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

Abstract

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 $F_{\text{eff}}$ for such measurements with a Bayesian post-correction and show that the Cramer-Rao bound for the final precision is $\frac{1}{\sqrt{F_{\text{eff}}}}$. 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 $F_{\text{eff}} \sim \frac{N^2}{a+bN^2}$ 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, $L\geq \ell+2$, we show that the effective Fisher information has a Heisenberg scaling $F_{\text{eff}}= O(N^2)$ when optimized over the Hilbert space. Finally, we study the density of the effective Fisher information in the Hilbert space and show that when $L\geq \ell+2$, almost any Haar random state has a Heisenberg scaling, i.e., $F_{\text{eff}}=O(N^2)$. 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.

Figures (3)

• Figure 1: In situ error detection in quantum sensing:a Effect of random errors ($\bm{\epsilon}$) on the precision of estimating an unknown parameter $\varphi$. Each datapoint $i$ is sampled from a conditional $P(i|\varphi, \bm{\epsilon})$. If the errors are unknown, one has access only to the average distribution $P(i|\varphi)$, which is less sensitive than the conditional to the unknown $\varphi$. b. Errors in a system of two-mode bosons are indistinguishable from the signal; in multi-mode bosons, where we can simultaneously readout the occupancies $\hat{n}_i$, the "extra information" can be used to detect and post correct the errors. c. Post corrected precision $\Delta^2\varphi$ saturates to a constant, if there are too many sources of error ($\ell$) and too few modes ($L$) to correct them all. We show that if $L\geq \ell+2$, this precision has a Heisenberg scaling $\sim\frac{1}{N^2}$ with the atom number.
• Figure 2: System and setup:a. Controls available in multi-mode bosons, i.e., BEC trapped in an optical lattice (see text). b. Schematic of noisy quantum sensing, with the relevant example of gravimetry with multi-mode bosons. c. Example of noise in the on-site potential.
• Figure 3: Numerical illustrations:a. The effective Fisher information $F_{\text{eff}}$ for $\ell=4$ channels of error saturates for any $L<\ell+2=6$ due to a singularity in the QFI. For $L\geq \ell+2$, it scales asymptotically as $\mathcal{O}(N^2)$. b.$F_{\text{eff}}$ at a fixed $N=10^4$, showing that one needs at least $L=\ell+2$ to mitigate the errors. Each curve corresponds to a different $\ell$ and saturates at $L=\ell+2$. c, d. Minimum eigenvalue of the $F_Q$ of a state obtained after a random experimental sequence. The markers are numerical results and the solid line is the formula in Eq. (\ref{['typ']}). This shows that we can generate a random pulse sequence on the fly and produce a Heisenberg scaling in error mitigated quantum sensing of gravity.