and observing powerlaw behaviour. We demonstrate that the Cram\'er-Rao bound can be saturated inour protocol through the standard homodyne detection scheme. We explicitly showits advantage using the archetypal setting of the Dicke model and explore aquantum gas coupled to a single-mode cavity field as a potential platform. Inthis case an additional exponential enhancement of the quantum Fisherinformation can in practice be observed with the number of atoms $N$ in thecavity, even in the absence of $N$-body coupling terms.

Understanding and Improving Critical Metrology. Quenching Superradiant Light-Matter Systems Beyond the Critical Point