Gravitational parameter estimation in a waveguide

We investigate the intrinsic uncertainty in the accuracy to which a static spacetime can be measured from scattering experiments. In particular, we focus on the Schwarzschild black hole and a spatially kinked metric that has some mathematical resemblance to an expanding universe. Under selected conditions we find that the scattering problem can be framed in terms of a lossy bosonic channel, which allows us to identify shot-noise scaling as the ultimate scaling limit to the estimation of the spacetimes. Fock state probes with particle counting measurements attain this ultimate scaling limit and the scaling constants for each spacetime are computed and compared to the practical strategies of coherent state probes with heterodyne and homodyne measurements. A promising avenue to analyze the quantum limit of the analogue spacetimes in optical waveguides is suggested.