{"id":"8bcb218d-4a7e-4bda-9307-f21db044cc24","arxiv_id":"2608.07456","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The optimal observable for the nucleon-coupling ratio in a dual-alkali comagnetometer is the complex inter-channel ratio; its phase is calibration-free and near-sufficient above ~100 Hz, while the amplitude ratio adds information at low frequencies only if the gain is calibrated.","lead":"This paper shows that in a two-colour alkali comagnetometer, the best way to read out how dark matter couples to protons versus neutrons is a complex ratio of the two channels, and that the phase difference alone is nearly as good above 100 Hz but not at lower frequencies. It tells experimentalists which observable to record and when a calibration-free phase measurement is sufficient, which matters for upcoming axion dark matter searches.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The abstract's 'above ~100 Hz' sufficiency claim rests on the small-noise split of Eq. (13); at the low SNR of a search the phase-only Fisher information is far smaller, so the claim is unestablished and needs an SNR floor.","rationale":"The structural proof that the complex inter-channel ratio Z is the optimal observable, and that the differential phase Δφ is insensitive to a real positive relative gain, is clean and survives scrutiny. The load-bearing weakness is the quantitative headline about the 100 Hz sufficiency threshold, which is exactly the part a practical search would use. The paper is honest in footnote 6 about the small-noise requirement for Eq. (13), but the abstract and figure 3(f) present the sufficiency result without the SNR qualifier, so a reader operating at a detection SNR of order 10 would be misled about how much information Δφ actually retains. The proposed test settles this by computing the exact phase-only Fisher information at finite SNR; if the concern lands, the paper should either derive the finite-SNR phase-only information explicitly or state clearly that the 'above ~100 Hz' claim applies only for SNR ≳ 80. This does not overturn the central structural results, so the appropriate verdict is CONDITIONAL rather than REJECT. The reader's weakest_assumption mentions the small-noise linearization as a caveat, but does not elevate it to the central concern, hence 'partial' agreement.","tokens_in":13441,"tokens_out":31535,"duration_ms":293584,"concrete_test":"For a representative operating point (α=0.578, transfer functions of ref [17], R=0 and R=±0.01), evaluate the exact Fisher information of the phase-only estimator at f=40, 100, 200 Hz for SNR = 5, 10, 20, 50, 80 by numerical integration or Monte Carlo over the two-channel complex Gaussian likelihood (10^6 draws per point). Compute I^{Δφ}_{exact}/I^x and compare with sin^2(arg ∂θ Z) from figure 3(f). If the exact ratio falls more than 20% below the linearized value at SNR≤20 while agreeing within 10% at SNR≥80, the abstract must quote an SNR floor or restrict the sufficiency claim to the high-SNR regime.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 5's differential-phase sufficiency maps and the abstract's claim that 'above ~100 Hz the phase difference alone captures most of the coupling-ratio information' are computed from I^{Δφ}/I^x = sin^2(arg ∂θ Z), which follows from the split I^x = I^{Δφ}+I^{amp} in Eq. (13). Appendix A derives that split by linearizing the measured log-ratio Z and treating its real and imaginary parts as independent Gaussians of common variance w_eff. Footnote 6 concedes this requires the small-noise limit ε = |∂θ Z| σθ ≲ 0.3, i.e. SNR ≳ 80 at 200 Hz for the best intrinsic precision. At finite SNR the marginal phase of a complex Gaussian is not independent of the amplitude: the radius carries additional information about the phase through the radial–angular correlation, so the exact phase-only Fisher information is strictly smaller than (Im ∂R Z)^2/w_eff, and the gap grows as SNR falls. Thus Eq. (12)'s total information is exact at all SNR, but the phase/amplitude split and the sufficiency maps are not valid in the low-SNR regime. Since dark matter searches typically detect signals near threshold, the unqualified 'above ~100 Hz' statement is not supported; the frequency above which Δφ is near-sufficient, and the factor ≳2 improvement below 40 Hz, are themselves SNR-dependent.