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REVIEW 4 major objections 6 minor 58 references

Quantum gravitational decoherence of a mechanical oscillator from spacetime fluctuations

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A fluctuating Planck-scale deformation parameter makes quantum superpositions decohere in momentum space, and a 16 µg oscillator bounds the effect.

desk verdict Solid master-equation derivation, but the β bound rests on an uncalibrated 2% ellipticity and the κ bound direction is flipped. read the letter →

arxiv 2411.13523 v2 pith:BRGXZ6KO submitted 2024-11-20 quant-ph gr-qchep-phhep-th

Sandro Donadi , Matteo Fadel This is my paper · ORCID
classification quant-phgr-qchep-phhep-th PACS 03.65.Yz04.60.-m
keywords generalizeduncertaintyprincipledeformedcommutatormomentumdecoherencemechanicaloscillatorspacetimefluctuationsHBARresonatorquantumgravityphenomenologynonlocaldynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a stochastic component in the Planck-scale deformation of the position–momentum commutator turns spacetime into a source of momentum-basis decoherence for any quantum system. Starting from $[\hat X,\hat P]=i\hbar(1+\beta\ell_P^2\hat P^2/\hbar^2)$ with $\beta(t)$ fluctuating around a mean $\bar\beta$, the authors derive a master equation whose non-unitary term is a double commutator with $\hat K^2$, so superpositions of different momenta decay. They apply the model to a $16\,\mu\mathrm g$ high-frequency mechanical oscillator prepared in quantum states and use measured energy relaxation and Ramsey dephasing times to bound the fluctuation amplitude $\kappa\le 4.0\times10^{46}\,\mathrm s$ and the mean deformation $\bar\beta<2.2\times10^{30}$; the same ground-state data also bound the nonlocality length scale $\ell_k\le5.9\times10^{-20}\,\mathrm m$. A sympathetic reader would care because these are quantum-regime bounds, which do not require the contested assumption that a deformed commutator also deforms classical Poisson brackets.

What carries the argument

The load-bearing object is the fluctuating deformation parameter $\beta(t)$ inside the deformed commutator $[\hat X,\hat P]=i\hbar(1+\beta(t)\ell_P^2\hat P^2/\hbar^2)$, characterized by its mean and by an autocorrelation $\kappa f(t-t')$. The calculation works through the transformation $\hat X=\hat x$, $\hat P=(1+\beta\ell_P^2\hat p^2/\hbar^2)\hat p$, which maps the modified algebra to the canonical one plus a perturbation $\hat H_\beta=4a_P\bar\beta\hat K^2$; a Born–Markov average over the noise then produces the double-commutator master equation. For the oscillator, the rotating-wave approximation reduces the unitary part to a Kerr-like number-squared term, and the collapse operator $\hat K^2$ is evaluated in the Fock basis to give the short-time decay rates used in the data analysis.

What would settle it

Measure the T2 excess and ground-state ellipticity of the same HBAR mode after an independent calibration of the readout chain, and repeat on a second oscillator with a different effective mass: the model predicts a GUP decoherence rate proportional to $m^2\omega^4$ and a squeezing $\varepsilon\propto m$, so observed scalings that disagree would falsify the $\kappa$ and $\bar\beta$ bounds.

Watch

Extended reading notes

Core claim

The central discovery is that allowing the deformation parameter in the generalized uncertainty principle to fluctuate turns the GUP correction into a genuine open-quantum-system effect: after averaging over the noise, the reduced dynamics contains a Lindblad term proportional to $[\hat K^2,[\hat K^2,\hat\rho]]$, which damps coherences between momentum eigenstates while leaving energy eigenstates of a free particle decohered only if their kinetic energies differ. For a harmonic oscillator in the white-noise limit, a superposition $(|0\rangle+|1\rangle)/\sqrt2$ loses coherence as $1-\frac{30}{8}t/\tau_G$ with $1/\tau_G\equiv 8a_P^2\kappa\hbar^2\omega^4$, and the ground state itself acquires a slow heating term $1-\frac{6}{8}t/\tau_G$. Applying these formulas to $T_1$ and $T_2$ data from a $16.2\,\mu\mathrm g$ sapphire HBAR mode yields the upper bound $\kappa\le4.0\times10^{46}\,\mathrm s$ and, from the measured ground-state Wigner-function ellipticity $\varepsilon=0.020(5)$, the bound $\bar\beta<2.2\times10^{30}$. The paper also corrects a factor of two and an omitted unitary term in the earlier white-noise master equation, and re-derives bounds for the metric-fluctuation model, giving $\tau_c\le3.7\times10^{-18}\,\mathrm s$.

