# Quantum parameter-estimation of frequency and damping of a harmonic-oscillator

## TL;DR

This paper derives the quantum Cramér-Rao bound for estimating the frequency and damping of a damped quantum harmonic oscillator in Gaussian states, providing a comprehensive solution and practical implications for nanoscale resonator measurements.

## Contribution

It extends quantum parameter estimation to include damping, offers a unified approach for frequency estimation in Gaussian states, and suggests feasible experimental sensitivities with current nanotechnology.

## Key findings

- Derived the quantum Cramér-Rao bound for damping and frequency estimation.
- Unified previous partial results into a comprehensive solution.
- Proposed that current nanotube resonators can achieve electron-mass sensitivity.

## Abstract

We determine the quantum Cram\'er-Rao bound for the precision with which the oscillator frequency and damping constant of a damped quantum harmonic oscillator in an arbitrary Gaussian state can be estimated. This goes beyond standard quantum parameter estimation of a single mode Gaussian state for which typically a mode of fixed frequency is assumed. We present a scheme through which the frequency estimation can nevertheless be based on the known results for single-mode quantum parameter estimation with Gaussian states. Based on these results, we investigate the optimal measurement time. For measuring the oscillator frequency, our results unify previously known partial results and constitute an explicit solution for a general single-mode Gaussian state. Furthermore, we show that with existing carbon nanotube resonators (see J. Chaste et al.~Nature Nanotechnology 7, 301 (2012)) it should be possible to achieve a mass sensitivity of the order of an electron mass $\text{Hz}^{-1/2}$.