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We consider the general problem of estimating an unknown control parameter ofan open quantum system. We establish a direct relation between the evolution ofboth system and environment and the precision with which the parameter can beestimated. We show that when the open quantum system undergoes a first-orderdynamical phase transition the quantum Fisher information (QFI), which givesthe upper bound on the achievable precision of any measurement of the systemand environment, becomes quadratic in observation time (cf. "Heisenbergscaling"). In fact, the QFI is identical to the variance of the dynamicalobservable that characterises the phases that coexist at the transition, andenhanced scaling is a consequence of the divergence of the variance of thisobservable at the transition point. This identification allows to establish thefinite time scaling of the QFI. Near the transition the QFI is quadratic intime for times shorter than the correlation time of the dynamics. In the regimeof enhanced scaling the optimal measurement whose precision is given by the QFIinvolves measuring both system and output. As a particular realisation of theseideas, we describe a theoretical scheme for quantum enhanced phase estimationusing the photons being emitted from a quantum system near the coexistence ofdynamical phases with distinct photon emission rates.

Dynamical phase transitions as a resource for quantum enhanced metrology