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 reduce the noise induced by vacuum fluctuations, via injection of squeezed states of light. Future detectors, such as Einstein Telescope, may require the use of two filter cavities or a three-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 optical layouts and their resilience with respect to different degradation sources (optical losses, mismatching, locking precision), analytically and numerically. 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 further highlight the role of mode-mismatch phases in limiting squeezing. Finally, 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.