Here we report on the realization of a Michelson-Sagnac interferometer whose purpose is the precise characterization of the motion of membranes showing significant light transmission. Our interferometer has a readout noise spectral density (imprecision) of 3E-16 m/sqrt(Hz) at frequencies around the fundamental resonance of a SiN_x membrane at about 100 kHz, without using optical cavities. The readout noise demonstrated is more than 16 dB below the peak value of the membranes standard quantum limit (SQL). This reduction is significantly higher than those of previous works with nano-wires [Teufel et al., Nature Nano. 4, 820 (2009); Anetsberger et al., Nature Phys. 5, 909 (2009)]. We discuss the meaning of the SQL for force measurements and its relation to the readout performance and conclude that neither our nor previous experiments achieved a total noise spectral density as low as the SQL.
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