{"id":"04932c31-2055-4666-aa20-028871f9f4e1","arxiv_id":"2504.13744","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Out-of-phase coupling between two rotational wobble modes of a levitated, non-spinning ferromagnet reveals the Einstein-de Haas spin-rotation coupling, giving a gyromagnetic g-factor close to the rare-earth-magnet expectation.","lead":"A non-spinning magnet can still act like a gyroscope because the electron spins inside it carry real angular momentum, and this experiment detects that hidden coupling by watching tiny wobbles in a magnet levitated above a superconductor. The measured wobble pattern yields the magnet's spin content and gyromagnetic ratio, supporting proposals for ultra-sensitive magnetometers and quantum-stabilized levitation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unmeasured azimuthal rotation about the easy axis can produce the same out-of-phase signal; the thermal estimate alone does not exclude a residual mechanical spin comparable to a fraction of ωI.","rationale":"The reader identified the same load-bearing concern: the unmeasured rotation angle γ around the spin axis. I agree that this is the weakest assumption in the central argument. I considered other candidates: SQUID phase mismatch is addressed by the stated <0.01° differential phase calibration and by the null check on the auto-correlation out-of-phase components; anisotropic inertia is shown in Eq. (S4) to produce only in-phase coupling; damping is negligible over the 0.2–1 s records; and the g-factor discrepancy is a consistency concern rather than a falsifier of the coupling itself. The γ channel is the only alternative that reproduces the exact π/2 phase signature of the observed elliptical trajectories. The manuscript explicitly acknowledges this limitation, which strengthens the need to flag it. The proposed test would settle it by checking whether the inferred coupling is insensitive to deliberately injected spin. The reader's CONDITIONAL verdict is appropriate and my analysis does not change it, so the verdict is UNCHANGED.","tokens_in":11455,"tokens_out":17572,"duration_ms":197085,"concrete_test":"For the same particle, intentionally impart a controlled mechanical rotation about the easy axis before levitation (e.g., by rolling the particle on the micromanipulator surface or by a short directed gas pulse along a tangent), then measure the inferred fI immediately after loading and again after several rotational damping times; if the deliberately spun configuration yields the same fI as the normal loading within statistical error, residual mechanical rotation is negligible, while a shift would calibrate the γdot channel and show that it must be monitored.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing assumption is that the particle has no mechanical rotation γ about its magnetization/easy axis. As shown in Eq. (S3), a constant γdot enters exactly as (ωI+γdot), so an unobserved rotation from magnetization, loading, or trapped gas would produce the same elliptical librational trajectories and the same out-of-phase cross-correlation as the claimed intrinsic spin-rotation coupling. The paper only estimates the thermal value, γdot_rms = sqrt(kBT/I) < 0.01ωI, and reasons that the two librational modes are thermalized; it does not measure γ or include a control that would exclude a nonthermal residual spin. This matters quantitatively: the inferred g factors are already 10–15% below the 1.28 model value, so a residual γdot ≈ -0.1ωI would fully explain the discrepancy without any error in the intrinsic-spin interpretation. The size scaling and the two-cooldown reproducibility are useful partial checks, but they do not separate ωI from γdot because the same constant offset would shift all measurements coherently. Unless γ is directly constrained, the central claim of measuring the intrinsic angular momentum and g-factor of a non-spinning magnet remains conditional.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports an experiment in which a hard ferromagnetic microsphere is levitated in a superconducting trap and undergoes small librational oscillations around two transverse axes, α and β. The authors measure the cross-correlation between two SQUID readout channels and observe an out-of-phase component, which they interpret as evidence of gyroscopic spin-rotation coupling between the librational modes. For a non-spinning magnet, this coupling is attributed to the intrinsic spin angular momentum of the electron system, i.e., the Einstein-de Haas effect. From the measured coupling they extract the Einstein-de Haas frequency fI and a gyromagnetic g-factor, obtaining fI in