Optimal Phase-Insensitive Force Sensing with Non-Gaussian States

TL;DR

This work develops a framework for phase-insensitive force sensing with a single bosonic mode subjected to phase randomization, showing that non-Gaussian $N$-spaced states nearly saturate the FI bound for small displacements and can outperform Gaussian probes under realistic decoherence. Using excitation-number-resolving measurements, the authors derive the FI and establish a universal bound $f_{\alpha} \le 1+2\langle \hat{n} \rangle$, with pure $N$-spaced states approaching it for small $\alpha$; 1-spaced states yield no gain while 2-spaced (parity) states exhibit favorable scaling. By solving a Lindblad master equation and performing decoherence-aware quantum optimal control in a minimal spin-boson model, they identify a transition where number-squeezed cat (moon) states or Fock-like states maximize force sensitivity under losses and heating, illuminating a path toward robust, high-precision sensing across mechanical, optical, and microwave platforms. The findings highlight the trade-offs between state spacing, phase-space features, and decoherence, and provide a practical route to enhance weak-force sensing with diverse continuous quantum systems.

Abstract

Quantum metrology enables sensitivity to approach the limits set by fundamental physical laws. Even a single continuous mode offers enhanced precision, with the improvement scaling with its occupation number. Due to their high information capacity, continuous modes allow for the engineering of quantum non-Gaussian states, which not only improve metrological performance but can also be tailored to specific experimental platforms and conditions. Recent advancements in control over continuous platforms operating in the quantum regime have renewed interest in sensing weak forces, also coupling to massive macroscopic objects. In this work, we investigate a force-sensing scheme where a physical process completely randomizes the direction of the induced phase-space displacement, and the unknown force strength is inferred through excitation-number-resolving measurements. We find that $N$-spaced states, where only every $N^{\text{th}}$ Fock state occupation is nonzero, approach the achievable sensing bound. Additionally, non-Gaussian states are shown to be more resilient against decoherence than their Gaussian counterparts with the same occupation number. While Fock states typically offer the best protection against decoherence, we uncover a transition in the metrological landscape -- revealed through a tailored decoherence-aware Fisher-information-based reward functional -- where experimental constraints favor a family of number-squeezed Schrödinger cat states. Specifically, by implementing quantum optimal control in a minimal spin-boson system, we identify these states as maximizing force sensitivity under lossy dynamics and finite system controllability. Our results provide a pathway for enhancing force sensing in a variety of continuous quantum systems, ranging from massive systems like mechanical oscillators to massless systems such as quantum light and microwave resonators.

Figures (15)

• Figure 1: (a) Squeezed, narrow phase-space features along the known force direction are necessary for sensing schemes to manifest quantum advantage. (b) If the direction of phase displacement is completely randomized during the sensing process, the narrow features need to be present along all directions. (c) Quantum non-Gaussian states with discrete rotational symmetry Grimsmo2020, such as compass Shukla2023, grid Gottesman2001a, or $\ket{mn}$ states Asenbeck2025Lachman2025 (from left to right), approach the Heisenberg limit with excitation-number-resolving measurements. (d) In an optimally controlled spin-boson model with finite controllability and decoherence, a family of number-squeezed cat states maximizes achievable sensitivity for weak force sensing.
• Figure 2: Metrological gain $f_{\alpha}$ for Fock (solid) and Gaussian (dashed) states under loss [$\ell$; (a,c,e,f)] or heating [h; (b,d)]. (a)-(d) $f_{\alpha}$ for various $\left\langle \hat{n} \right\rangle$ at fixed $\alpha = 0.005$ and $\alpha = 0.07$ [(a,c) and (b,d), respectively], computed via the approximation \ref{['small-time']} [(a,b)] or full dynamics \ref{['full-master']} [(c,d)]. Fock states outperform Gaussian ones. (e,f) $f_{\alpha}^{\ell}$ for Gaussian and Fock states with $\left\langle \hat{n} \right\rangle=5$ at finite $\gamma \tau$, plotted against $\alpha$. The shaded region marks sup-vacuum displacements $\alpha>0.5$. Dashed line indicates $\gamma \tau = 0$. Fock states maintain gain across the whole range, unlike Gaussian ones.
• Figure 3: (a) Metrological gain $f_{\alpha}$ in a spin-boson system maximized via an optimal drive. The solutions for $\alpha \leq 0.1$ are identical for a fixed $\tau_\text{max}$. The low-opacity curve corresponds to the states optimized for $\alpha = 0.005$, but evaluated for $\alpha = 0.2$. (b) Corresponding occupation numbers $\left\langle \hat{n} \right\rangle$. The transition for $\alpha = 0.2$ is visible. (c,d) Wigner functions of the optimal solutions for $\alpha = 0.005, 0.05, 0.1$ (c) and $\alpha = 0.2$ (d). (e,f) Metrological gain $f_{\alpha}$ at the end of the state preparation for the optimal states under loss with $\alpha = 0.1$ (e) and heating with $\alpha = 0.2$ (f). Red lines signify the maximum achievable FI at a fixed decoherence strength. (g,h) Maximally achievable FI for various values of $\alpha$. Curves are constructed analogously to the red lines from (e,f). At lower decoherence levels, consistent with (a), optimal solutions yield higher FI for smaller $\alpha$, with the trend reversing at larger values of $\gamma$.
• Figure A1: (a) Number occupation of $N$-spaced state---only every $N^{\text{th}}$ Fock state is occupied. (b) Effect of the action of channel \ref{['mixing']} in the closest vicinity to some initially occupied $\rho_n \neq 0$. In the limit of small $\alpha$, the change in probability distribution becomes smaller as the distance from the $n$-site grows. (c) Contributions to the FI from respective sites. Nonzero values come only from $n+1$- and $n$-sites.
• Figure C1: Metrological gain $f_\alpha$ for different families of non-Gaussian states as a function of $\alpha$ (a,b) and $\left\langle \hat{n} \right\rangle$ (c,d). The families include 2-spaced states (a,c): moon (green-to-blue solid lines), Gaussian (dashed purple), GKP (dashed brown), number-phase (dashed sky blue), and 4-spaced states (b,d): $\ket{3}+\ket{7}$ (solid purple), compass (solid brown), and number-phase (solid blue sky) states. Solid black lines represent Fock state. Either $\langle \hat{n} \rangle=5$ is fixed (a,b) or $\alpha=0.05$ (c,d). The considered non-Gaussian states outperform the Gaussian ones, with 4-spaced states performing almost as well as Fock states for small values of $\alpha$ and over a broad range of $\langle \hat{n} \rangle$.