Load-bearing premise

The β bound rests on attributing the measured ground-state Wigner-function ellipticity entirely to GUP-induced squeezing; any part of that ellipticity coming from calibration or other systematics would invalidate the bound.

Editorial extensions

If this is right

  • If the model is right, any massive quantum superposition decoheres in momentum space at a rate set by $\kappa$, so the $T_1$ and $T_2$ data already exclude spacetime fluctuations of this type as the dominant decoherence source for a $16\,\mu\mathrm g$ oscillator.
  • The ground-state ellipticity gives a quantum-regime bound $\bar\beta<2.2\times10^{30}$ that does not assume deformed Poisson brackets, and a heavier oscillator should tighten it because the predicted squeezing grows with mass.
  • The same variance measurement bounds the quantum-gravity nonlocality scale to $\ell_k\le5.9\times10^{-20}\,\mathrm m$, comparable to the LHC-derived bound but obtained on a tabletop quantum device.
  • For the independent metric-fluctuation model, the same data give $\tau_c\le3.7\times10^{-18}\,\mathrm s$, and probing $\tau_c\sim t_P$ would demand the very large product $\omega^2/\gamma\sim10^{43}\,\mathrm s^{-1}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the GUP-squeezing interpretation is right, the ground-state ellipticity of the same oscillator should change linearly with its effective mass at fixed frequency; measuring that scaling would separate genuine spacetime squeezing from readout calibration artifacts.
  • The predicted leakage out of the ground state is a distinctive signature of this model, and a long-time population measurement of a deeply cooled oscillator could test it because ordinary thermalization acts in the opposite direction.
  • The master-equation derivation is not tied to white noise, so a non-Markovian spectrum $f(t-t')$ would imprint a frequency dependence on the decoherence rate that could distinguish this mechanism from ordinary environmental dephasing.
  • Because the quantum-regime $\bar\beta$ bound is much weaker than classical-oscillator bounds, combining the two types of measurement could test whether a deformed commutator must also deform classical dynamics.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper studies decoherence induced by a fluctuating spacetime at low energies. In the first model, the deformation parameter β in the GUP-type commutator [X,P]=iℏ(1+βℓ_P²P²/ℏ²) is promoted to a stochastic process with mean β and covariance κf(t−t′). The authors derive a non-Markovian master equation for the momentum-basis decoherence, recover a Markovian limit, and compute the decay of Fock-state coherences and populations for a harmonic oscillator, including amplitude damping. In a second model, isotropic metric fluctuations à la Breuer et al. lead to a K-covariance master equation with amplitude τ_c. Using published T1 and T2 data from a 16 µg HBAR oscillator, they extract τ_G and γ and claim κ≤(4.0±0.9)×10^46 s; using the ground-state Wigner function ellipticity (ϵ=0.020(5)) they claim β<2.2(6)×10^30 and l_k≤5.9(8)×10^-20 m, and from the same T1/T2 data they claim τ_c≤(3.7±0.8)×10^-18 s.

