Theoretical Analysis of Mechanical Displacement Measurement Using a Multiple Cavity Mode Transducer

We present an optomechanical displacement transducer that relies on three cavity modes parametrically coupled to a mechanical oscillator and whose frequency spacing matches the mechanical resonance frequency. The additional resonances allow reaching the standard quantum limit at a substantially lower input power (compared to the case of a single cavity mode), as both sensitivity and quantum backaction are enhanced. Furthermore, it is shown that in the case of multiple cavity modes, coupling between the modes is induced via reservoir interaction, e.g., enabling quantum backaction noise cancellation. Experimental implementation of the schemes is discussed in both the optical and microwave domain.

Figure 1

Illustration of the triple mode transducer scheme. Auxiliary cavity resonances at ${\omega }_0\pm {\Omega }_m$ permit resonant motional sideband buildup.

Figure 2

The measurement noise spectrum $S_{xx,\theta =\pi /2}^{\mathrm{tot}}({\Omega }_m)$ [normalized to the zero point motion spectral density $S_{xx}^{\mathrm{zpf}}({\Omega }_m)=\hslash |{\chi }_0({\Omega }_m)|$] for the triple cavity mode transducer as a function of the input power (with $\kappa ={\Omega }_m/10$) compared to the single mode case. The traces in the horizontal plane follow the minima as a function of the normalized cavity lifetime.

Figure 3

The normalized quantum backaction force spectral density $S_{FF}^{\mathrm{dual}}(\Omega )/S_{FF}^{\mathrm{dual}}(+{\Omega }_m)$ (thick solid line) is plotted together with the result from a classical model (thick dashed line). The reservoir interaction between the modes leads to complete noise cancellation at $\Omega ={\Omega }_m/2$. The thin dashed lines indicate the backaction force coming from two uncorrelated modes.

Figure 4

(a) Three degenerate optical modes are coupled via semitransparent mirrors. (b) Three LC oscillators are capacitively coupled in series. While mode $\widehat{a}$ is connected to an external drive, mode $\widehat{c}$ is parametrically coupled to a mechanical oscillator. When the coupling rate $g_c$ exceeds the individual decay rate, the spectrum exhibits normal mode splitting.