  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.