Significance. The master-equation derivation is careful and self-contained: it generalizes the white-noise model of Ref. [12] to non-Markovian noise, corrects a factor-2 error in the Markovian Lindblad coefficient, and correctly separates the mean-deformation unitary term from the fluctuation-induced decoherence. The use of genuinely quantum mechanical oscillator data (T1, T2, and Wigner tomography) rather than classical measurements is a real step forward for constraining GUP-type models without the deformed Poisson-bracket assumption. The κ and τ_c bounds, if properly one-sided, are useful, and the metric-fluctuation bound is significantly tighter than the GUP one. However, the β and l_k bounds are not yet secure: they rest on a single 2% ellipticity extracted from a Wigner function with no quadrature-gain calibration or systematics budget, and the derivation of the variance formula is not shown. The paper also contains a concrete factor-4 error in the RWA Hamiltonian and a misstated bound direction for τ_G. These issues affect central claims and require correction.

major comments (4)
  1. [Ground state deformation] The derivation of β<2.2(6)×10^30 from the ground-state Wigner function ellipticity rests entirely on identifying the measured Δx²_max/Δx²_min=(2+ϵ)/(2−ϵ) with the GUP-induced squeezing of the mechanical mode. No calibration of the two quadrature gains is reported, no control measurement of a known circular state is described, and no systematic uncertainty budget is given; a 2% detection anisotropy in the tomography pipeline would produce exactly the same signal as the claimed physical ellipticity. The stated uncertainty 0.005 on ϵ is therefore not sufficient to support a one-sided bound. In addition, the variance formula Δx_θ²=1/2−1/4ϵ cos(2θ) is asserted without derivation. A standard first-order perturbative treatment of H_β=4a_PβK² in the canonical quadratures yields Δx_θ²=1/2+(1/4)ϵ cos(2θ) with ϵ=6βa_Pℏω, i.e. the opposite sign of the ellipticity; the paper should justify its sign and specify whether the measured quadratures are canonical or physical momentum operators. Because the l_k bound in the following paragraph is built on the same ϵ measurement, it inherits the same fragility.
  2. [Experimental tests] The text states that the fitted values γ^{-1}=169.9±47.5 µs and τ_G=975.2±237.4 µs 'should be considered as upper bounds' because T2 has additional technical-noise contributions. If the only unmodelled noise is extra dephasing in T2, then the solved 1/τ_G is overestimated, so τ_G is underestimated, not overestimated; the quoted τ_G is then a lower bound, and the quoted γ^{-1} is also a lower bound. The claimed bound κ≤(4.0±0.9)×10^46 s is conservative in that particular scenario, but only if the unmodelled noise is absent from the T1 (population) channel. If the T1 channel also contains additional relaxation/heating, the solved 1/τ_G can move in the opposite direction and the one-sided κ bound is not guaranteed. The authors need to state this assumption explicitly and provide a sensitivity analysis. Furthermore, the uncertainty ±0.9×10^46 s is not obtained from a visible propagation of the quoted T1 and T2 fit errors; since κ∝1/τ_G, the propagated error is asymmetric.
  3. [Eq. (10) and SM Eq. (28)] Equation (10) and the corresponding SM Eq. (28) contain a factor-4 error in the RWA of the modified Hamiltonian. With K=p²/2m and the oscillator expression p²=(mℏω/2)(2N+1−a†²−a²), the number-conserving part of K² is (3ℏ²ω²/8)(N²+N+1/2). Multiplying by 4a_Pβ from Eq. (5) gives a correction (3/2)a_Pβℏ²ω²(N²+N+1/2), not (3/8)a_Pβℏ²ω²(N²+N+1/2). A consistency check is that ⟨0|H_RWA|0⟩ would be (3/16)a_Pβℏ²ω², whereas the exact ground-state expectation of H_β is (3/4)a_Pβℏ²ω². This error affects the unitary evolution and the non-Markovian results, even though the white-noise decoherence rates in Eqs. (12,13) are independent of the RWA coefficient.
  4. [Ground state deformation, l_k paragraph] The bound l_k≤5.9(8)×10^-20 m is presented without any derivation of the mapping between the measured ground-state ellipticity and l_k. The text only says that the quartic x^4 term in the nonlocal Hamiltonian 'plays a similar role' to the p^4 term in H_β. Since this bound is quoted as on par with LHC constraints, the explicit relation between the measured ϵ and l_k must be shown. As written, the l_k result cannot be checked, and it also inherits the calibration uncertainty of the same ϵ measurement.
minor comments (6)
  1. [Experimental tests] The phrase 'upper bounds for the corresponding parameters' is ambiguous because the quoted numbers are inverse rates; please restate in terms of γ and 1/τ_G or correct the direction as discussed in the major comments.
  2. [Experimental tests / Metric fluctuations model] The paper presents κ≤(4.0±0.9)×10^46 s and τ_c≤(3.7±0.8)×10^-18 s as one-sided bounds with symmetric errors; since the conversion from measured times to these parameters is nonlinear, a one-sided confidence interval at a stated confidence level would be more appropriate.
  3. [Ground state deformation] The value ϵ=0.020(5) is quoted without stating whether 0.005 is statistical, systematic, or total; please clarify what is included in this uncertainty.
  4. [Modified dynamics, after Eq. (5)] The definition of a_P contains a typo: 'e−1 P' should read 'e_P^{-1}'. Please also give the dimensions of a_P explicitly, since several later formulas depend on its units.
  5. [Experimental tests, note [41]] The note says the measurements in Fig. 2 are taken for t/τ_G<0.2; because τ_G is itself extracted from the same data, this is a post-fit consistency condition rather than an a priori validity check. Stating t/T1 and t/T2 would make the short-time assumption transparent.
  6. [Eq. (9)] The quantities ∆E_k used in Eq. (9) are defined only in the Supplementary Material; please define them in the main text at first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the master equations are derived from the assumed deformed commutator, and the bounds on κ, β, τ_c, and l_k are inferred from independent, previously published experimental measurements.

