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.
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