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arXiv:1703.09056 · Physical Review A

Precision bounds for gradient magnetometry with atomic ensembles

Iagoba Apellaniz, Iñigo Urizar-Lanz, Zoltán Zimborás, Philipp Hyllus, Géza Tóth

Published May 8, 2018• 45 citations
arXivHTMLPDF

Abstract

We study gradient magnetometry with an ensemble of atoms with arbitrary spin. We calculate precision bounds for estimating the gradient of the magnetic field based on the quantum Fisher information. For quantum states that are invariant under homogeneous magnetic fields, we need to measure a single observable to estimate the gradient. On the other hand, for states that are sensitive to homogeneous fields, a simultaneous measurement is needed, as the homogeneous field must also be estimated. We prove that for the cases studied in this paper, such a measurement is feasible. We present a method to calculate precision bounds for gradient estimation with a chain of atoms or with two spatially separated atomic ensembles. We also consider a single atomic ensemble with an arbitrary density profile, where the atoms cannot be addressed individually, and which is a very relevant case for experiments. Our model can take into account even correlations between particle positions. While in most of the discussion we consider an ensemble of localized particles that are classical with respect to their spatial degree of freedom, we also discuss the case of gradient metrology with a single Bose-Einstein condensate.

DOI
10.1103/physreva.97.053603

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We study gradient magnetometry with an ensemble of atoms with arbitrary spin. We calculate precision bounds for estimating the gradient of the magnetic field based on the quantum Fisher information. For quantum states that are invariant under homogeneous magnetic fields, we need to measure a single observable to estimate the gradient. On the other hand, for states that are sensitive to homogeneous fields, a simultaneous measurement is needed, as the homogeneous field must also be estimated. We prove that for the cases studied in this paper, such a measurement is feasible. We present a method to calculate precision bounds for gradient estimation with a chain of atoms or with two spatially separated atomic ensembles. We also consider a single atomic ensemble with an arbitrary density profile, where the atoms cannot be addressed individually, and which is a very relevant case for experiments. Our model can take into account even correlations between particle positions. While in most of the discussion we consider an ensemble of localized particles that are classical with respect to their spatial degree of freedom, we also discuss the case of gradient metrology with a single Bose-Einstein condensate.
We develop a general theory to estimate magnetic field gradients in quantum metrology. We consider a system of NNN particles distributed on a line whose internal degrees of freedom interact with a magnetic field. Usually gradient estimation is based on precise measurements of the magnetic field at two different locations, performed with two independent groups of particles. This approach, however, is sensitive to fluctuations of the offset field determining the level splitting of the particles and results in collective dephasing. In this work, we use the framework of quantum metrology to assess the maximal accuracy for gradient estimation. For arbitrary positioning of particles, we identify optimal entangled and separable states, allowing the estimation of gradients with maximal accuracy, quantified by the quantum Fisher information. We also analyze the performance of states from the decoherence-free subspace (DFS), which are insensitive to the fluctuations of the magnetic offset field. We find that these states allow us to measure a gradient directly, without the necessity of estimating the magnetic offset field. Moreover, we show that DFS states attain a precision for gradient estimation comparable to the optimal entangled states. Finally, for the above classes of states, we find simple and feasible measurements saturating the quantum Cram\'er-Rao bound.
We present a method for measuring magnetic field gradients with macroscopic singlet states realized with ensembles of spin-jjj particles. While the singlet state is completely insensitive to homogeneous magnetic fields, the variance of its collective spin components is highly sensitive to field gradients. We compute the dynamics of this variance analytically for a chain of spins and also for an ensemble of particles with a given density distribution. We find an upper bound on how precisely the field gradient can be estimated from the measured data. Based on our calculations, differential magnetometry can be carried out with cold atomic ensembles using a multipartite singlet state obtained via spin squeezing. On the other hand, comparing the metrological properties of the experimentally prepared state to that of the ideal singlet can be used as further evidence that a singlet state has indeed been created.
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.
ABSTRACT Quantum sensors are devices that utilize quantum systems' properties to achieve high‐precision measurements of physical quantities, such as magnetic fields and temperature. The metrological precision of quantum sensors refers to their ability to provide accurate and reliable measurements with minimal uncertainty. In this paper, we propose a quantum sensor consisting of multiple atoms arranged in a spin chain, exposed to a thermal bath at a temperature and an external weak magnetic field with a magnetic‐field gradient for quantum thermometry and magnetometry. Employing the Hilbert‐Schmidt speed (HSS), a popular measure of quantum statistical speed to quantify precision limits in parameter estimation theory, we have investigated in detail how the magnetic‐field gradient and spin chain position influence the quantum sensor's metrological capabilities. We show that the performance and precision of our quantum sensor depend on the spin chain position and the magnetic field and its gradient; increasing these factors' values reduces estimation precision and weakens quantum sensor performance. Finally, increasing the number of atoms in a quantum sensing structure improves the accuracy of temperature and field estimation.
We propose a method to detect the amplitude gradient of a microwave field by using a pair of entangled two-component Bose-Einstein condensates. We consider the two spatially separated condensates to be coupled to the two different magnetic fields. The magnetic-field gradient can be determined by measuring the variances of population differences and relative phases between the two-component condensates in two wells. The precision of measurement can reach the Heisenberg limit. We study the effects of one-body and two-body atom losses on the detection. We find that the entangled atoms can outperform the uncorrelated atoms in probing the magnetic fields in the presence of atom losses. The effect of atom-atom interactions is also discussed.
In this thesis, first, we investigate the metrological usefulness of a family of states known as unpolarized Dicke states, which turn to be very sensitive to the magnetic field. Quantum mechanics plays a central role in achieving such a high precision. Second, we investigate possible lower bounds on the quantum Fisher information, a quantity that characterizes the usefulness of a state for quantum metrology, using the theory of Legendre transforms such that we obtain tight lower bounds based on few measurements of the initial quantum state that will be used for metrology. And last but not least, we investigate gradient magnetometry, i.e., we develop a theory to study the sensitivity of some states on the change in space of the magnetic field.