full rationale

This paper is a parameter-constraint analysis, not a circular derivation. The chain is: assume the deformed commutator (1) with fluctuating β(t); derive the stochastic Schrödinger equation (4) and master equations (7)-(8) by a standard Born-Markov expansion (SM Section I); solve for the oscillator decoherence rates (12)-(13); invert the measured T1 and T2 to bound κ. Separately, the constant-β deformed commutator gives ground-state squeezing Δx̂²_θ = 1/2 − (1/4)ϵ cos(2θ) with ϵ ≡ 6βa_Pℏω, and the measured ground-state Wigner ellipticity is inverted to bound β and l_k. At no point is a parameter fitted to a subset of data and then 'predicted' from the same data: the T1 and T2 rates are independently measured decay constants, and the model provides a distinct two-unknown/two-equation inversion for γ⁻¹ and τ_G; the Wigner ellipticity is a directly measured shape of the state, not an output of the model. The experimental inputs are previously published, externally verifiable measurements (von Lüpke et al., Nature Physics 2022; Bild et al., Science 2023; Marti et al., Nature Physics 2024). Some of these papers share an author with the present work, but the citation is to published, externally falsifiable data, so it is real evidence rather than a circular self-citation. The ground-state variance formula is attributed to external works [43,44]; the metric-fluctuation model is attributed to Breuer et al. [26]; and the comparison with [12] is transparent (a factor-2 correction, an extra unitary term, and a free κ). The skeptic's concern, that a 2% quadrature-gain calibration anisotropy could mimic the 2% ellipticity, is a systematics and correctness risk, not a circularity: the prediction and the measurement are distinct quantities connected by an independently derived formula. No equation in the paper reduces to its own input by construction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The model postulates a stochastic deformation parameter β(t) and, for the metric model, white-noise metric fluctuations; these are not derived from first principles. The free parameters κ, β, τ_c, and l_k are constrained by the HBAR data. No new particles or forces are introduced.

