The extraordinary sensitivity of the output field of an optical cavity to small quantum-scale displacements has led to breakthroughs such as the first detection of gravitational waves cite{LIGO,LIGODC} and of the motions of quantum ground-state cooled mechanical oscillators cite{Teufel2011,Chan2011}. While heterodyne detection of the cavity field preserves asymmetries which provide a key signature that mechanical oscillators has attained the quantum regime, detection of a rotating quadrature of the light averages out important quantum correlations, yielding a weaker signal and lower sensitivity than homodyne detection. In turn, homodyning, detects a single optical quadrature, but loses the important quantum sideband asymmetries. In the present work we present and experimentally demonstrate a technique, involving judicious construction of the autocorrelators of the output current using filter functions, which can restore the lost correlations (whether classical or quantum), drastically augmenting the useful information extracted: the filtering adjusts for moderate errors in the locking phase of the local oscillator, allowing efficient single-shot measurement of hundreds of different field quadratures and rapid mapping of detailed features from a simple heterodyne trace. One may also control whether the correlations are recovered in isolation or interfere with the usual stationary heterodyne sidebands. In the latter case we obtain a spectrum of hybrid homodyne-heterodyne character, with motional sidebands of combined amplitudes comparable to homodyne. We term such recovery of lost heterodyne correlations with filter functions r-heterodyning: although investigated here in a thermal regime, its robustness and generality represents a promising new approach to sensing of quantum-scale displacements.
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