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper formulates the extraction of the nucleon-coupling ratio R = ξ_n/ξ_p in a dual-alkali 87Rb–39K–3He comagnetometer as a statistical estimation problem. It shows that, after profiling the common unknown drive out of the two-channel likelihood, the complex inter-channel ratio Z = ln(Tb_K/Tb_Rb) is the variance-optimal observable, and that its imaginary part (the differential phase Δφ) is calibration-free while its real part (the log-amplitude ratio) is not. Using Fisher information, the paper derives closed-form expressions for the attainable precision (Eqs. 12 and 13), maps the intrinsic precision and the differential-phase sufficiency over the 20–1000 Hz band and the full coupling-angle range, and concludes that above ~100 Hz Δφ captures most of the coupling-ratio information, while below ~40 Hz the amplitude ratio would improve precision by a factor ≳2 if the relative gain were accurately known.","tokens_in":13701,"tokens_out":7542,"duration_ms":65437,"significance":"If the results hold, the paper provides a valuable conceptual clarification: the optimal observable is not Δφ but the complex ratio, and only its phase is calibration-free. The exact reduction of the profiled Fisher information (Eq. 12 and Appendix A) is elegant, self-contained, and appears correct; the structural observations (i)–(iii) of Section 4 are clearly argued and exact within the stated Gaussian model. The numerical maps are potentially useful for experimental design, subject to the adopted response model and operating parameters of ref. [17]. However, the headline quantitative claims—the ~100 Hz sufficiency threshold and the ≳2 factor below ~40 Hz—are derived from the small-noise phase/amplitude split (Eq. 13) and are not valid at the low SNR typical of a dark matter search; the paper's own footnote 6 concedes this. The discrepancy between the exact structural results and the SNR-limited quantitative predictions is the main weakness, and it is fixable by explicitly qualifying the claims or extending the analysis to finite SNR.","major_comments":[{"comment":"The abstract's statements that 'above ~100 Hz the phase difference alone captures most of the coupling-ratio information' and that below ~40 Hz the amplitude ratio improves precision by a factor ≳2 rest on the phase/amplitude split of Eq. (13), which Appendix A derives under the small-noise linearization. Footnote 6 states that this split requires ε = |∂_θ Z| σ_θ ≲ 0.3, i.e., SNR ≳ 80 at 200 Hz for the best intrinsic precision. Dark matter searches typically operate near the detection threshold (SNR of order a few), where the marginal phase of a complex Gaussian is not independent of the amplitude and the phase-only Fisher information is strictly smaller than (Im ∂_R Z)^2/w_eff. Therefore the frequency thresholds and the factor ≳2 are SNR-dependent and are not established for low-SNR runs. The unqualified abstract claims should be revised to state the required SNR floor, or the analysis should be extended to finite SNR using the exact marginal phase distribution.","section":"Abstract and Section 5, Eq. (13), footnote 6"},{"comment":"The derivation of the phase/amplitude split linearizes the log-ratio Z and treats its quadratures as independent Gaussians of common variance w_eff/|N|^2. This is a first-order approximation that ignores the radial–angular correlation of a complex Gaussian at finite SNR; the independence holds only in the limit ε → 0. Since the sufficiency maps in Figures 3(c), 3(e), and 3(f) and the precision factors quoted in the abstract are precisely statements that weigh phase against amplitude, the paper should either restrict all such claims to the regime ε ≲ 0.3 (and state this condition in the relevant figure captions and the abstract) or compute the exact Fisher information of the marginal phase and log-amplitude distributions and show how the sufficiency maps change as a function of SNR. Without this, a reader cannot assess whether the claimed sufficiency percentages are accurate for a realistic weak-signal experiment.","section":"Appendix A, Eq. (13) and Section 5, Figure 3"},{"comment":"The 'intrinsic precision' κ_θ is defined as σ_θ SNR. For the full complex-ratio information this quantity is SNR-independent because both σ_θ and SNR scale inversely with |N|. However, for the phase-only observable, σ_θ^{Δφ} is taken from the small-noise CR bound based on the approximate linearized likelihood, so κ_θ^{Δφ} is not the exact intrinsic precision for finite SNR. The text should clarify that the phase-only curves in Figures 3(c) and 3(e) are CR bounds under the small-noise approximation, not exact attainable precisions, and that their SNR-independence holds only in that limit.","section":"Section 5, Eq. (15)"}],"minor_comments":[{"comment":"The entry for 39K in the σ_n and σ_p columns appears as '0.034−0.131', which is visually ambiguous; it should be typeset as '0.034' and '−0.131' with appropriate spacing or a clear minus sign.","section":"Table 1"},{"comment":"The frequency thresholds (~100 Hz and ~40 Hz) are stated without reference to the operating parameters α = 0.578 and the noise levels taken from ref. [17]; adding a qualifier such as 'for the baseline parameters of ref. [17]' would make the domain of validity explicit.","section":"Abstract and Section 5"},{"comment":"The caption says the sufficiency