Quantum noise of a Michelson-Sagnac interferometer with a translucent mechanical oscillator

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 ($\approx$100 ng) with high mechanical $Q$ values at room temperature ($⩾$10${}^6$) have attracted attention as low thermal noise mechanical oscillators. However, the power reflectance of these membranes is much lower than unity ($<$0.4 at a wavelength of 1064 nm) 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 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.

Figure 1

Schematic of a power- and signal-recycling Michelson-Sagnac interferometer. The translucent mechanical oscillator forms the joint end mirror of the two Michelson arms. While the displacement of the oscillator is measured via the Michelson mode, the transmitted light is stored in the Sagnac mode. The combination of Michelson and Sagnac modes enables the resonant enhancement of both the laser power and the displacement signal by placing power- and signal-recycling mirrors in the input and output ports of the interferometer, respectively. The parameter $L$ is the length of the Michelson interferometer arms, and $L_{\mathrm{SR}}$ is the distance between the beam splitter and the signal-recycling mirror.

Figure 2

The four arrows represent the incident-light fields ($E_{\mathrm{A}},E_{\mathrm{B}}$) and outgoing interference of the reflected and transmitted light ($E_{\mathrm{C}},E_{\mathrm{D}}$). These four fields give rise to radiation pressure effects at the membrane.

Figure 3

Goal sensitivity of the Michelson-Sagnac interferometer to measure radiation pressure noise based on the parameters in Table . The graphs show the sensitivity with and without signal recycling, as labeled. Thick solid and dashed lines (red) are the radiation pressure noise and shot noise, respectively. Thin solid and dashed lines (blue) are thermal noise at 1 K in the cases of the viscous damping and structural damping [30].