Optimal estimation of matter-field coupling strength in the dipole approximation

Abstract

This paper is devoted to the study of Bayesian-inference approach in the context of estimating the dipole coupling strength in matter–field interactions. In particular, we consider the simplest model of a two-level system interacting with a single-mode of the radiation field. Our estimation strategy is based on the emerging state of the two-level system, whereas we determine both the minimum mean-square error and maximum likelihood estimators for uniform and Gaussian prior probability density functions. In the case of the maximum likelihood estimator, we develop a mathematical method which extends the already existing approaches to the variational problem of the average cost function. We demonstrate that long interaction times, large initial mean photon numbers, and non-zero detuning between two-level system transition and the frequency of the electromagnetic field mode have a deleterious effect on the optimality of the estimation scenario. We also present several cases where the estimation process is inconclusive, despite many ideal conditions being met.