the range 0.33-0.88 Hz and g ≈ 1.10-1.19, compared with a model expectation of geff ≈ 1.28 for Nd2Fe14B. The analysis is based on a calibration-free relation that is independent of the coil coupling coefficients, and the results are shown for four configurations including three particle radii and two cooldowns of one particle.","tokens_in":11664,"tokens_out":17944,"duration_ms":161002,"significance":"If the interpretation holds, this is the first direct observation of gyroscopic spin-rotation coupling in the mechanical librations of a non-spinning macroscopic ferromagnet, a regime that connects to proposals for ultrasensitive magnetometry and quantum-spin-stabilized levitation. The extraction of ωI via Eq. (9) is genuinely parameter-free with respect to the detection cross-couplings, which is a strong methodological strength. The nonzero out-of-phase component in the cross-correlation is a specific, falsifiable signature, and the consistency of the inferred fI across sizes and cooldowns supports the internal validity of the measurement. The paper is clearly written and the experimental controls (including the measured differential phase delay between channels and the zero out-of-phase autocorrelation) are appropriate. The main weakness is that the interpretation as an intrinsic, non-spinning Einstein-de Haas signal rests on an unmeasured mechanical rotation degree of freedom about the easy axis.","major_comments":[{"comment":"The paper's central quantitative conclusion relies on the assumption ˙γ = 0, i.e., no mechanical rotation about the spin axis. Equation (S3) shows explicitly that a constant ˙γ enters the equations of motion in exactly the same additive way as ωI, so the quantity extracted from Eq. (9) is actually (ωI + ˙γ)^2, not ωI^2 alone. The thermal estimate ˙γrms = sqrt(kBT/I) < 0.01ωI quoted in the main text bounds the standard deviation of a zero-mean stochastic process; it does not exclude a nonzero mean or a slowly decaying residual spin imparted during magnetization or loading. This is quantitatively relevant because the inferred g-factors in Table I are systematically 10-15% below the geff = 1.28 model value, so a residual counter-rotation of about -0.1ωI would fully account for the observed discrepancy without any error in the intrinsic-spin interpretation. Since the claim of observing the Einstein-de Haas effect is specifically about a non-spinning magnet, this degeneracy is load-bearing. Please add a direct or indirect constraint on ˙γ, for example by reporting the waiting time after loading relative to the rotational damping time, measuring fI as a function of time after loading, or comparing the extracted coupling at different gas pressures; these would clarify whether the system is in rotational equilibrium around the easy axis.","section":"Supplemental Information, Eq. (S3)"},{"comment":"The consistency of fI across the four configurations is presented as evidence for the intrinsic-spin interpretation, but this argument has limited discriminating power against a constant relative mechanical rotation of the form ˙γ = c ωI: such an offset would leave the R-dependence of fI and the run-to-run scatter essentially unchanged, while shifting all inferred g-values coherently. The two-cooldown measurement on the same particle (Table I, rows 3 and 4) is the closest available control, since any spin imparted during the initial loading would presumably have decayed before the second cooldown; however, the paper does not state the time elapsed between loading and each measurement, nor between the two cooldowns. Please state these times and the estimated rotational damping time, and discuss explicitly what the reproducibility in Table I does and does not constrain about a possible residual ˙γ.","section":"Main text, Table I and Eq. (9)"}],"minor_comments":[{"comment":"The word 'cylidrically' should be 'cylindrically'.","section":"Introduction, paragraph 2"},{"comment":"The statement that the data 'will be available in the future' is vague; please specify where and when the data will be deposited, or state that they are available from the authors upon request.","section":"Main text, data availability statement"},{"comment":"The sentence 'A value of only slightly higher 1.36 is obtained by replacing Nd with Pr' is grammatically awkward; please rephrase, e.g., 'Replacing Nd by Pr gives a value of 1.36, only slightly higher.'","section":"Main text, after Eq. (12)"},{"comment":"The envelope 1 - A1|τ| in the correlation fit is stated to be an artifact of the finite integration window; since the expected value is A1 = 1/T, it would be useful to state explicitly that A1 is left free as a consistency check and to confirm that the fitted A1 is consistent with 1/T within the statistical uncertainty.","section":"Supplemental Information, Eq. (S11)"},{"comment":"Reference [17] is a placeholder ('see Supplemental Information at URL'); in the published version the actual DOI or URL should be provided.