panel is fixed by the transfer functions alone, but the panel's color scale shows percentages that derive from the small-noise split of Eq. (13); it should mention that this definition assumes the small-noise regime.","section":"Figure 3, panel (f) caption"},{"comment":"The small-noise condition ε ≲ 0.3 is a central validity condition for Eq. (13) and should be moved from the footnote into the main text, ideally alongside Eq. (13), so that the reader encounters it before the sufficiency claims in Section 5.","section":"Footnote 6"},{"comment":"The title 'Optimal Calibration-Free Observable' could be misread as asserting that the optimal observable itself is calibration-free; since the optimal complex ratio is only partially calibration-free, consider rewording to something like 'Optimal Observable and the Calibration-Free Phase Readout for the Nucleon-Coupling Ratio'.","section":"Title"}],"recommendation":"major_revision","confidential_remarks":"The paper's central formal results are sound and well presented, but the abstract overstates the quantitative conclusions by omitting the small-SNR caveat that the authors themselves document in footnote 6. Given that the paper's own text acknowledges the limitation, a major revision focused on qualifying the sufficiency claims and optionally adding a finite-SNR analysis is feasible within the manuscript's scope. I would advise the editor that the paper is promising but not acceptable in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nWhat's new here: Rosenzweig et al. turn the extraction of the coupling ratio R in a dual-alkali comagnetometer into a proper estimation problem. They prove that the complex inter-channel ratio Z = ln(T_K/T_Rb) is the information-sufficient observable; that it splits into a differential phase and an amplitude ratio; and that only the phase is immune to the relative gain calibration. Then they map, over the 20–1000 Hz band, how much of the coupling information the phase retains. This goes beyond their previous proposal (ref [17]) which used Δφ as a calibration-free readout but didn't ask whether it was optimal or how much information it throws away.\n\nThe math is clean. The profiled Fisher information reduces to |∂R Z|^2/w_eff (Eq. 12) and the Appendix A derivation is straightforward and correct under the stated model: affine transfer functions, independent circular Gaussian readout noise, fully correlated magnetic background. The structural claims (single channel is blind, Z is optimal) are exact, not approximations. The paper is also honest: footnote 6 admits the phase/amplitude split (Eq. 13) requires a small-noise limit ε≲0.3, i.e., detection SNR ≳80 at 200 Hz.\n\nThat footnote is where the trouble starts. The abstract and Section 5 say 'above ~100 Hz the phase difference alone captures most of the coupling-ratio information' without that qualifier. At finite SNR, the exact phase-only Fisher information is strictly smaller than (Im ∂R Z)^2/w_eff, because the amplitude and phase of a noisy complex Gaussian are correlated; the gap grows as the SNR falls. Since DM searches typically look for signals near threshold, the unqualified sufficiency statement is not supported. The factor ≳2 improvement below 40 Hz has the same issue. This is fixable: either add an SNR floor to the abstract and to the relevant sentences, or plot the sufficiency as a function of SNR, not just frequency. As it stands, a reader could take the abstract literally and skip measuring the amplitude ratio when it would actually help.\n\nLesser concerns: the numerical maps rely entirely on the parameters of ref [17] (α=0.578, noise levels), so they inherit any errors there. No code or data are shipped, but the analytic formulas are explicit enough for reimplementation.\n\nWho should read this: groups building or analyzing dual-alkali comagnetometers for axion-like particle searches, and anyone interested in a concise example of profiled Fisher information in a two-channel sensor.\n\nRecommendation: send it to peer review. The central result is correct, useful, and a genuine advance over the prior proposal. With a revised abstract that includes the SNR condition, the paper is publishable. I'd be skeptical of accepting the current version's headline claim as stated, but the fix is small.","headline":"Clean Fisher-information derivation showing the complex inter-channel ratio is the optimal coupling-ratio observable, but the abstract's 'above ~100 Hz' phase sufficiency claim needs an SNR floor.","tokens_in":14276,"tokens_out":8027,"would_cite":true,"duration_ms":67506,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Optimal dark-matter readout is the complex channel ratio","keywords":["axionlike dark matter","comagnetometer","coupling ratio","Fisher information","calibration-free observable","differential phase","spin couplings","dark matter