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.
Highlights
Metrology plays an important role in many areas of physics and engineering [1]
We introduce the basics of multiparameter quantum metrology, and we adapt that formalism to our problem
We investigated the precision limits of measuring the gradient of a magnetic field with atomic ensembles arranged in different geometries and initialized in different states
Summary
Metrology plays an important role in many areas of physics and engineering [1]. With the development of experimental techniques, it is possible to realize metrological tasks in physical systems that cannot be described well by classical physics and instead quantum mechanics must be used for their modeling. We calculate lower bounds on the precision of estimating B1 based on a measurement on the state after it passed through the unitary dynamics U = exp(−iH t), where t is the time spent by the system under the influence of the magnetic field. In schemes in which the gradient is calculated based on measurements on two separate atomic ensembles or different atoms in a chain, the measuring operators can always commute with each other [14,15,46].
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