A MIMO system identification approach for the longitudinal control of the filter cavity of the advanced virgo gravitational-wave detector

The sensitivity of the second generation ground-based gravitational-wave detectors is mostly limited by quantum noise (QN). The injection of frequency-dependent squeezed vacuum states into the output port of the interferometer has been shown to reduce QN across the entire detector bandwidth. Frequency dependent squeezed states are generated by reflecting a frequency independent squeezed states off a detuned optical cavity: the phase response of the cavity rotates the squeeze angle as a function of frequency. The precision of the longitudinal control of the such cavity, known as filter cavity, is one of the key parameters affecting the QN suppression factor. The target longitudinal control precision was achieved by simultaneously acting on both the cavity length and the frequency of the squeezing main laser. In this scenario, the analysis of this system requires a Multiple-Input Multiple-Output (MIMO) system model. In this work, we demonstrate that a MIMO model is required and show that a MIMO system identification technique is effective to characterize the system and improve its robustness. Ultimately we show that these techniques allow the design of robust filters that can keep the cavity residual length fluctuations below 1 pm, allowing for a QN reduction of 4.5 dB at high frequencies and 2 dB at low frequencies in the Advanced Virgo interferometer.