free parameters (4)
  • κ (fluctuation amplitude of β) = κ ≤ 4.0×10^46 s
    Determined from T1 and T2 data using Eqs. (12)-(13); treated as an upper bound because uncharacterized technical noise may shorten T2.
  • β (mean deformation parameter) = β < 2.2×10^30
    Extracted from ground state ellipticity ϵ=0.020(5) via ϵ=6βa_Pℏω; the bound direction is questionable.
  • τ_c (metric fluctuation noise amplitude) = τ_c ≤ 3.7×10^-18 s
    Determined from the same T1 and T2 data using the Breuer et al. model, Eqs. (16)-(17).
  • l_k (nonlocality length scale) = l_k ≤ 5.9×10^-20 m
    Derived from the same ground state ellipticity measurement using the effective Hamiltonian of Ref. [25].
assumptions (4)
  • ad hoc to paper Deformed commutator [X,P]=iℏ(1+β(t)ℓ_P²P²/ℏ²) with β(t) a stochastic process with mean β and covariance κf(t−t′)
    This is the postulated model for spacetime fluctuations; it is not derived from a more fundamental theory.
  • domain assumption Born-Markov approximation: f(t) has characteristic time τ such that f(t≥τ)≈0 and Hamiltonian evolution is negligible during τ
    Used to derive Eq. (7); the experimental bounds adopt the white-noise limit.
  • domain assumption Non-relativistic limit of the Klein-Gordon equation with a fluctuating metric yields H_B=√τ_c w(t) K with white noise w(t)
    Taken from Breuer et al. [26] without re-derivation.
  • domain assumption The measured ground state ellipticity is entirely due to the p^4 term in the modified Hamiltonian
    Used to convert the measured Wigner function ellipticity into bounds on β and l_k; no other source of squeezing is considered.

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Cite this review

Pith. "Pith review of Quantum gravitational decoherence of a mechanical oscillator from spacetime fluctuations." pith.science (2026). https://pith.science/paper/BRGXZ6KO

@misc{pith2026241113523,
  author       = {Pith},
  title        = {Pith review of: Quantum gravitational decoherence of a mechanical oscillator from spacetime fluctuations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRGXZ6KO}},
  note         = {Machine review of arXiv:2411.13523}
}
read the original abstract

We consider the scenario of a fluctuating spacetime due to a deformed commutation relation with a fluctuating deformation parameter, or to a fluctuating metric tensor. By computing the resulting dynamics and averaging over these fluctuations, we find that a system experiences a decoherence in the momentum basis. We studied the predictions of the model for a free particle and an harmonic oscillator. Using experimental data taken from a mechanical oscillator prepared in quantum states of motion, we put a bound on the free parameters of the considered model. In addition, we comment on how these measurements can also provide bounds to other phenomenological quantum gravity models, such as the length scale for nonlocal dynamics.

Figures

Figures reproduced from arXiv: 2411.13523 by the authors.

Figure 1
Figure 1. FIG. 1. Numerical simulation of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Left: measurement of the mechanical resonator energy relaxation rate. Fitting the data with a decaying exponential allows us to extract [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Wigner function measurement of the mechanical resonator [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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    Note that ω2/γ ∼ Q f, the product of the oscillator’s quality factor and frequency

  43. [51]

    QUANTUM GRA VITATIONAL DECOHERENCE OF A MECHANICAL OSCILLATOR FROM SPACETIME FLUCTUATIONS

    L. Susskind, Phys. Rev. D 49, 6606 (1994). 7 SUPPLEMENTARY MATERIALS FOR “QUANTUM GRA VITATIONAL DECOHERENCE OF A MECHANICAL OSCILLATOR FROM SPACETIME FLUCTUATIONS” I. DERIV ATION OF THE MASTER EQUATION (7) The starting point of our analysis is Eq. (4) of the main text. First,...

    1994

  44. [52]

    The goal is to find an approximate solution where only the second term of Eq

    + aPβ3ℏ2ω2 8 " ˆN2 + ˆN + 1 2 # (28) and ˆK2I(t)≡ ei ˆHRW At/ℏ ˆK2e−i ˆHRW At/ℏ, (29) where ˆK = ˆp2 2m. The goal is to find an approximate solution where only the second term of Eq. (27) is treated perturbatively. Note also that while ˆHRW Ais the Hamiltonian under the rotati...

