Abstract
Continuously monitored atomic spin-ensembles allow, in principle, for real-time sensing of external magnetic fields beyond classical limits. Within the linear-Gaussian regime, thanks to the phenomenon of measurement-induced spin-squeezing, they attain a quantum-enhanced scaling of sensitivity both as a function of time, t, and the number of atoms involved, N. In our work, we rigorously study how such conclusions based on Kalman filtering methods change when inevitable imperfections are taken into account: in the form of collective noise, as well as stochastic fluctuations of the field in time. We prove that even an infinitesimal amount of noise disallows the error to be arbitrarily diminished by simply increasing N, and forces it to eventually follow a classical-like behaviour in t. However, we also demonstrate that, ‘thanks’ to the presence of noise, in most regimes the model based on a homodyne-like continuous measurement actually achieves the ultimate sensitivity allowed by the decoherence, yielding then the optimal quantum-enhancement. We are able to do so by constructing a noise-induced lower bound on the error that stems from a general method of classically simulating a noisy quantum evolution, during which the stochastic parameter to be estimated—here, the magnetic field—is encoded. The method naturally extends to schemes beyond the linear-Gaussian regime, in particular, also to ones involving feedback or active control.
Highlights
Optical magnetometers based on atomic spin-ensembles [1] are considered today as stateof-the-art magnetic-field sensors competing head to head in sensitivity with SQUIDbased devices [2] without need of cryogenic cooling, while being already miniaturised to chip scales [3]
We have studied the problem of sensing a magnetic field in real time within the canonical atomic magnetometry setting—a polarised spin-ensemble is being continuously probed in the perpendicular direction to induce spin-squeezing of the atoms, so that a quantumenhanced precision in estimating the field can be maintained
As the so-obtained limit applies to any type of statepreparation and continuous-measurement scheme—as long as the conditional dynamics of the magnetometer in between subsequent measurements is not changed—it has allowed us to prove that the continuous measurement model of our interest may often be considered to be optimal in presence of any, even infinitesimal, noise
Summary
Optical magnetometers based on atomic spin-ensembles [1] are considered today as stateof-the-art magnetic-field sensors competing head to head in sensitivity with SQUIDbased devices [2] without need of cryogenic cooling, while being already miniaturised to chip scales [3]. One can answer fundamental questions using techniques of quantum metrology [14, 15], in particular, by how much can the sensitivity (1) be improved by allowing for arbitrary quantum states of the atomic ensemble and measurements more general than the natural light-probing scheme based on the Faraday effect [16] This has lead to the seminal observation that “Equation One” can be breached—in particular, its 1/N -behaviour commonly referred to as the Standard Quantum Limit (SQL)—by preparing the atomic ensemble in an entangled state [17], so that the MSE can in principle attain the ultimate Heisenberg Limit ∼ 1/N 2 [18].
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