Multiparameter estimation with an array of entangled atomic sensors

TL;DR

This work demonstrates quantum-enhanced multiparameter sensing with arrays of entangled atomic sensors. By distributing a globally spin-squeezed state across multiple sensors and dynamically redistributing squeezing with local rotations, the authors realize flexible, high-precision estimation of multiple local parameters and their nonlocal linear combinations, surpassing the standard quantum limit. They develop a unified framework based on a joint Cramer-Rao bound and show that the proposed estimators can approach the bound in ideal conditions, achieving significant gains for two- and three-sensor configurations and for generic linear combinations. The results establish a practical route toward quantum-enhanced field imaging and sensor networks, with potential applications in vector magnetometry, imaging, and clock networks, and provide a foundation for scaling to larger sensor arrays and compressed-sensing-inspired sensing tasks.

Abstract

In quantum metrology, entangled states of many-particle systems are investigated to enhance measurement precision of the most precise clocks and field sensors. While single-parameter quantum metrology is well established, many metrological tasks require joint multiparameter estimation, which poses new conceptual challenges that have so far only been explored theoretically. We experimentally demonstrate multiparameter quantum metrology with an array of entangled atomic ensembles. By splitting a spin-squeezed ensemble, we create an atomic sensor array featuring inter-sensor entanglement that can be flexibly configured to enhance measurement precision of multiple parameters jointly. Using an optimal estimation protocol, we achieve significant gains over the standard quantum limit in key multiparameter estimation tasks, thus grounding the concept of quantum enhancement of field sensor arrays and imaging devices.

Figures (6)

• Figure 1: Array of entangled atomic sensors for multiparameter estimation. An array of $M$ sensors, each consisting of a collective spin $\mathbf{S}_k$ of $N_k$ two-level atoms, is used to determine $M$ parameters $\theta_1$, $\theta_2$, … , $\theta_M$ that are encoded on the sensors as local spin rotations. The sensor spins are prepared by coherently splitting a two-component BEC in a spin-squeezed state, resulting in entanglement between atoms within each sensor and between different sensors. In combination with individual spin rotations and detection, the entanglement enables a statistical gain in the determination of the $M$ parameters compared to the case without quantum correlations.
• Figure 2: Joint estimation of two parameters with two entangled atomic sensors. (A) Parameters $\theta_1$ and $\theta_2$ are imprinted on the two sensor spins. (B) Absorption image of the two atomic clouds with $N_1\approx N_2$. (C) Correlation plot of simultaneous measurements of $\theta_1$ and $\theta_2$, showing strong correlations due to the inter-sensor entanglement. Two datasets are shown for two different values of $\theta_2$, each with $1200$ repetitions (purple and blue color, respectively). (D) Histograms obtained from the measurements in (C) for $\theta_2$ (top), $\theta_+$ (middle), and $\theta_-$ (bottom). The measurement of $\theta_+$ exploits the inter-sensor entanglement, resulting in the smallest variances. Dashed lines: distribution for an ideal coherent spin state. (E) Correlation plot similar to (C), but for measurements with a $\pi$-pulse applied to $\mathbf{S}_2$ prior to parameter imprinting. (F) Histograms for the data in (E). Now, the measurement of $\theta_-$ shows minimal variance due to the entanglement.
• Figure 3: Joint estimation of two local parameters enhanced by nonlocal squeezing. (A) Measurement results of $\theta_+$ (teal) and $\theta_-$ (violet) for different applied rotations $\theta_2$. Error bars: standard deviations (SD) of measurement outcomes. Solid lines: linear fit to SD. Shaded areas: SQL for an ideal coherent spin state. (B) Joint estimation of $\theta_1$ (orange) and $\theta_2$ (green) from properly weighted measurements of $\theta_+$ and $\theta_-$ as described in the text, with error bars and solid lines indicating SD as in (A). Dashed lines: SD obtained if inter-sensor entanglement is ignored. Shaded areas: SQL. (C) Comparison of quantum gains for estimating $\theta_1$ and $\theta_2$ using different strategies: Unentangled atoms (gray and light gray), local measurements ignoring inter-sensor entanglement (pink and blue), and joint estimation using non-local entanglement (orange and green). Solid lines indicate the corresponding theoretical expectations. The square points in teal and violet show the quantum gain for estimating only $\theta_+$ or only $\theta_-$, respectively. Teal line: initial squeezing. All error bars are standard errors of the mean. (D) Histograms of the measurements of $\theta_1$ and $\theta_2$ at an applied $\theta_2=0$ using colors as in C. The top histograms show results for unentangled atoms. The histograms in the middle and bottom rows show data from the same experimental runs with entangled atoms. In the middle row, local estimators make use only of local entanglement within each sensor. In the bottom row, joint estimation makes use also of the non-local entanglement between the sensors.
• Figure 4: Optimizing the estimation of different linear combinations of parameters. Linear combinations of two parameters $\mathbf{n}\cdot\boldsymbol{\theta}=\cos (\alpha)\,\theta_1 + \sin (\alpha)\,\theta_2$ are characterized by the mixing angle $\alpha$. For a given $\alpha$, we engineer the quantum correlations between the two sensors in order to harness the entanglement in an optimal way. The resulting quantum gain is shown for ten different linear combinations (data points in main plot). Six examples are shown in the insets, colour code matching the data points.
• Figure 5: Joint multiparameter estimation with $M=3$ entangled atomic sensors. (A) Schematic of the three entangled sensor spins on which three local parameters are imprinted (top) and absorption image of the three atomic clouds (bottom). (B) Matrix of metrological gains compared to the SQL for four different sensor preparations and four estimated parameter combinations. Each row corresponds to a different preparation with $\pi$-pulses applied to the spins $(\mathbf{S}_1,\mathbf{S}_2,\mathbf{S}_3)$ as indicated. Each colum corresponds to the estimation of a different linear combination $(\pm 0.644\, \theta_1\pm 0.431\,\theta_2 + 0.632\,\theta_3)$ with signs $(\pm,\pm,+)$ as indicated. Quantum gain is observed on the diagonal, where the sensor configuration matches the parameter combination.