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arXiv:quant-ph/0402158 · Physical Review A

Estimation of a classical parameter with Gaussian probes: Magnetometry with collective atomic spins

Klaus Mølmer, Lars Bojer Madsen

Published Nov 1, 2004• 57 citations
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Abstract

We present a theory for the estimation of a classical magnetic field by an atomic sample with a Gaussian distribution of collective spin components. By incorporating the magnetic field and the probing laser field as quantum variables with Gaussian distributions on equal footing with the atoms, we obtain a very versatile description which is readily adapted to include probing with squeezed light, dissipation, and loss and additional measurement capabilities on the atomic system.

DOI
10.1103/physreva.70.052102

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We present a theory for the estimation of a classical magnetic field by an atomic sample with a Gaussian distribution of collective spin components. By incorporating the magnetic field and the probing laser field as quantum variables with Gaussian distributions on equal footing with the atoms, we obtain a very versatile description which is readily adapted to include probing with squeezed light, dissipation, and loss and additional measurement capabilities on the atomic system.
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 consider an ensemble of trapped atoms interacting with a continuous-wave laser field. For sufficiently polarized atoms and for a polarized light field, we may approximate the nonclassical components of the collective spin angular momentum operator for the atoms and the Stokes vectors of the field by effective position and momentum variables for which we assume a Gaussian state. Within this approximation, we present a theory for the squeezing of the atomic spin by polarization rotation measurements on the probe light. We derive analytical expressions for the squeezing with and without inclusion of the noise effects introduced by atomic decay and by photon absorption. The theory is readily adapted to the case of inhomogeneous light-atom coupling [A. Kuzmich and T.A.B. Kennedy, Phys. Rev. Lett. 92, 030407 (2004)]. As a special case, we show how to formulate the theory for an optically thick sample by slicing the gas into pieces, each having only small photon absorption probability. Our analysis of a realistic probing and measurement scheme shows that it is the maximally squeezed component of the atomic gas that determines the accuracy of the measurement.
We study the problem of estimating a time-dependent magnetic field by continuous optical probing of an atomic ensemble. The magnetic field is assumed to follow a stochastic Ornstein-Uhlenbeck process and it induces Larmor precession of the atomic ground-state spin, which is read out by the Faraday polarization rotation of a laser field probe. The interactions and the measurement scheme are compatible with a hybrid quantum-classical Gaussian description of the unknown magnetic field, and the atomic and field variables. This casts the joint conditional quantum dynamics and classical parameter estimation problem in the form of update formulas for the first and second moments of the classical and quantum degrees of freedom. Our hybrid quantum-classical theory is equivalent to the classical theory of Kalman filtering and to the quantum theory of Gaussian states. By reference to the classical theory of smoothing and with the quantum theory of past quantum states, we show how optical probing after time ttt improves our estimate of the value of the magnetic field at time ttt, and we present numerical simulations that analyze and explain the improvement over the conventional filtering approach.
We consider the effective dynamics obtained by double-passing a far-detuned laser probe through a large atomic spin system. The net result of the atom-field interaction is a type of coherent positive feedback that amplifies the values of selected spin observables. An effective equation of motion for the atomic system is presented, and an approximate 2-parameter model of the dynamics is developed that should provide a viable approach to modeling even the extremely large spin systems, with N>>1 atoms, encountered under typical laboratory conditions. Combining the nonlinear dynamics that result from the positive feedback with continuous observation of the atomic spin offers an improvement in quantum parameter estimation. We explore the possibility of reaching the Heisenberg uncertainty scaling in atomic magnetometry without the need for any appreciable spin-squeezing by analyzing our system via the quantum Cramer-Rao inequality. Finally, we develop a realistic quantum parameter estimator for atomic magnetometry that is based on a two-parameter family of Gaussian states and investigate the performance of this estimator through numerical simulations. In doing so, we identify several issues, such as numerical convergence and the reduction of estimator bias, that must be addressed when incorporating our parameter estimation methods into an actual laboratory setting.
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.
The Gaussian state description of continuous variables is adapted to describe the quantum interaction between macroscopic atomic samples and continuous-wave light beams. The formalism is very efficient: a non-linear differential equation for the covariance matrix of the atomic system explicitly accounts for both the unitary evolution, the dissipation and noise due to the atom-light interaction, and the back-action due to homodyne optical detection on the beam after its interaction with the atoms. Applications to atomic spin squeezing and estimation of unknown classical parameters are presented, and extensions beyond the Gaussian states are discussed.