Ultimate precision limit of noise sensing and dark matter search

A fundamental question that puzzles us today is the nature of the hypothetical dark matter (DM) that makes up a large portion of the entire Universe's energy density, as inferred from multiple astrophysical and cosmological observations and simulations. Due to its weak interaction with ordinary matter, DM is extremely challenging to search for. Moreover, as the frequency of DM is unknown, a search requires a scan over a huge frequency range from terahertz to hertz across systems ranging from opto-mechanical and microwave, which can easily take hundreds of years with the state-of-the-art technology. As much attention is on utilizing quantum metrology, empowered by quantum resources such as squeezing and entanglement, to boost DM search, it is crucial to understand the ultimate precision limits allowed by quantum physics. Axion dark matter search relies on microwave haloscopes—microwave cavities in a magnetic field—that allow axions to convert to microwave photons. Such searches can be modeled as quantum sensing over phase-covariant bosonic Gaussian channels, where the additive noise level encodes the DM presence. While ultimate limits of phase, displacement, loss, and amplifier-gain sensing have been explored, little is known about additive-noise sensing limits under energy constraints. This work derives the ultimate precision limit for energy-constrained noise sensing and thus for axion DM searches, showing entanglement assistance is necessary for optimal performance and that homodyne-based strategies are generally suboptimal compared to photon-counting-based nulling receivers.

The paper situates its contribution within quantum metrology for DM searches that leverage squeezing and entanglement to enhance sensitivity in microwave haloscopes. Prior ultimate limits have been established for phase sensing, displacement sensing, loss sensing, and amplifier gain sensing, but additive-noise sensing under energy constraints remained largely unexplored. Earlier experimental proposals and demonstrations (e.g., HAYSTAC) primarily used squeezed vacuum with homodyne detection, improving over vacuum-homodyne baselines but not reaching the true quantum limit set by photon counting on vacuum or by entanglement-assisted strategies. Teleportation-stretching bounds for noise estimation without energy constraints were known; this work develops a tighter bound relevant for finite-energy regimes and compares it to the teleportation bound. The study also connects to broader literature on entanglement-assisted sensing and sensor networks, highlighting potential scaling advantages and clarifying that more exotic non-Gaussian resources (e.g., GKP states) are unnecessary under energy constraints.

- Model DM search as additive-noise estimation in a phase-covariant bosonic Gaussian channel with transmissivity κ(ω) and additive noise n_g(ω) that includes thermal background and a small axion-induced contribution via cavity susceptibilities χ_mm(ω) and χ_ma(ω). - Use quantum Fisher information (QFI) to bound root-mean-square estimation error via the quantum Cramér-Rao bound. Derive an energy-constrained per-mode QFI upper bound via a unitary-extension (UE) approach, additive across modes/frequencies, and combine it with an energy-unconstrained teleportation-stretching (TP) bound to obtain the tightest applicable bound across photon-number regimes. - Optimize over entanglement-assisted inputs allowing an ancilla A jointly measured with the returned signal R, under a total mean photon-number constraint N_s per mode. Consider Gaussian sources analytically: vacuum, single-mode squeezed vacuum (SV), and two-mode squeezed vacuum (TMSV). Derive closed-form QFI expressions for each and compare to the UE/TP upper bounds. - Design practical measurement protocols to achieve the source QFIs: (1) homodyne detection on vacuum or SV; (2) photon counting (on vacuum); (3) nulling receivers combining (anti-)squeezing operations with photon-number-resolving detection. For SV, apply anti-squeezing to null the return (identity channel limit) then photon count; for TMSV, apply two-mode anti-squeezing to null the return under pure-loss, then joint photon counting on signal and ancilla. - Map Fisher information for channel noise to DM density Fisher information via a parameter-change rule I_{n_a}(ω) ∝ χ_ma(ω)^2 I_{n_g}(ω). Define total (scan-rate-like) Fisher information as an integral over frequency, justified by taking the continuum limit of frequency stepping during scans. - Analyze scan-rate upper bounds and achievable performance for realistic haloscope parameters (temperatures ~35–61 mK, frequencies ~7–10 GHz, practical squeezing up to ~20 dB), including coupling regimes (critical and over-coupling) and low-temperature approximations n_T ≪ 1. Evaluate robustness to practical source thermal noise and discuss measurement configurations measuring both modes vs. signal-only.

