Towards gravimetry enhancement with squeezed states

TL;DR

This work investigates gravimetry with squeezed Gaussian probes, focusing on how the squeezing phase affects estimation of the gravitational acceleration $g$. By solving the nonrelativistic dynamics in a uniform gravitational field and applying Gaussian-state quantum metrology, the authors derive a closed-form QFI $\mathcal{F}_{g}(\tau,r,\theta)$ that depends on both squeezing amplitude and phase, showing that phase-engineered squeezing can surpass the shot-noise limit even when canonical quadrature squeezing fails. A position–momentum correlated squeezing phase ($\theta=\pi/4$) yields a robust quantum advantage for all evolution times, and a projective momentum measurement (with a time-dependent $\theta_{\text{opt}}(\tau)$) saturates the quantum limit. These results emphasize the crucial role of phase control in CV gravimetry and provide guidance for designing experiments that reach the quantum limit in gravity sensing.

Abstract

We investigate the estimation sensitivity of gravitational acceleration using squeezed probe states within a quantum metrology framework. In particular, we analyze how the squeezing phase, beyond its amplitudes, of the probes affects the attainable precision. We find that probes squeezed along the canonical phase-space quadrature can fail to achieve a quantum Fisher information (QFI) surpassing the shot-noise limit, regardless of the interaction time with the gravitational field. In contrast, position-momentum correlated input states with the squeezing amplitude can overcome this limit. Furthermore, we show that optimal sensitivity is attained through projective momentum measurements combined with a time-dependent adjustment of the squeezing phase. Our results are important to highlight the fundamental role of phase-engineered squeezing in experimental gravimetry protocols.

Figures (5)

• Figure 1: QFI for estimation of $g$ as a function of the evolution time $\tau/\tau_{0}$ for different amplitudes and phases of the squeezed input state. Panels (a-c) correspond to the short-time evolution regime, while panels (d-f) depict the dynamics at the long-time limit. Curves represent squeezing intensities $r=0.4$ (solid black line), $r=0.5$ (dashed red line), $r=0.6$ (dotted blue line), for distinct squeezing phases $\theta$ of the squeezed input state (\ref{['GSS']}) and vacuum state $r=0$ (dot-dashed gray line). Both panels illustrate that the quantum advantage in estimating gravitational acceleration with squeezed probe states initialized in canonical directions, such as position ($\theta = 0$) and momentum ($\theta=\pi/2$), is limited to a certain time [shaded area in Fig. \ref{['fig01']}(a, e)]. In contrast, probes with position-momentum correlation offer a consistent improvement in the QFI over any vacuum state, regardless of the evolution time [shaded area in Fig. \ref{['fig01']}(c, f)].
• Figure 2: The Relative Quantum Fisher Information (RQFI) $\mathcal{Q}$, with values $\mathcal{Q} > 1$ implies an advantage using squeezed states, $\mathcal{Q} < 1$ indicates no gain, and $\mathcal{Q} = 1$ denotes equal performance with vacuum states in the estimation of $g$. Panel (a) illustrates the temporal evolution of $\mathcal{Q}$ for different squeezing directions in phase space. The solid black line corresponds to squeezing applied along the canonical position quadrature direction ($\theta = 0$), the dashed red line represents squeezing along the momentum quadrature ($\theta = \pi/2$), and the dotted blue line depicts a probe exhibiting position-momentum correlations, with the squeezing direction chosen at an intermediate phase of $\theta = \pi/4$. In (b), we display a contour plot of $\mathcal{Q}$ as a function of time and squeezing phase $\theta$. It highlights that multiple directions beyond $\theta = \pi/4$ yield $\mathcal{Q} > 1$, demonstrating the broad flexibility in optimizing squeezing orientation for enhanced metrological performance. In all plots, we fixed the squeezing amplitude at $r=0.5$.
• Figure 3: Behavior of the ratio $\mathcal{R}(\tau,r,\theta,s) = \mathcal{I}_{g}(\tau,r,\theta,s)/\mathcal{F}_g(\tau,r,\theta)$ between the CFI and the QFI bound for different measurement schemes, as a function of the squeezing phase $\theta$ and the evolution time $\tau$. Here, the squeezing parameter $r=0.5$ is kept fixed across all panels and the measurement setting $s$ varies from one panel to another. Panel (a) shows the case $s \rightarrow \infty$ (projective momentum measurement), (b) corresponds to $s \rightarrow 0$ (projective position measurement), and (c) illustrates the heterodyne scheme with $s = 1$. Note that only the momentum measurement strategy can saturate the QFI bound, yielding the maximum possible information [$\mathcal{R} = 1$, yellow region in Fig. \ref{['fig3']}(a)]. Yet, the optimal choice of squeezing phase is time-dependent, and squeezing along canonical quadratures, position ($\theta=0$) or momentum ($\theta=\pi/2$), does not always provide the best sensitivity at all times.
• Figure S1: Wigner function of the squeezed input state (\ref{['eq01_SM']}) for different values of amplitude $r$ and phase $\theta$, showing that, for example, in the specific case $\theta = \pi/4$, the state exhibits position-momentum correlations, resulting in an elliptical shape with the quadratures correlated along this direction in phase space.
• Figure S2: Sensitivity for estimating gravitational acceleration $g$ as a function of evolution time $\tau$. The plot compares probes prepared with a fixed squeezing of 4.3 dB ($r=0.5$). The solid black line corresponds to squeezing applied along the canonical position quadrature direction ($\theta = 0$), the dashed red line represents squeezing along the momentum quadrature ($\theta = \pi/2$), and the dotted blue line depicts a probe exhibiting position-momentum correlations, with the squeezing direction chosen at an intermediate phase of $\theta = \pi/4$. The shot-noise limit (indicated by the dot-dashed gray line) is shown for reference. We simulate an ensemble of Cesium atoms with a mass of $m=2.21\times 10^{-25}$ kg, a position uncertainty of $\sigma_0=30$ nm, and for a free-fall time of 500 ms Gerginov_2025Metrologia, yielding a sensitivity of $1.0\times 10^{-7}$ m$\cdot$s$^{-2}/\sqrt{\text{Hz}}$ that is consistent with current technology Qvarfort2018.