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arXiv:0708.2730

Magnetic Field Estimation at and beyond 1/N Scaling via an Effective Nonlinearity

Bradley A. Chase, Heather L. Partner, Brigette D. Black, Ben Q. Baragiola, Robert L. Cook, J. M. Geremia

Published Aug 20, 2007• 2 citations
arXivHTMLPDF

Abstract

We provide evidence, based on direct simulation of the quantum Fisher information, that 1/N scaling of the sensitivity with the number of atoms N in an atomic magnetometer can be surpassed by double-passing a far-detuned laser through the atomic system during Larmor precession. Furthermore, we predict that for N>>1, the proposed double-pass atomic magnetometer can essentially achieve 1/N scaling without requiring any appreciable amount of entanglement.

DOI
10.48550/arxiv.0708.2730

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We provide evidence, based on direct simulation of the quantum Fisher information, that 1/N scaling of the sensitivity with the number of atoms N in an atomic magnetometer can be surpassed by double-passing a far-detuned laser through the atomic system during Larmor precession. Furthermore, we predict that for N>>1, the proposed double-pass atomic magnetometer can essentially achieve 1/N scaling without requiring any appreciable amount of entanglement.
We argue that it is possible in principle to reduce the uncertainty of an atomic magnetometer by double passing a far-detuned laser field through the atomic sample as it undergoes Larmor precession. Numerical simulations of the quantum Fisher information suggest that, despite the lack of explicit multibody coupling terms in the system's magnetic Hamiltonian, the parameter estimation uncertainty in such a physical setup scales better than the conventional Heisenberg uncertainty limit over a specified but arbitrary range of particle number. Using the methods of quantum stochastic calculus and filtering theory, we demonstrate numerically an explicit parameter estimator (called a quantum particle filter) whose observed scaling follows that of our calculated quantum Fisher information. Moreover, the quantum particle filter quantitatively surpasses the uncertainty limit calculated from the quantum Cram\'er-Rao inequality based on a magnetic coupling Hamiltonian with only single-body operators. We also show that a quantum Kalman filter is insufficient to obtain super-Heisenberg scaling and present evidence that such scaling necessitates going beyond the manifold of Gaussian atomic states.
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.
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We demonstrate a precise magnetic microscope based on direct imaging of the Larmor precession of a 87Rb^{87}\mathrm{Rb}87Rb spinor Bose-Einstein condensate. This magnetometer attains a field sensitivity of 8.3  pT/Hz1/28.3\text{ }\text{ }\mathrm{pT}/{\mathrm{Hz}}^{1/2}8.3  pT/Hz1/2 over a measurement area of $120\text{ }\text{ }\ensuremath{\mu}{\mathrm{m}}^{2}$, an improvement over the low-frequency field sensitivity of modern SQUID magnetometers. The achieved phase sensitivity is close to the atom shot-noise limit, estimated as 0.15  pT/Hz1/20.15\text{ }\text{ }\mathrm{pT}/{\mathrm{Hz}}^{1/2}0.15  pT/Hz1/2 for a unity duty cycle measurement, suggesting the possibilities of spatially resolved spin-squeezed magnetometry. This magnetometer marks a significant application of degenerate atomic gases to metrology.
We consider the limitations due to noise (e.g., quantum projection noise and photon shot-noise) on the sensitivity of an idealized atomic magnetometer that utilizes spin squeezing induced by a continuous quantum nondemolition measurement. Such a magnetometer measures spin precession of N atomic spins by detecting optical rotation of far-detuned light. We show that for very short measurement times, the optimal sensitivity scales as N-3/4; if strongly squeezed probe light is used, the Heisenberg limit of N-1 scaling can be achieved. However, if the measurement time exceeds tau(rel)/N-1/2 in the former case, or tau(rel)/N in the latter, where tau(rel) is the spin relaxation time, the scaling becomes N-1/2, as for a standard shot-noise-limited magnetometer.
In this paper, a new type of high-sensitivity magnetometer that can surpass SQUID magnetometers operating in liquid-helium in sensitivity and spatial resolution is discussed. The sensitivity of atomic magnetometers is determined by the electron spin relaxation time and the number of alkali-metal atoms. The relaxation time is usually limited by spin-exchange collisions between atoms and drops at higher alkali-metal density. Spin-exchange collisions preserve the total spin of the atoms, but can change their hyperfine state, which reverses the direction of spin precession in the magnetic field. Thus, spin-exchange collisions usually cause spin decoherence. However, this relaxation process can be suppressed if the spin precession frequency is much smaller than the rate of spin-exchange collisions. These conditions lead to 'motional narrowing', where the spin precession angle between the collisions is so small that the atoms do not lose coherence. A magnetometer operating near zero magnetic field can simultaneously achieve high alkali-metal density and long spin-relaxation time, resulting in very high sensitivity.