# Optimum parameter estimation of shaped phase objects

## TL;DR

This paper presents a general quantum estimation method for optimally determining parameters of phase objects, deriving precision bounds and practical measurement strategies, with applications to nanostructure characterization.

## Contribution

It introduces a universal approach to achieve optimal parameter estimation of phase objects using quantum light, including mode engineering and bounds derivation.

## Key findings

- Derived the quantum Cramér-Rao bounds for phase object parameters.
- Identified conditions where classical and quantum illumination models are equivalent.
- Demonstrated optimal measurement strategies for precise parameter estimation.

## Abstract

We show a general method to estimate with optimum precision, i.e., the best precision determined by the light-matter interaction process, a set of parameters that characterize a phase object. The method derives from ideas presented by Pezze et al., [Phys. Rev. Lett. 119, 130504 (2017)]. Our goal is to illuminate the main characteristics of this method as well as its applications to the physics community, probably not familiar with the formal quantum language usually employed in works related to quantum estimation theory. First, we derive precision bounds for the estimation of the set of parameters characterizing the phase object. We compute the Cr\`amer-Rao lower bound for two experimentally relevant types of illumination: a multimode coherent state with mean photon number N, and N copies of a multimode single-photon quantum state. We show under which conditions these two models are equivalent. Second, we show that the optimum precision can be achieved by projecting the light reflected/transmitted from the object onto a set of modes with engineered spatial shape. We describe how to construct these modes, and demonstrate explicitly that the precision of the estimation using these measurements is optimum. As example, we apply these results to the estimation of the height and sidewall angle of a cliff-like nanostructure, an object relevant in semiconductor industry for the evaluation of nanofabrication techniques.