  Quantum fluctuations in the radiation pressure of light can excite stochastic
motions of mechanical oscillators thereby realizing a linear quantum
opto-mechanical coupling. When performing a precise measurement of the position
of an oscillator, this coupling results in quantum radiation pressure noise. Up
to now this effect has not been observed yet. Generally speaking, the strength
of radiation pressure noise increases when the effective mass of the oscillator
is decreased or when the power of the reflected light is increased. Recently,
extremely light SiN membranes with high mechanical Q-values at room temperature
have attracted attention as low thermal noise mechanical oscillators. However,
the power reflectance of these membranes is much lower than unity which makes
the use of advanced interferometer recycling techniques to amplify the
radiation pressure noise in a standard Michelson interferometer inefficient.
Here, we propose and theoretically analyze a Michelson-Sagnac interferometer
that includes the membrane as a common end mirror for the Michelson
interferometer part. In this new topology, both, power- and signal-recycling
can be used even if the reflectance of the membrane is much lower than unity.
In particular, signal-recycling is a useful tool because it does not involve a
power increase at the membrane. We derive the formulas for the quantum
radiation pressure noise and the shot-noise of an oscillator position
measurement and compare them with theoretical models of the thermal noise of a
SiN membrane with a fundamental resonant frequency of 75 kHz and an effective
mass of 125 ng. We find that quantum radiation pressure noise should be
observable with a power of 1 W at the central beam splitter of the
interferometer and a membrane temperature of 1 K.