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"This is an impressive calibration-free measurement of a small gyroscopic coupling, and the out-of-phase cross-correlation signature is convincing. The main obstacle is the unmeasured rotation angle γ about the easy axis, which is exactly degenerate with the Einstein-de Haas frequency. I believe this can be addressed without a new experiment if the authors can add a solid argument for rotational equilibrium (waiting times, damping times, and possibly a time-resolved check of fI), but as written the claim of observing the Einstein-de Haas effect in a non-spinning magnet is conditional on that assumption. If the authors can provide the requested constraint, the paper would be suitable for a high-impact journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper reports the first direct observation of gyroscopic spin-rotation coupling in a non-spinning levitated ferromagnet. The key experimental move is the use of cross-correlation between two SQUID channels to extract a tiny out-of-phase component in the librational motion. The extraction of the Einstein-de Haas frequency ωI is calibration-free with respect to the coil coupling coefficients, which is a genuine strength. The inferred fI scales with particle radius as expected, and two cooldowns of the same particle give consistent values. The paper is also honest about the main alternative interpretation: a mechanical rotation γ about the easy axis enters the equations in exactly the same way as ωI. The authors estimate the thermal value of γdot to be under 1% of ωI and argue that any residual rotation would have been damped. That argument is plausible, though not airtight, because γ is not directly measured.\n\nWhere are the soft spots? First, the g-factors are 10-15% below the 1.28 expected for Nd2Fe14B. The paper attributes this to material composition but does not quantitatively explain it. This is not fatal, but a more detailed analysis of the alloy composition or a discussion of systematic errors in the inferred M and R would strengthen the claim. Second, there is no null control. A non-magnetic particle cannot be levitated in this trap, so that path is closed, but an experiment with a deliberately non-spherical particle or a different material would isolate the effect. Third, the differential phase delay between the two SQUID channels is checked to below 0.01 degrees, but the possibility of frequency-dependent crosstalk is not explicitly addressed.\n\nOverall, the central argument holds. The cross-correlation signal is nonzero, the calibration-free extraction is robust, and the internal consistency checks are convincing. The unmeasured γ is a caveat, not a fatal flaw.\n\nThis paper deserves serious peer review. It will be of interest to the levitated optomechanics, precision magnetometry, and fundamental physics communities. I would recommend acceptance with minor revisions, focusing on the g-factor discrepancy and making the γ limitation more explicit.","headline":"First convincing observation of gyroscopic coupling in a non-spinning levitated magnet, with calibration-free extraction of the Einstein-de Haas frequency; the unmeasured γ is a caveat but not fatal.","tokens_in":12174,"tokens_out":6494,"would_cite":true,"duration_ms":59740,"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":"A non-spinning levitated ferromagnet shows gyroscopic spin-rotation coupling between its librational modes, yielding a gyromagnetic g-factor of about 1.1–1.2.","keywords":["Einstein-de Haas effect","spin-rotation coupling","gyroscopic coupling","levitated ferromagnet","librational modes","superconducting trap","gyromagnetic ratio","SQUID cross-correlation"],"falsifier":"Directly measure the rotation angle γ around the spin axis while repeating the cross-correlation measurement; if γ̇ is comparable to or larger than ωI, the out-of-phase component would persist even with the intrinsic spin contribution removed, falsifying the gyroscopic interpretation. Alternatively, reverse the particle's magnetization and check that the sign of the out-of-phase correlation reverses as predicted for intrinsic spin coupling.","tokens_in":11239,"feed_emoji":"🧲","tokens_out":5297,"duration_ms":46889,"temperature":0.7,"pith_summary":"The paper reports experimental signatures that a permanent ferromagnet carries real