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Physical Review A

Optimal estimation of the magnetic field in atomic magnetometry with arbitrary shape of the probe light beam

Jaromı́r Fiurášek

Published Apr 2, 2008• 0 citations
Publisher

Abstract

We investigate atomic magnetometry based on the rotation of a collective spin of atomic ensemble in a magnetic field. The rotation is monitored by a quantum nondemolition interaction of the atomic cloud with a probe light beam. We derive the optimal shape of the light pulse that for a given fixed measurement time and pulse energy minimizes the measurement error. We show that the error can be somewhat reduced by using a properly shaped pulse, but the asymptotic scaling of the error cannot be improved with respect to the scheme with constant intensity of the light beam.

DOI
10.1103/physreva.77.045801

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We investigate atomic magnetometry based on the rotation of a collective spin of atomic ensemble in a magnetic field. The rotation is monitored by a quantum nondemolition interaction of the atomic cloud with a probe light beam. We derive the optimal shape of the light pulse that for a given fixed measurement time and pulse energy minimizes the measurement error. We show that the error can be somewhat reduced by using a properly shaped pulse, but the asymptotic scaling of the error cannot be improved with respect to the scheme with constant intensity of the light beam.
Sensing a magnetic field with an atomic magnetometer operated in real time presents significant challenges, primarily due to sensor nonlinearity, the presence of noise, and the need for one-shot estimation. To address these challenges, we propose a comprehensive approach that integrates measurement, estimation, and control strategies. Specifically, this involves implementing a quantum nondemolition measurement based on continuous light probing of the atomic ensemble. The resulting photocurrent is then directed into an extended Kalman filter to produce instantaneous estimates of the system’s dynamical parameters. These estimates, in turn, are utilized by a linear quadratic regulator, whose output is applied back to the system through a feedback loop. This procedure automatically steers the atomic ensemble into a spin-squeezed state, yielding a quantum enhancement in precision. Furthermore, thanks to the feedback proposed, the atoms exhibit entanglement even when the measurement data are discarded. To prove that our approach constitutes the optimal strategy in realistic scenarios, we derive ultimate bounds on the estimation error applicable in the presence of both local and collective decoherence, and show that these are indeed attained. Additionally, we demonstrate that, for large ensembles, the extended Kalman filter not only reliably predicts its own estimation error in real time, but also accurately estimates spin squeezing at short timescales.
We present a theory for the estimation of a scalar or a vector magnetic field by its influence on an ensemble of trapped spin-polarized atoms. The atoms interact off resonantly with a continuous laser field, and the measurement of the polarization rotation of the probe light, induced by the dispersive atom-light coupling, leads to spin squeezing of the atomic sample which enables an estimate of the magnetic field that is more precise than that expected from standard counting statistics. For polarized light and polarized atoms, a description of the non classical components of the collective spin angular momentum for the atoms and the collective Stokes vectors of the light field in terms of effective Gaussian position and momentum variables is practically exact. The Gaussian formalism describes the dynamics of the system very effectively and accounts explicitly for the back action on the atoms due to measurement and for the estimate of the magnetic field. Multicomponent magnetic fields are estimated by the measurement of suitably chosen atomic observables and precision and efficiency is gained by dividing the atomic gas in two or more samples which are entangled by the dispersive atom-light interaction.
We address the estimation of the magnetic field B acting on an ensemble of atoms with total spin J subjected to collective transverse noise. By preparing an initial spin coherent state, for any measurement performed after the evolution, the mean-square error of the estimate is known to scale as , i.e. no quantum enhancement is obtained. Here, we consider the possibility of continuously monitoring the atomic environment, and conclusively show that strategies based on time-continuous non-demolition measurements followed by a final strong measurement may achieve Heisenberg-limited scaling and also a monitoring-enhanced scaling in terms of the interrogation time. We also find that time-continuous schemes are robust against detection losses, as we prove that the quantum enhancement can be recovered also for finite measurement efficiency. Finally, we analytically prove the optimality of our strategy.
We address the estimation of the magnetic field B acting on an ensemble of atoms with total spin J subjected to collective transverse noise. By preparing an initial spin coherent state, for any measurement performed after the evolution, the mean-square error of the estimate is known to scale as 1/J1/J1/J, i.e. no quantum enhancement is obtained. Here, we consider the possibility of continuously monitoring the atomic environment, and conclusively show that strategies based on time-continuous non-demolition measurements followed by a final strong measurement may achieve Heisenberg-limited scaling 1/J21/{J^2}1/J2 and also a monitoring-enhanced scaling in terms of the interrogation time. We also find that time-continuous schemes are robust against detection losses, as we prove that the quantum enhancement can be recovered also for finite measurement efficiency. Finally, we analytically prove the optimality of our strategy.
We address metrological protocols for the estimation of the intensity and the orientation of a magnetic field, and show that quantum-enhanced precision may be achieved by probing the field with an arbitrary spin at thermal equilibrium. A general expression is derived for the ultimate achievable precision, as given by the quantum Fisher information. The optimal observable is shown to correspond to the spin projection along a temperature-dependent direction, and allows a maximally precise parameter estimation also through ensemble measurements. Finally, we prove the robustness of our scheme against deviations of the measured spin projection from optimality.