Enhancing the Bandwidth of Gravitational-Wave Detectors with Unstable Optomechanical Filters

Advanced interferometric gravitational-wave detectors use optical cavities to resonantly enhance their shot-noise-limited sensitivity. Because of positive dispersion of these cavities—signals at different frequencies pick up different phases, there is a tradeoff between the detector bandwidth and peak sensitivity, which is a universal feature for quantum measurement devices having resonant cavities. We consider embedding an active unstable filter inside the interferometer to compensate the phase, and using feedback control to stabilize the entire system. We show that this scheme in principle can enhance the bandwidth without sacrificing the peak sensitivity. However, the unstable filter under our current consideration is a cavity-assisted optomechanical device operating in the instability regime, and the thermal fluctuation of the mechanical oscillator puts a very stringent requirement on the environmental temperature and the mechanical quality factor.

Figure 1

(a) The typical laser interferometer configuration for advanced GW detectors, with an unstable optomechanical filter (blue) embedded. (b) The tradeoff between the detector bandwidth and peak sensitivity (black and red curves), and the effect of adding an unstable filter in the ideal scenario (blue). (c) A simple flow chart for the entire scheme.

Figure 2

The top panel shows the shot-noise-only strain sensitivity of the unstable-filter scheme, compared with the broadband and narrow-band tuned cases without the filter. The power and arm cavity length are the same as for Advanced LIGO. At high frequencies, the sensitivity deviates from the ideal scenario, because of imperfect phase cancellation of higher-order terms of $\mathrm{\Omega }$ in $e^{i{\varphi }_{\text{arm}}(\mathrm{\Omega })}$ (bottom panel).

Figure 3

Effect of the thermal fluctuation of the mechanical oscillator on the sensitivity.

Figure 4

(a) Schematics of the optomechanical filter. (b) Frequencies of interest: cavity resonance ${\omega }_0$ (red), laser frequency ${\omega }_0+{\omega }_m$ (blue), and its lower sideband (black), with the upper sideband around ${\omega }_0+2{\omega }_m$ (not shown).

Figure 5

The difference between ${\varphi }_{\text{filter}}$ obtained without using the RWA and $-2\arctan (\mathrm{\Omega }/{\gamma }_{\text{opt}})$. The parameter regime with ${\omega }_m\gg {\gamma }_f\gg {\gamma }_{\text{opt}}$ is thus preferred for matching the required frequency dependence.

Figure 6

The figure shows the total quantum noise (including radiation pressure noise) of the unstable filter scheme at different temperatures. The color code for different curves is the same as in Fig. 2. The test mass is assumed to be 40 kg, the same as for Advanced LIGO.