mechanical angular momentum from its electron spins even when the object is not rotating, so its rocking motion is coupled like a gyroscope's. The authors levitate a hard ferromagnetic microsphere in a superconducting trap, watch two orthogonal librational modes, and find an out-of-phase cross-correlation between two SQUID channels that matches the predicted spin-rotation coupling. From that coupling they extract the Einstein-de Haas frequency and a gyromagnetic g-factor of about 1.1–1.2, close to the 1.28 expected for Nd2Fe14B. The result matters because a non-spinning ferromagnet in this regime behaves like a giant atomic spin, opening a route to precession-based magnetometry and quantum-spin-stabilized levitation.","feed_headline":"Non-spinning magnet shows gyroscopic coupling","feed_subtitle":"Out-of-phase wobble in a levitated ferromagnet reveals the Einstein-de Haas effect and a g-factor of about 1.2.","key_machinery":"The central object is the spin-rotation coupling term in the linearized Euler equations for a hard ferromagnet. With total angular momentum J = IΩ + S and the intrinsic spin S rigidly attached to the crystal easy axis, the librational modes obey üα + ωα²α + ωI β̇ = 0 and üβ + ωβ²β − ωI α̇ = 0. Because the coupling is kinetic, involving first derivatives, a mode excited along one axis drives the other axis in quadrature, creating elliptical motion with a π/2 phase shift. The two-SQUID cross-correlation technique extracts the sine (out-of-phase) part of this motion, and the product rα rβ cancels the unknown coil coupling constants, giving a calibration-free measurement of ωI.","core_discovery":"The central claim is that gyroscopic effects appear in the rigid-body dynamics of a non-spinning permanent hard ferromagnet: the intrinsic angular momentum S locked to the crystal lattice couples the two librational modes through the Einstein-de Haas frequency ωI = S/I, producing elliptical trajectories in the α-β plane with a π/2 phase shift between the two axes. The authors show that the ratio of out-of-phase to in-phase cross-correlation components in two imperfectly selective SQUID channels, combined for the quasi-α and quasi-β modes, gives a calibration-free expression for ωI. Applying this to four configurations with three particles yields Einstein-de Haas frequencies from 0.33 to 0.88 Hz and g-factors of about 1.10–1.19, consistent with the expected ~1.28 for the rare-earth alloy. The paper argues this is the first direct observation of gyroscopic spin-rotation coupling in the dynamics of a macroscopic non-spinning permanent magnet.","pith_inferences":["The main unmeasured loophole is rotation around the magnetization axis: because a classical γ̇ would produce exactly the same out-of-phase coupling, a direct measurement of γ̇, rather than a thermal estimate, would close the last alternative explanation.","The systematic ~10% shortfall of the measured g-factor relative to the Nd2Fe14B prediction could encode information about the alloy's Co and Pr substitution, or about small shape anisotropy, and could be tested by independent magnetization measurements on the same particles.","The same calibration-free rα rβ product could serve as a metrological tool: once ωI is calibrated, the cross-correlation phase would report changes in magnetization, moment of inertia, or surface mass, suggesting applications to adsorption and temperature sensing.","Pushing the trap toward the partial gyroscopic regime should make the out-of-phase component grow relative to the in-phase component, providing a clear experimental test of the model's scaling with ωI."],"forward_implications":["The measured intrinsic angular momentum confirms that electron spin contributes mechanically to a macroscopic rigid body's dynamics, extending the Einstein-de Haas equivalence to the quasi-static librational regime.","The calibration-free extraction gives an absolute estimate of S/I without knowing the SQUID coupling coefficients, so the same protocol can be transferred to other levitated ferromagnet platforms.","The inferred g-factors support identifying the commercial alloy's magnetism with Nd2Fe14B-like rare-earth spin-plus-orbital angular momentum, and deviations from 1.28 become a quantitative test of composition.","If scaled to nanomagnets with radius below about 500 nm, the full gyroscopic regime ωI ≫ ωα, ωβ becomes accessible, making Larmor-precession magnetometry and quantum-spin-stabilized levitation experimentally reachable.","A partial gyroscopic regime with ωβ ≫ ωI ≫ ωα may be achievable in the existing trap by compensating stray fields, allowing a modified Larmor precession to be observed."],"supporting_citations":[{"why":"The