Estimating spacetime fluctuation strength in SU(1,1) and SU(2) interferometers

High-precision laser interferometers are commonly used to search for signatures of random spacetime fluctuations, in an attempt to understand the fundamental nature of gravity. Conventionally, $SU(2)$ interferometers have been used in such experimental investigations. Motivated by [K. Zheng \textit{et al.}, Photon. Res. \textbf{8}, 1653 (2020)], I assess if there is any advantage to be gained by using an $SU(1,1)$ interferometer instead of the $SU(2)$ interferometer. To this end, I compute the quantum Fisher information for estimating the strength and the correlation length of the spacetime fluctuations, and the corresponding classical Fisher information considering different experimentally relevant measurement schemes, in both types of interferometers. I compare the information metrics corresponding to different parameter regimes that are relevant to both current and possible future experimental setups. This helps me assess if and when the $SU(1,1)$ interferometer offers any advantage over the $SU(2)$ interferometer for estimating either the fluctuation strength or correlation length of the spacetime fluctuations.