# Recovering correlations in optomechanical heterodyne spectra for high-precision quantum displacement sensing

## TL;DR

This paper introduces a post-processing method called r-heterodyning that recovers lost correlations in heterodyne spectra for quantum displacement sensing, enhancing spectral information without extra experimental constraints.

## Contribution

The paper presents a novel post-processing technique to recover correlations in heterodyne spectra, improving quantum displacement sensing capabilities.

## Key findings

- r-heterodyning effectively recovers spectral correlations
- Simulations show excellent agreement with quantum noise spectra
- Method enhances spectral features without additional experimental constraints

## Abstract

Homodyne and heterodyne detection represent "twin-pillars" of quantum displacement sensing using optical cavities, having permitted major breakthroughs including detection of gravitational waves and of the motion of quantum ground-state cooled mechanical oscillators. Both can suffer disadvantages as diagnostics in quantum optomechanics, either through symmetrisation (homodyne), or loss of correlations (heterodyne). We show that, for modest heterodyne beat frequencies ($\Omega \sim \omega_M/10 \gg \Gamma$), judicious construction of the autocorrelation of the measured current can either recover (i) a spectrum with strong sidebands but without an imprecision noise floor (ii) a spectrum which is a hybrid, combining both homodyne and heterodyne sideband features. We simulate an experimental realisation with stochastic numerics and find excellent agreement with analytical quantum noise spectra. We term such retrospective recovery of lost heterodyne correlations "r-heterodyning": as the method simply involves post-processing of a normal heterodyne time signal, there is no additional experimental constraint other than on the magnitude of $\Omega$.