Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit

Optomechanics allows the transduction of weak forces to optical fields, with many efforts approaching the standard quantum limit. We consider force sensing in a mirror-in-the-middle optomechanical setup and use two coupled cavity modes originated from normal mode splitting for separating pump and probe fields. We find that this two-mode model can be reduced to an effective single-mode model if we drive the pump mode strongly and detect the signal from the weak probe mode. We show that the optimal force sensitivity at zero frequency (dc) beats the standard quantum limit via squeezing and is limited by mechanical thermal noise and optical losses. We also find that the bandwidth is proportional to the cavity damping in the resolved sideband regime. Finally, the squeezing spectrum of the output signal is calculated and it shows that almost perfect squeezing at dc is possible by using a high-quality factor and high-frequency mechanical oscillator. Considering a homodyne measurement efficiency of 99%, the squeezing will be limited to 23 dB, with requires input power ${10}^{-2}$ smaller than conventional single-mode optomechanical systems.

Figure 1

(a) Symmetric mirror-in-the-middle optomechanical system comprising a Fabry-Pérot cavity of length $2L$ with a high-reflectivity mirror mounted in the middle and coupled to a mechanical oscillator. Displacements of the middle mirror (via mechanical oscillations) couple the two normal modes $a$ (red) and $b$ (blue) as the left-right symmetry is broken. (b) Normal mode frequencies ${\omega }_{\text{cav},\pm }$ (green) as a function of middle-mirror displacement $x$. (c) Transmission spectrum of the three-mirror cavity, showing pairs of normal modes. We drive mode $a$ strongly (long red line) and detect mode $b$ (short blue line).

Figure 2

(a) Rescaled force sensitivity at dc as a function of $\Delta /\kappa$ for $\xi =0.1,0.01,0.001$. At $\Delta /2\kappa$, the minimum value is achieved. In (b)–(d) we choose ${\omega }_m=2\pi \times {10}^6\mathrm{Hz},\kappa =0.2{\omega }_m,\Delta =2\kappa$, and $\xi =0.01$ as the base values to plot the rescaled sensitivity as a function of frequency. The bandwidth is the frequency range where the best sensitivity is maintained. (b) Bandwidth dependence on $\kappa$, where ${\omega }_m$ is fixed and $\kappa$ varies and the ratio $\kappa /{\omega }_m$ is shown in plot. In the resolved sideband regime the bandwidth increases with $\kappa$ linearly, while in the unresolved regime it does not increase much. (c) Bandwidth dependence on $\xi$, which is approximately linear. (d) Bandwidth dependence on $\Delta /\kappa$, where $\kappa$ is fixed and $\Delta$ varies. The dependence is approximately in the form $\sqrt{1+{(\Delta /\kappa )}^2}$ and suggests a narrower bandwidth than our estimate (24).