Estimating a fluctuating magnetic field with a continuously monitored atomic ensemble

TL;DR

This paper develops a hybrid quantum-classical Gaussian framework for estimating a time-varying magnetic field using continuous optical measurements of an atomic ensemble, demonstrating improved estimation via smoothing techniques.

Contribution

It introduces a unified hybrid quantum-classical model for magnetic field estimation that connects Kalman filtering and quantum Gaussian state theory, including smoothing methods.

Findings

Hybrid model aligns with classical Kalman filtering and quantum Gaussian states.

Smoothing improves magnetic field estimates compared to filtering.

Numerical simulations confirm enhanced estimation accuracy.

Abstract

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 with the classical theory of Kalman filtering and with 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 $t$ improves our estimate of the value of the magnetic field at time $t$, and we present numerical simulations that analyze and explain the improvement over the conventional filtering approach.