Sequential quantum-enhanced measurement with an atomic ensemble

We propose a quantum-enhanced iterative (with $K$ steps) measurement scheme based on an ensemble of $N$ two-level probes which asymptotically approaches the Heisenberg limit ${\ensuremath{\delta}}_{K}\ensuremath{\propto}{R}^{\ensuremath{-}K/(K+1)}$, where $R$ is the number of quantum resources. The protocol is inspired by Kitaev's phase estimation algorithm and involves only collective manipulation and measurement of the ensemble. The iterative procedure takes the shot-noise-limited primary measurement with precision ${\ensuremath{\delta}}_{1}\ensuremath{\propto}{N}^{\ensuremath{-}1/2}$ to increasingly precise results, ${\ensuremath{\delta}}_{K}\ensuremath{\propto}{N}^{\ensuremath{-}K/2}$. We propose an implementation of the algorithm for the measurement of a magnetic field using a two-component atomic cloud of bosons.