detection"],"falsifier":"A controlled experiment could inject a known oscillating pseudomagnetic field whose coupling orientation $\\theta$ is varied, then measure the transfer functions of the two channels. If $T_b^j(\\theta)$ deviates from $A_j\\cos\\theta+B_j\\sin\\theta$, or if the per-channel noise is not circular and independent at the operating point, the claimed optimality of $Z$ and the greater-than-100 Hz phase sufficiency would fail. Alternatively, at a frequency below roughly 40 Hz, compare the variance of an estimator using $\\Delta\\varphi$ alone with one using the full complex ratio: if the ratio does not improve precision by approximately a factor of two, the Fisher-information reduction is not realized.","tokens_in":13222,"feed_emoji":"🧲","tokens_out":5154,"duration_ms":42537,"temperature":0.7,"pith_summary":"The paper asks which measurement built from the two optical channels of a dual-alkali 87Rb–39K–3He comagnetometer extracts the ratio $R=\\xi_n/\\xi_p$ of neutron to proton spin couplings of an axionlike dark matter field with the smallest statistical error. Treating extraction as a parameter-estimation problem with the common drive amplitude profiled out, it shows the optimal observable is the complex inter-channel ratio $Z=\\ln(T_b^K/T_b^{Rb})$, which splits into the differential phase $\\Delta\\varphi$ and a log-amplitude ratio. Only $\\Delta\\varphi$ is immune to the relative optical gain between channels, and therefore calibration-free. For the chosen operating parameters, above roughly 100 Hz the phase difference alone retains most of the coupling-ratio information; below roughly 40 Hz the amplitude ratio could improve precision by a factor of about two or more, but only if the relative gain is known accurately.","feed_headline":"Optimal dark-matter readout is the complex channel ratio","feed_subtitle":"In a dual-alkali comagnetometer, phase alone is near-optimal above 100 Hz; amplitude can help only with calibrated gain.","key_machinery":"The central object is the profiled two-channel Fisher information, reduced by Lagrange's identity to $I^x_R(S_B)=|\\partial_R Z|^2/w_{\\mathrm{eff}}$. Here $Z=\\ln(s_K/s_{\\mathrm{Rb}})=\\ln(T_b^K/T_b^{\\mathrm{Rb}})$ is the field-referred log ratio, and $w_{\\mathrm{eff}}=w+S_B|1/s_K-1/s_{\\mathrm{Rb}}|^2$ combines the independent per-channel readout variances with the correlated magnetic background weighted by a misalignment factor. The argument rests on the response being affine in the coupling angle, $s_j(\\theta)=a_j\\cos\\theta+b_j\\sin\\theta$, so that differentiating $Z$ separates the information into the squared real part (amplitude) and imaginary part (phase). This object identifies the optimal observable, quantifies the magnetic-background penalty, and yields the phase sufficiency ratio $\\sin^2(\\arg\\partial_\\theta Z)$.","core_discovery":"On the paper's own terms, for a single cell with 87Rb and 39K read out through two optical-rotation channels, all information about the coupling ratio $R$ survives profiling the common unknown drive out of the likelihood, and it is fully contained in the complex ratio $Z=\\ln(T_b^K/T_b^{Rb})$. Writing $Z$ in Cartesian form, $\\operatorname{Re}Z=\\ln|T_b^K/T_b^{Rb}|$ is the amplitude-ratio information and $\\operatorname{Im}Z=\\Delta\\varphi$ is the phase information, and the two contributions to the Fisher information add. The inter-channel gain multiplies only the amplitude, so $\\Delta\\varphi$ is the calibration-free part. Numerically, with the operating point of the companion experimental proposal, $\\Delta\\varphi$ alone is near-sufficient above roughly 100 Hz, while below roughly 40 Hz the amplitude ratio would improve the precision on $R$ by a factor of about two or more, at the price of gain calibration.","pith_inferences":["If the affine-response assumption generalizes, the same drive-profiling construction could apply to cells with more than two readout species, where the calibration-free subspace would be the phase differences around the loop.","The roughly 100 Hz sufficiency threshold is tied to the specific alkali pair and operating point; a similar analysis for other pairs or compensation settings could shift where the amplitude ratio becomes informative.","The amplitude ratio, though not calibration-free, could serve in a calibrated run as a consistency check: disagreement between $R$ from $\\Delta\\varphi$ and from the amplitude ratio would flag gain drift or model failure."],"forward_implications":["Above roughly 100 Hz and over most coupling angles, the differential phase $\\Delta\\varphi$ alone achieves near-minimal variance on $R$, so a calibration-free readout is also statistically near-optimal.","Below roughly 40 Hz, the full complex ratio beats $\\Delta\\varphi$ by at least a factor of two in precision on $R$; recovering that gain requires an accurate inter-channel gain