- Ultimate precision bounds: Derived an additive per-mode QFI upper bound under energy constraints (UE bound) and combined it with an energy-unconstrained teleportation-stretching bound to obtain the tightest overall bound across N_s. The UE bound is tight for small N_s; the TP bound dominates at very large N_s. - Source optimality: TMSV entanglement-assisted probing is optimal (or near-optimal) across broad parameters; it achieves the TP bound at large squeezing and is optimal in the weak-noise, lossless limit κ=1. SV is optimal only in the lossless case; with loss (κ<1), SV’s QFI saturates to a finite value and can underperform the vacuum limit when n_g<1. - Measurements: Homodyne (including Bell measurement for TMSV) is generally suboptimal. A nulling receiver (anti-squeezing plus photon counting) achieves the SV or TMSV source QFIs in the low-noise limit. For vacuum input, simple photon counting attains the vacuum limit J_VL = 1/[n_g(n_g+1)], vastly outperforming vacuum-homodyne (I_Vac-hom = 1/(1+2 n_g)^2), with I_Vac-hom/J_VL ≈ 2 n_g in the weak-noise limit. - Quantitative benchmarks: - Even with 20 dB of SV, the achievable scan rate remains below the vacuum limit obtained by photon counting on vacuum; SV + homodyne needs ~15 dB of squeezing (lossless, ideal) just to surpass the vacuum photon-counting baseline locally; reaching the vacuum-limit scan rate across the band can require nearly 40 dB of squeezing. - Nulling receivers deliver near-optimal performance; vacuum + photon counting can yield up to ~30 dB advantage over vacuum-homodyne in weak-noise regimes. - There is a constant ~4.1 dB gap between SV-homodyne and the SV QFI limit in total Fisher information analyses. - TMSV + nulling achieves large, near-optimal advantage over the vacuum limit for arbitrary squeezing within practical ranges, often saturating the derived upper bound in over-coupled regimes and low temperature. - Scan-rate interpretation: The total Fisher information over frequency corresponds to the scan rate (information acquisition rate). Under this metric, TMSV + nulling provides linear-in-N_s improvements and can attain the ultimate bound, while SV + homodyne offers only limited gains and can be worse than vacuum + photon counting. - Practicality: More exotic non-Gaussian resources (e.g., GKP states) are unnecessary under energy constraints, as TMSV saturates the bounds; priority should be given to realizing quantum-limited microwave photon-number-resolving detectors.

By modeling axion haloscope searches as additive-noise estimation over bosonic Gaussian channels and optimizing under energy constraints, the study establishes the ultimate precision limits relevant to scan-rate performance. The findings show that entanglement assistance via TMSV, combined with nulling receivers employing photon counting, is necessary to realize near-optimal sensitivity and scan rates. This directly addresses the key challenge in DM searches—rapidly scanning large frequency ranges with limited integration time—by maximizing Fisher information per unit time. The work clarifies that strategies centered on homodyne detection, even when improved by single-mode squeezing, are fundamentally limited and may not surpass the vacuum photon-counting baseline in realistic lossy settings. The results also recast the scan rate as an integrated Fisher information, unifying prior homodyne-based visibility metrics with rigorous quantum estimation theory. For the field, this highlights a concrete technology roadmap: implement microwave photon-number-resolving detectors and deploy TMSV entanglement to reach ultimate scan-rate limits, while avoiding unnecessary complexity of non-Gaussian states. The insights extend to sensor networks, where coherent combination yields an additional M^2 scaling in scan rate on top of quantum advantages.

The paper derives ultimate, energy-constrained precision limits for additive-noise sensing in phase-covariant bosonic Gaussian channels and applies them to axion dark matter searches. It proves that entanglement-assisted TMSV sources with nulling photon-counting receivers achieve near-optimal performance and can saturate the ultimate bounds, while single-mode squeezing is only optimal in the lossless case and often underperforms vacuum photon counting under loss. Homodyne-based strategies are suboptimal by orders of magnitude in weak-noise regimes. In terms of scan rate, TMSV + nulling offers large, close-to-optimal advantages over the vacuum limit with practical squeezing, whereas SV + homodyne requires impractically large squeezing to match vacuum counting. The work provides clear guidance: prioritize development of quantum-limited microwave photon counting and deploy entanglement assistance; exotic non-Gaussian sources are unnecessary under energy constraints. Future research includes experimental realization of nulling receivers in microwave haloscopes, robust photon-number-resolving detection, extension to and demonstration with sensor networks, and detailed studies of robustness to realistic imperfections (e.g., finite ancilla storage, detector inefficiencies, excess input noise).

- Many optimality results are derived in the low-noise (n_g → 0) and low-temperature (n_T ≪ 1) limits; performance away from these limits can deviate. - The TMSV nulling receiver that perfectly nulls the return is optimal for pure-loss channels and in the ng → 0 limit; it does not work for amplifier channels or for sizable ng. - Analyses assume known transmissivity κ from prior calibration and perfect ancilla storage (identity channel) for entanglement-assisted strategies. - Achieving the theoretical limits requires ideal photon-number-resolving detectors and precise nulling (anti-squeezing) operations; practical detector inefficiencies and parameter deviations can degrade performance, though measuring both modes improves robustness. - Some bounds (UE vs. TP) are tight in complementary regimes; at very large N_s the UE bound becomes loose due to assumed environment access, while the TP bound ignores energy constraints. - Upper-bound derivations at times assume ideal input engineering (no extra thermal noise at the input port); practical source preparation noise reduces achievable performance and introduces constant gaps to the bound (~6.9 dB reported). - Arrays with correlated sensors are not explicitly analyzed in the main derivations (though arguments suggest reduction to an effective single-sensor model and potential M^2 scaling).