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

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 $\Omega$.