Performance of multiple filter-cavity schemes for frequency-dependent squeezing in gravitational-wave detectors

Gravitational-wave detectors use state-of-the-art quantum technologies to circumvent vacuum fluctuations via squeezed states of light. Future detectors such as Einstein Telescope may require the use of two filter cavities or a 3-mirror, coupled filter cavity to achieve a complex rotation of the squeezing ellipse in order to reduce the quantum noise over the whole detector bandwidth. In this work, we compare the theoretical feasibility and performances of these two systems and their resilience with respect to different degradation sources (optical losses, mismatching, locking precision). We provide both analytical models and numerical insights. We extend previous analysis on squeezing degradation and find that the coupled cavity scheme provides similar or better performances than the two-cavity option, in terms of resilience with respect to imperfections and optical losses. We propose a possible two-step implementation scheme for Einstein Telescope using a single filter cavity that can be possibly upgraded into a coupled filter cavity.