calibration.","Because the correlated magnetic background only rescales $w_{\\mathrm{eff}}$, it changes the overall precision but not the phase–amplitude split or the differential-phase sufficiency.","The same $Z$ construction removes information about signal amplitude, making coupling-ratio extraction and field detection complementary uses of the two channels.","At fixed frequency, the coupling angles with the strongest response have the worst intrinsic precision, an inverse relation that holds over the mapped band."],"supporting_citations":[{"why":"Supplies the dual-alkali single-cell comagnetometer scheme, the transfer-function response model, and the operating parameters that the statistical analysis builds on.","marker":"[17]"},{"why":"Provides the nuclear spin content fractions for 39K, 87Rb, and 3He that define the coupling ratio $R$ through eq. (1).","marker":"[15]"},{"why":"Standard reference for Fisher information and the Cramér–Rao bound used to define attainable precision throughout the paper.","marker":"[28]"},{"why":"Estimation-theory reference underlying the likelihood profiling and information reduction that produces the optimal-observable result.","marker":"[29]"},{"why":"Earlier work showing how to disentangle nucleon couplings with a set of comagnetometers, which the single-cell approach extends and contrasts with.","marker":"[16]"}],"fun_headline_variants":["Complex channel ratio is optimal comagnetometer dark-matter readout","Phase alone near-optimal above 100 Hz; amplitude helps low-frequency","Calibration-free phase wins; amplitude needs gain calibration","Optimal coupling-ratio readout: complex ratio, phase robust","Dark-matter ratio: phase near-sufficient high-freq, amplitude low-freq"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The optimality of $Z$ and the frequency thresholds assume the comagnetometer response is affine in the coupling angle and the noise is independent circular Gaussian per channel with a fully correlated magnetic background, together with the specific operating parameters inherited from the companion proposal.","fun_headline_variants_meta":{"raw":{"variants":["Complex channel ratio is optimal comagnetometer dark-matter readout","Phase alone near-optimal above 100 Hz; amplitude helps low-frequency","Calibration-free phase wins; amplitude needs gain calibration","Optimal coupling-ratio readout: complex ratio, phase robust","Dark-matter ratio: phase near-sufficient high-freq, amplitude low-freq"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000173,"raw_usage":{"total_tokens":1307,"prompt_tokens":1003,"completion_tokens":304,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":619,"completion_tokens_details":{"reasoning_tokens":212}},"tokens_in":619,"tokens_out":304,"duration_ms":2998,"temperature":1.0,"reasoning_tokens":212,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:26:37.022260+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A controlled experiment could inject a known oscillating pseudomagnetic field whose coupling orientation $\\theta$ is varied, then measure the transfer functions of the two channels. If $T_b^j(\\theta)$ deviates from $A_j\\cos\\theta+B_j\\sin\\theta$, or if the per-channel noise is not circular and independent at the operating point, the claimed optimality of $Z$ and the greater-than-100 Hz phase sufficiency would fail. Alternatively, at a frequency below roughly 40 Hz, compare the variance of an estimator using $\\Delta\\varphi$ alone with one using the full complex ratio: if the ratio does not improve precision by approximately a factor of two, the Fisher-information reduction is not realized.","supporting_citations":[{"cited_title":"Correlated comagnetometry for precision measurements","cited_arxiv_id":"2607.18221","evidence_quote":"Supplies the dual-alkali single-cell comagnetometer scheme, the transfer-function response model, and the operating parameters that the statistical analysis builds on."},{"cited_title":"Nuclear spin content and constraints on exotic spin-dependent couplings.New J","cited_arxiv_id":null,"evidence_quote":"Provides the nuclear spin content fractions for 39K, 87Rb, and 3He that define the coupling ratio $R$ through eq. (1)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Standard reference for Fisher information and the Cramér–Rao bound used to define attainable precision throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Estimation-theory reference underlying the likelihood profiling and information reduction that produces the optimal-observable result."},{"cited_title":"Atomic probe of dark matter differential interactions with subatomic particles.Phys","cited_arxiv_id":null,"evidence_quote":"Earlier work showing how to disentangle nucleon couplings with a set of comagnetometers, which the single-cell approach extends and contrasts with."}],"review_version":2}