Energy-efficient quantum frequency estimation

The problem of estimating the frequency of a two-level atom in a noisy environment is studied. Our interest is to minimise both the energetic cost of the protocol and the statistical uncertainty of the estimate. In particular, we prepare a probe in a "GHZ-diagonal" state by means of a sequence of qubit gates applied on an ensemble of $n$ atoms in thermal equilibrium. Noise is introduced via a phenomenological time-nonlocal quantum master equation, which gives rise to a phase-covariant dissipative dynamics. After an interval of free evolution, the $n$-atom probe is globally measured at an interrogation time chosen to minimise the error bars of the final estimate. We model explicitly a measurement scheme which becomes optimal in a suitable parameter range, and are thus able to calculate the total energetic expenditure of the protocol. Interestingly, we observe that scaling up our multipartite entangled probes offers no precision enhancement when the total available energy $\mathcal{E}$ is limited. This is at stark contrast with standard frequency estimation, where larger probes---more sensitive but also more "expensive" to prepare---are always preferred. Replacing $\mathcal{E}$ by the resource that places the most stringent limitation on each specific experimental setup, would thus help to formulate more realistic metrological prescriptions.