  The measurement of extremely small displacements is of utmost importance,
both for fundamental studies [1-4], and practical applications [5-7]. One way
to estimate a small displacement is to measure the Doppler shift generated in
light reflected off an object moving with a known periodic frequency. This
remote sensing technique converts a displacement measurement into a frequency
measurement, and has been considerably successful [8-14]. The displacement
sensitivity of this technique is limited by the Doppler frequency noise floor
and by the velocity of the moving object. Other primary limitations are hours
of integration time [12,13] and optimal operation only in a narrow Doppler
frequency range. Here we show a sensitive device capable of measuring
$\mu$Hz/$\sqrt{\text{Hz}}$ Doppler frequency shifts corresponding to tens of fm
displacements for a mirror oscillating at 2 Hz. While the Doppler shift
measured is comparable to other techniques [12.13], the position sensitivity is
orders of magnitude better, and operates over several orders of magnitude of
Doppler frequency range. In addition, unlike other techniques which often rely
on interferometric methods, our device is phase insensitive, making it
unusually robust to noise.
