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arXiv:quant-ph/0605237 · Physical Review A

Estimation of fluctuating magnetic fields by an atomic magnetometer

Vivi Petersen, Klaus Mølmer

Published Oct 2, 2006• 21 citations
arXivHTMLPDF

Abstract

We present a theoretical procedure to estimate with an atomic magnetometer the time dependence of a magnetic field that fluctuates according to an Ornstein-Uhlenbeck process. The magnetometer applies the detected polarization rotation of an optical probe to measure a collective atomic spin, which precesses due to the magnetic field. Based on the noisy optical detection record, our consistent Gaussian update formalism provides an estimator for the magnetic fields, and we identify analytically the steady-state performance of this estimator. We show that the estimate of the current value of the magnetic field is further improved if noisy measurement data obtained also at later times are taken into account.

DOI
10.1103/physreva.74.043802

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We present a theoretical procedure to estimate with an atomic magnetometer the time dependence of a magnetic field that fluctuates according to an Ornstein-Uhlenbeck process. The magnetometer applies the detected polarization rotation of an optical probe to measure a collective atomic spin, which precesses due to the magnetic field. Based on the noisy optical detection record, our consistent Gaussian update formalism provides an estimator for the magnetic fields, and we identify analytically the steady-state performance of this estimator. We show that the estimate of the current value of the magnetic field is further improved if noisy measurement data obtained also at later times are taken into account.
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.
Atomic magnetometers use the coherent precession of polarized atomic spins to detect and measure magnetic fields. When an off-resonant, linearly polarized probe beam propagates through the atomic vapor cell, the precession will cause a rotation of its polarization plane. Measuring the angle of the rotation, the magnetic field can be calculated. In this paper, we introduce the basic principles of atomic magnetometers and describe the method to measure the angle. Then a dual-phase lock-in amplifier (LIA) is proposed to detect the weak signal of the atomic magnetometer. And we design a simulation for the LIA based on Matlab/Simulink. Our theoretical analysis and simulation results show that the detecting system is well working and it is possible to detect weak signal of magnetic fields in an accurate way.
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.
Quantum entanglement, in the form of spin squeezing, is known to improve the sensitivity of atomic sensors to static or slowly varying fields. Sensing transient events presents a distinct challenge, requires different analysis tools, and has not been shown to benefit from entanglement in practically important scenarios such as spin-precession magnetometry. To address this, we apply concepts from continuous quantum measurements and estimation theory to optical atomic magnetometers, aiming to accurately model these devices, interpret their measurement data, control their dynamics, and achieve optimal sensitivity. Quantifying this optimal performance requires determining a fundamental quantum limit on sensitivity. We derive this limit, imposed by noise, and show that it scales at best linearly with sensing time and atom number N, ruling out any super-classical scaling. This limit is independent of the initial state, measurement, estimator, and measurement-based feedback, and depends only on the decoherence model and the strength of field fluctuations. Thus, finding an estimator that attains this bound proves the sensing strategy optimal. To approach this limit, we develop a quantum dynamical model scalable with N, based on a co-moving Gaussian approximation of the stochastic master equation, which includes measurement backaction and decoherence. This enables a real-time estimation and control architecture integrating an extended Kalman filter with a linear quadratic regulator. Simulating the magnetometer with our model and EKF+LQR strategy shows that quantum-limited tracking of constant and fluctuating fields is within reach of current atomic magnetometers. Our sensing strategy can also track biologically relevant signals, such as heartbeat-like waveforms, and drive the atomic ensemble into an entangled state, even when the measurement record is used for feedback but later discarded.
This study proposes an in situ method for measuring the equilibrium spin polarization and its longitudinal spatial distribution in a spin-exchange relaxation-free (SERF) atomic magnetometer. The total relaxation rate was determined by analyzing the steady-state response of the magnetometer to the triaxial magnetic field. The transfer coefficient between spin polarization along the probe beam axis and the steady-state response was derived from the optical depth of the probe light and the incident/transmitted power through the vapor cell. By applying a DC magnetic field along the sensitive axis and combining the total relaxation rate with the transfer coefficient, the equilibrium spin polarization value under zero-field conditions was obtained. The equilibrium spin polarization distribution along the pump beam propagation direction was measured, showing consistent trends with simulation results, with a maximum relative error of less than 5%. The expanded uncertainty of the proposed equilibrium spin polarization measurement method ranges from 0.8% to 9.7%. The advantages of this method include no requirement for additional optical field or magnetic field modulation, immunity to temperature and temperature gradients, and the capability to directly determine the equilibrium spin polarization value.
We propose to employ a recurrent neural network to estimate a fluctuating magnetic field from continuous optical Faraday rotation measurement on an atomic ensemble. We show that an encoder-decoder architecture neural network can process measurement data and learn an accurate map between recorded signals and the time-dependent magnetic field. The performance of this method is comparable to Kalman filters while it is free of the theory assumptions that restrict their application to particular measurements and physical systems.