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arXiv 2603.03426quant-phcond-mat.quant-gas

玻色子晶格重力仪中非马尔可夫误差的贝叶斯后修正

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

Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；New Mexico Consortium（新墨西哥联盟）

Bharath Hebbe Madhusudhana, Andrew Harter, Avadh Saxena

备注：
14 pages, 3 figures
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AI总结：

研究了存在随机空间非均匀性的玻色子囚禁原子重力仪中的误差问题，提出利用贝叶斯推断进行后修正，分析了有效费舍尔信息的标度行为及在不同模式数下的表现。

摘要：

我们研究了存在随机空间非均匀性的玻色子囚禁原子重力仪中的误差问题。由于随机、次次波动的空间非均匀性导致的误差是量子非马尔可夫性的。我们证明在具有L>2模式（即囚禁站点）的系统中，这些误差可通过贝叶斯推断进行后修正。后修正通过原位测量误差并根据测量误差优化数据处理来实现。我们定义了具有贝叶斯后修正的此类测量的有效费舍尔信息F_eff，并证明最终精度的克拉默-拉奥下界为1/sqrt(F_eff)。通过研究有效费舍尔信息与原子数N的标度关系，我们发现当误差源过多而模式数过少时，它趋于常数。即，对于ℓ个独立误差源，我们证明当模式数较小时（L<ℓ+2），有效费舍尔信息的标度为F_eff ~ N²/(a + bN²)（其中常数a, b>0），即使在最大化希尔伯特空间的情况下。当模式数较大（L≥ℓ+2）时，我们证明有效费舍尔信息在优化希尔伯特空间时具有海森堡标度F_eff=O(N²)。最后，我们研究了希尔伯特空间中有效费舍尔信息的密度，并证明当L≥ℓ+2时，几乎任何哈达玛随机态都具有海森堡标度，即F_eff=O(N²)。基于这些结果，我们开发了一种类似洛施密特回声的实验序列用于误差抑制的重力仪和梯度仪，并讨论了潜在的实现方式。最后，我们论证有效费舍尔信息可以解释为等效非厄米特演化对应的费舍尔信息。

英文摘要：

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.

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