# Multi-parameter estimation with multi-mode Ramsey interferometry

## TL;DR

This paper demonstrates that multi-parameter estimation at the standard quantum limit can be achieved using multi-mode unentangled states in Ramsey interferometry, provided the unitary transformation is orthogonal, with practical implementation considerations.

## Contribution

It generalizes single-parameter Ramsey interferometry to multi-parameter estimation using unentangled states and orthogonal transformations, offering a scalable and experimentally feasible approach.

## Key findings

- Multi-parameter estimation at SQL with unentangled states is possible.
- Orthogonal unitary transformations enable multi-parameter interferometry.
- Particle number fluctuation and noise effects are analyzed.

## Abstract

Estimating multiple parameters simultaneously is of great importance to measurement science and application. For a single parameter, atomic Ramsey interferometry (or equivalently optical Mach-Zehnder interferometry) is capable of providing the precision at the standard quantum limit (SQL) using unentangled probe states as input. In such an interferometer, the first beam splitter represented by unitary transformation $U$ generates a quantum phase sensing superposition state, while the second beam splitter $U^{-1}$ recombines the phase encoded paths to realize interferometric sensing in terms of population measurements. We prove that such an interferometric scheme can be directly generalized to estimation of multiple parameters (associated with commuting generators) to the SQL precision using multi-mode unentangled states, if (but not iff) $U$ is orthogonal, i.e. a unitary transformation with only real matrix elements. We show that such a $U$ can always be constructed experimentally in a simple and scalable manner. The effects of particle number fluctuation and detection noise on such multi-mode interferometry are considered. Our findings offer a simple solution for estimating multiple parameters corresponding to mutually commuting generators.