  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
membrane's 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.