  45. [53]

    + aPβ3ℏ2ω2 8 " n2 j + n j + 1 2 # (45) for j = 1, 2. Eq. (43) is of the form ∂tr(t) = f (t)r(t) + g(t), (46) which has solution r(t) = r(0)e R t 0 f (t′)dt′ + Z t 0 dse R t s f (t′)dt′ g(s) ds. (47) This implies that the solution of Eq. (43) is ⟨n1|ˆρ1(t)|n2⟩ =⟨n1|ˆρ1(0)|n2⟩e ...

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    General structure of the calculation Since this is the term that we want to compute perturbatively, when applying the superoperator eL1t we stop to the first term of the series in Eq. (49). Hence ⟨0|eL1t h ˆK2I(τ), h ˆK2I(t′), ˆρ(0) ii |1⟩≃ e− i ℏ (E0−E1)te−γ 2 tC(τ, t′) (53) ...

  47. [55]

    Computation of D 1 We can now compute D1(τ, t′) =⟨0| ˆK2I(τ) ˆK2I(t′)ˆρ(0)|1⟩ = 1 2 ⟨0| ˆK2I(τ) ˆK2I(t′)|0⟩ +⟨0| ˆK2I(τ) ˆK2I(t′)|1⟩ (65) Since the operators ˆK2I(t′) contains even number of ladder operators, all matrix elements between an odd and an even state are zero, there...

  48. [56]

    (59) and (60) we get D2(τ, t′) = 3ℏ2ω2 16 15ℏ2ω2 16 2 = 45ℏ4ω4 512 (69) 12

    Computation of D 2 We proceed by computing: D2(τ, t′) =⟨0| ˆK2I(τ)ˆρ(0) ˆK2I(t′)|1⟩ = 1 2 n ⟨0| ˆK2I(τ)|0⟩⟨0| ˆK2I(t′)|1⟩ +⟨0| ˆK2I(τ)|0⟩⟨1| ˆK2I(t′)|1⟩ + (68) +⟨0| ˆK2I(τ)|1⟩⟨0| ˆK2I(t′)|1⟩ +⟨0| ˆK2I(τ)|1⟩⟨1| ˆK2I(t′)|1⟩ o = ⟨0| ˆK2I(τ)|0⟩ +⟨0| ˆK2I(τ)|1⟩ ⟨0| ˆK2I(t′)|1⟩ +⟨1|...

  49. [57]

    Using Eqs

    Computation of D 3 We finally compute D3(τ, t′) =⟨0|ˆρ(0) ˆK2I(t′) ˆK2I(τ)|1⟩ = 1 2 ⟨0| ˆK2I(t′) ˆK2I(τ)|1⟩ +⟨1| ˆK2I(t′) ˆK2I(τ)|1⟩ (70) = 1 2⟨1| ˆK2I(t′) ˆK2I(τ)|1⟩ = 1 2 ∞X n=0 ⟨1| ˆK2I(t′)|n⟩⟨n| ˆK2I(τ)|1⟩ = = 1 2 ⟨1| ˆK2I(t′)|1⟩⟨1| ˆK2I(τ)|1⟩ +⟨1| ˆK2I(t′)|3⟩⟨3| ˆK2I(τ)|1...

  50. [58]

    (74) Then, going back to Eq

    Computation of C (τ, t′) In summary we found D1(τ, t′) = 9ℏ4ω4 512 + 9ℏ4ω4 64 e i ℏ (E0−E2)(τ−t′) + 3ℏ4ω4 64 e i ℏ (E0−E4)(τ−t′) (72) D2(τ, t′) = 45ℏ4ω4 512 (73) D3(τ, t′) = 225ℏ4ω4 512 + 75ℏ4ω4 64 e i ℏ (E1−E3)(t′−τ) + 15ℏ4ω4 64 e i ℏ (E1−E5)(t′−τ). (74) Then, going back to E...

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