Theoretical study of the squeezed-light-enhanced sensitivity to gravity-induced entanglement via finite-time analysis

We investigate the advantage of using squeezed input light for generating gravity-induced entanglement (GIE) through Fourier-domain analysis. Based on the findings of Ref.~\cite{Miki2024}, which demonstrated the feasibility of detecting GIE in optomechanical systems under quantum control, we further demonstrate that squeezed input light can reduce the optical noise in the mechanical conditional state and enhance GIE. Furthermore, we estimate the systematic and statistical errors in the measurement of GIE using the Fourier transformation over a finite measurement time. Based on the error estimations using the signal-to-noise ratio (SNR) in GIE detection, we find that a total measurement time of $10^6\,\mathrm{s}$ is required to achieve ${\rm SNR} = 1$ when using squeezed input light, whereas $10^{6.8}\,\mathrm{s}$ is needed without squeezed input light. This result highlights the effectiveness of optomechanical systems and the critical role of squeezed input light in enhancing the detectability of GIE.