Bayesian post-correction of non-Markovian errors in bosonic lattice gravimetry

We study gravimetry with bosonic trapped atoms in the presence of random spatial inhomogeneity. The errors resulting from a random, shot-to-shot fluctuating spatial inhomogeneity are quantum non-Markovian. We show that in a system with L>2 modes (i.e., trapping sites), these errors can be post-corrected using a Bayesian inference. The post-correction is done via in situ measurements of the errors and refining the data-processing according to the measured error. We define an effective Fisher information F_eff for such measurements with a Bayesian post-correction and show that the Cramer-Rao bound for the final precision is frac{1}{sqrt{F_eff}}. Exploring the scaling of the effective Fisher information with the number of atoms N, we show that it saturates to a constant when there are too many sources of error and too few modes. That is, with ell independent sources of error, we show that the effective Fisher information scales as F_eff sim frac{N²}{a+bN²} for constants a, b>0 when the number of modes is small: L<ell+2, even after maximization over the Hilbert space. With larger number of modes, Lgeq ell+2, we show that the effective Fisher information has a Heisenberg scaling F_eff= O\(N²\) when optimized over the Hilbert space. Finally, we study the density of the effective Fisher information in the Hilbert space and show that when Lgeq ell+2, almost any Haar random state has a Heisenberg scaling, i.e., F_eff=O\(N²\). Based on these results, we develop a Loschmidt echo-like experimental sequence for error mitigated gravimetry and gradiometry and discuss potential implementations. Finally, we argue that the effective Fisher information can be interpreted as the Fisher information corresponding to an equivalent non-Hertimitian evolution.