Interferometer readout noise below the standard quantum limit of a membrane

Here we report on the realization of a Michelson-Sagnac interferometer whose purpose is the precise characterization of the motion of, i.e., membranes showing significant light transmission. Our interferometer has a readout noise spectral density (imprecision) of $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}\phantom{\rule{0.16em}{0ex}}\mathrm{m}/\sqrt{\mathrm{Hz}}$ at frequencies around the fundamental resonance of a SiN${}_{\mathrm{x}}$ membrane at about 100 kHz, without using optical cavities. The readout noise demonstrated is more than 16 dB below the peak value of the membrane's standard quantum limit (SQL). This margin is significantly bigger than those of previous works with nanowires [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 emphasize that neither our nor previous experiments achieved a total noise spectral density as low as the SQL due to thermal noise.