historical Einstein-de Haas experiment establishing that magnetization changes are accompanied by mechanical angular momentum, the effect this paper generalizes to a non-spinning hard ferromagnet.","marker":"[1]"},{"why":"Proposes the precessing ferromagnetic needle magnetometer and identifies the gyroscopic regime with Larmor precession of a ferromagnet.","marker":"[4]"},{"why":"Provides the theoretical framework for ferromagnetic gyroscopes, including the equations of motion and applications to tests of fundamental physics.","marker":"[8]"},{"why":"Models the librational dynamics of a levitated ferromagnetic particle in a superconducting trap, supplying the restoring-torque and mode-frequency description used here.","marker":"[10]"},{"why":"Earlier experiment with the same levitated ferromagnet platform demonstrating thermal librational modes and the detection system, the baseline this work extends.","marker":"[12]"},{"why":"Supplies the literature g-factors and spin values for Nd and Fe used to compute the expected g-factor of 1.28.","marker":"[18]"},{"why":"Observation of spin-rotation coupling mediated by long-axis spinning in a levitated nanoparticle, the classical γ̇ channel that could mimic the present signal.","marker":"[19]"}],"fun_headline_variants":["Levitated magnet's wobble reveals gyroscopic coupling","Non-spinning ferromagnet couples modes via intrinsic spin","Einstein-de Haas effect inferred from levitated magnet's motion","Gyroscopic coupling measured in non-spinning levitated magnet","Intrinsic angular momentum read from magnet's wobble"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the magnet does not rotate appreciably around its own magnetization axis: the authors estimate this rotation is thermal and about a hundred times smaller than the Einstein-de Haas frequency, but they do not measure it directly.","fun_headline_variants_meta":{"raw":{"variants":["Levitated magnet's wobble reveals gyroscopic coupling","Non-spinning ferromagnet couples modes via intrinsic spin","Einstein-de Haas effect inferred from levitated magnet's motion","Gyroscopic coupling measured in non-spinning levitated magnet","Intrinsic angular momentum read from magnet's wobble"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000742,"raw_usage":{"total_tokens":3269,"prompt_tokens":865,"completion_tokens":2404,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":2324}},"tokens_in":481,"tokens_out":2404,"duration_ms":17306,"temperature":1.0,"reasoning_tokens":2324,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:01:25.461511+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly measure the rotation angle γ around the spin axis while repeating the cross-correlation measurement; if γ̇ is comparable to or larger than ωI, the out-of-phase component would persist even with the intrinsic spin contribution removed, falsifying the gyroscopic interpretation. Alternatively, reverse the particle's magnetization and check that the sign of the out-of-phase correlation reverses as predicted for intrinsic spin coupling.","supporting_citations":[{"cited_title":"Einstein and W","cited_arxiv_id":null,"evidence_quote":"The historical Einstein-de Haas experiment establishing that magnetization changes are accompanied by mechanical angular momentum, the effect this paper generalizes to a non-spinning hard ferromagnet."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proposes the precessing ferromagnetic needle magnetometer and identifies the gyroscopic regime with Larmor precession of a ferromagnet."},{"cited_title":"Fadeev, C","cited_arxiv_id":null,"evidence_quote":"Provides the theoretical framework for ferromagnetic gyroscopes, including the equations of motion and applications to tests of fundamental physics."},{"cited_title":"Vinante, C","cited_arxiv_id":null,"evidence_quote":"Models the librational dynamics of a levitated ferromagnetic particle in a superconducting trap, supplying the restoring-torque and mode-frequency description used here."},{"cited_title":"Ahrens, W","cited_arxiv_id":null,"evidence_quote":"Earlier experiment with the same levitated ferromagnet platform demonstrating thermal librational modes and the detection system, the baseline this work extends."},{"cited_title":"Buschow and F","cited_arxiv_id":null,"evidence_quote":"Supplies the literature g-factors and spin values for Nd and Fe used to compute the expected g-factor of 1.28."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Observation of spin-rotation coupling mediated by long-axis spinning in a levitated nanoparticle, the classical γ̇ channel that could mimic the present signal."}],"review_version":1}