Phase Estimation from Amplitude Collapse in Correlated Matter-Wave Interference

TL;DR

Phase Estimation from Amplitude Collapse (PEAC) introduces a robust statistical approach to extract differential phases from correlated interferometers without requiring phase-stable fringes. By monitoring amplitude collapses and revivals in the combined signal $S_{\text{all}}$, PEAC uses histogram PDFs and the amplitude relationship $A_{\text{all}}^2 = \sum_i \lambda_i^2 + 2 \sum_{i<j} \lambda_i \lambda_j \cos(\theta_i-\theta_j)$ to infer $\theta$ and external accelerations, even in non-state-selective or three-component mixtures. Compared to ellipse fitting, PEAC substantially reduces bias near degeneracy points (up to ~80% improvement in trueness) at the expense of some precision, and remains applicable across fully phase-stable and noisy regimes. The method is demonstrated experimentally with magnetically sensitive substates of $^{87}$Rb in a Mach–Zehnder Bragg interferometer and is supported by numerical replication and bootstrapping, highlighting its potential to enhance accuracy in next-generation matter-wave sensors and related correlated interferometric platforms.

Abstract

Operating matter-wave interferometers as quantum detectors for fundamental physics or inertial sensors in real-world applications with unprecedented accuracies relies on noise rejection, often implemented by correlating two sensors. Such sensors can be spatially separated (gradiometry or gravitational-wave detection) or consist of different internal states (magnetometry or quantum clock interferometry), in which case a signal-amplitude modulation may serve as a signature of a differential phase. In this work, we introduce Phase Estimation from Amplitude Collapse (PEAC) by applying targeted fitting methods for different magnetically sensitive substates of an atom interferometer. We demonstrate that PEAC provides higher trueness (up to 80% bias reduction) than standard tools for perfectly correlated signals. At its working point near, but not exactly at phase settings resulting in vanishing amplitude, it achieves precision competitive with standard methods, contrasting prior claims of optimal operation at vanishing amplitude. PEAC presents a generally applicable complementary evaluation method for correlated interferometers without phase stability, increasing the overall accuracy and enabling applications beyond atom interferometry.

Figures (4)

• Figure 1: Spacetime diagram and signal extraction. (A) Mach--Zehnder atom interferometer (MZI) realised by three Bragg pulses separated by $T$, driving transitions between momentum states $\left|0 \hbar k_\text{eff}\right\rangle$ (purple) and $\left|1 \hbar k_\text{eff}\right\rangle$ (green). The phase of the final pulse is shifted by $\phi_\text{las}$, thus scanning the interference fringes at the two exit ports. The MZI is followed by a time of flight (TOF) that spatially separates the exits, during which an optional Stern--Gerlach (SG) field can separate the ${m_F}$ substates of the $F=1$ ground state within each momentum class in addition. (B) Fringe scans ($20$ settings of $\phi_\text{las}$, $15$ repetitions averaged, uncertainties omitted for visual clarity) at $T = 1.6\,\text{ms}$. Without the SG field, we observe the fringe $S_\text{all}$ (black), with the corresponding histogram from $300$ measurements (left marginal, also displayed as a density plot on the far left) has a double-peak structure that depends on the signal's amplitude. Applying an SG field allows resolving the individual signals $S_{m_F}$ per substate (blue, yellow, red), whose histogram distributions are shown at the right marginal.
• Figure 2: Interference fringes, amplitude collapse, and signal correlation. (A) Coloured dots in panels (i)--(iv) show the averaged interference fringes of $S_{+1}$ (red), $S_{-1}$ (blue), and $S_\text{sum}$ (orange) for four different values of $T$, i. e. for distinct differential phases $\theta$. The beat of $S_{+1}$ and $S_{-1}$ leads to a collapsing amplitude of $S_\text{sum}$ for $\theta\cong\pi$ (iii). Uncertainties are omitted for visual clarity. (B) Collapse and revival of the amplitude $A_\text{sum}$ (orange) as a function of $T$, inferred from the underlying histograms of $S_\text{sum}$ shown as density plot, where brighter regions correspond to higher bin counts. The collapses occur at the beat nodes around $T=1.7\,\text{ms}$ and $T=3\,\text{ms}$. Dashed white lines indicate the times (i)--(iv) from (A). (C) Bivariate scatter plots for times (i)--(iv) of the correlated signals $S_{+1}$ (vertical) and $S_{-1}$ (horizontal), with a colour coding as in (A) indicating the phase scan of $\phi_\text{las}$. The eccentricity of the emerging ellipse encodes the differential phase $\theta$. At $T=1.7\,\text{ms}$$(\theta\cong\pi)$ and $T=2.5\,\text{ms}$$(\theta\cong3\pi/2)$, the ellipses become degenerate and collapse to lines along the anti-diagonal and diagonal, respectively.
• Figure 3: Rotated bivariate scatter plots and histograms. Rotating the original correlation signals (insets) for $T=1.7\,\text{ms}$ ($\theta\cong\pi$, left) and $T=2.5\,\text{ms}$ ($\theta\cong3\pi/2$, right) by $\pi/4$ clockwise aligns the ellipses to their principal axes, which correspond to $S_\text{sum}$ (horizontal) and $S_\text{diff}$ (vertical). The marginal distributions show the histograms of $S_\text{sum}$ and $S_\text{diff}$, illustrating that the two exchange their roles at the respective degeneracy points of $\theta$. This representation visualises the intrinsic link between ellipse-based estimation based on bivariate fitting and PEAC applied to $S_\text{sum}$ and $S_\text{diff}$. Note that $S_\text{diff}$ is only accessible by state-selective detection.
• Figure 4: Performance of phase estimation: Ellipse fitting versus PEAC. Comparison of ellipse fitting (dark blue) and PEAC along the $S_\text{sum}$ (orange) and $S_\text{diff}$ (light blue) directions. Experimental data are shown as crosses (error bars omitted for clarity) or rectangular bars, with centres indicating the mean and heights equal to twice the bootstrapped standard deviation. Solid lines are values obtained from numerical replications of the experiment. The dashed lines correspond to the set phases and amplitudes as reference values free of fitting errors. We convert the set phase $\theta_\text{set}$ via Eq. \ref{['eq:theta']} to $T$. (A) Reconstructed phase. The inset shows that ellipse fitting leads to a large bias $\theta_\text{bias} = \theta_\text{rec}-\theta_\text{set}$ near degeneracy, despite high precision, while PEAC along the minor axis $S_\text{sum}$ exhibits larger uncertainty but a reduced bias. (B) Modulation of the amplitudes $A_\text{sum}$ and $A_\text{diff}$ obtained from histogram fits along the two principal axes. We observe a deviation at degeneracy, contributing to PEAC's remaining bias. (C) Bias obtained from numerical replications. PEAC along the favourable (minor) axis reduces the bias by $80\,\text{\%}$, substantially enhancing trueness. The semi-transparent bands give the uncertainties extracted from $1000$ replicated experiments. (D) Precision of phase estimation. Ellipse fitting maintains low uncertainty across all $\theta_\text{set}$, while PEAC of $S_{sum}$ achieves comparable precision only near, but not at the degeneracy point $\theta_\text{set} = \pi$, as would be expected from simple Gaussian uncertainty propagation. This behaviour arises from an ambiguity in the histogram's PDF for small amplitudes.