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

Abstract

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.

The paper asks whether an $SU(1,1)$ interferometer, using optical parametric amplifiers instead of beam splitters, improves estimation of stochastic spacetime-fluctuation strength $\Gamma$ or correlation length $\ell$ relative to an $SU(2)$ interferometer. It models correlated Gaussian phase fluctuations in both arms, derives output displacement vectors and covariance matrices including equal internal losses, and uses a Gaussian-state treatment to calculate quantum Fisher information (QFI) and classical Fisher information for homodyne and finite-resolution photon counting. Writing $j_s=\alpha^2\Gamma\sigma$, where $\sigma$ is the within-arm correlation integral, the reported limiting scaled QFI is $\Gamma^2Q_\Gamma=(1-\eta)j_s$ without squeezing and approaches $1/2$ when $e^{-2\beta}\ll(1-\eta)j_s$. The same limits hold for $\ell^2Q_\ell$ for the chosen correlation model. With current-setup parameters—coherent amplitudes $10^{10}$ versus $10^4$ and loss fractions $5\times10^{-6}$ versus $0.4$ for $SU(2)$ and $SU(1,1)$, respectively—the latter offers no advantage. Hypothetically matching amplitude, loss and wavelength gives $SU(1,1)$ strength-QFI advantages of approximately $1.79$ and $4.28$ at 6 and 10 dB squeezing; corresponding best single-quadrature information ratios are $1.90$ and $6.20$. The analysis retains dominant terms at weak fluctuation strength, assumes coherent photons dominate squeezed photons, and generally takes within-arm correlations much larger than cross-arm correlations. The discussion emphasizes that realistic weak signals favor the low-squeezing regime, where no significant QFI advantage emerges. Near zero squeezing, dark-port counting resolving only one photon captures half the total $SU(2)$ QFI; no comparably convenient optimal $SU(1,1)$ measurement is identified. Advantages require increased signal intensity and amplifier gain.