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Magnetic noise in macroscopic quantum spatial superposition

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Magnetic-gradient noise sets a concrete 10^-8 current-stability budget for nanodiamond Stern-Gerlach interferometers.

desk verdict A genuinely useful noise-budget framework for SG nanodiamond interferometers, but the headline δI/I bound and the Humpty-Dumpty claim are undercut by the paper's own Appendix D and by numerical/unit errors in the contrast section. read the letter →

arxiv 2504.13252 v3 pith:QUSGUF2W submitted 2025-04-17 quant-ph

Sneha Narasimha Moorthy (1 , 2) , Andrew Geraci (3) , Sougato Bose (4) , Anupam Mazumdar (5) ((1) National Institute of Science Education , Research , (2) Homi Bhabha National Institute , (3) Northwestern University
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(4) University College London (5) Van Swinderen Institute University of Groningen)
This is my paper · ORCID
classification quant-ph
keywords magneticfieldgradientnoiseStern-GerlachinterferometernanodiamondNVcentremacroscopicquantumsuperpositiondephasingrateflickerwhiteHumpty-Dumptyproblem
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

How much magnetic-field-gradient noise can a Stern-Gerlach nanodiamond interferometer tolerate before its macroscopic spatial superposition decoheres? The paper's answer is quantitative: for a levitated NV-centred nanodiamond split into a nanometer-scale superposition by a chip-generated gradient, keeping the dephasing rate below about 100 Hz requires relative current fluctuations $\delta I/I \lesssim 10^{-8}$, whether the noise is white or 1/f flicker noise. It then shows that at this noise level the noise-driven trajectory deviations are only about $10^{-19}$ m, so the Humpty-Dumpty position-momentum mismatch leaves the interference contrast essentially at one. The result matters because such superpositions are proposed platforms for tabletop tests of gravity-induced entanglement, quantum sensors, and equivalence-principle experiments; the paper converts magnetic noise from a feared background into a stated engineering constraint.

What carries the argument

The load-bearing object is the phase-variance integral $\Gamma = (8H^2/\omega_0^5)\int_{\omega_{\min}}^{\infty} S_{\eta\eta}(\omega) F_{\rm HO}(\omega/\omega_0)\,d\omega$, with $H = 4\gamma_e B_0 \eta_0 \chi_\rho/\mu_0$ and $F_{\rm HO}(\xi)$ the harmonic-oscillator transfer function whose apparent singularities at $\xi = 1,2$ are removable. This integral converts a noise power spectral density into a dephasing rate, and the relation $\eta_0 = \mu_0 I/(2\pi d^2)$ converts the dephasing budget into the current-stability ratio $\delta I/I$. For the Humpty-Dumpty analysis the central mechanism is the Gaussian overlap contrast $C = \exp[-\frac12((\Delta x/\sigma_x)^2 + (\Delta p/\sigma_p)^2)]$, evaluated with Fourier-domain solutions for the noise-driven deviations $\delta x_j$ and $\delta p_j$ from Appendix D.

What would settle it

Recompute the dephasing budget using the full phase expression Eq. (D1)–(D4), adding the trajectory-deviation transfer function $F_{\rm dev}(\xi)$ to $F_{\rm HO}(\xi)$ at the Sec. IV A parameters; if the $\delta I/I$ bound tightens significantly below $1.3\times 10^{-8}$, the paper's headline constraint is not the actual one. A laboratory check would be to measure the gradient-noise PSD of a niobium microchip wire at 4 K and compare its $K$ with the $0.7\times 10^{-13}\ \mathrm{T\,m^{-1}\,A^{-1}}$ threshold.

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Extended reading notes

Core claim

The central claim is that magnetic-gradient noise from the chip wires is both the dominant systematic of the protocol and a controllable one. Modelling the two interferometer arms as a spin-dependent harmonic oscillator whose Lagrangian coefficients are perturbed by stochastic gradient fluctuations, the paper derives a dephasing rate $\Gamma$ from the ensemble-averaged phase variance. With the parameters of a diamagnetically levitated nanodiamond (12 A current, 20 $\mu$m wire distance, nanodiamond susceptibility), the white-noise amplitude is bounded by $A \lesssim 2.9\times 10^{-6}\ \mathrm{T\,m^{-1}\,Hz^{-1/2}}$ and the flicker-noise constant by $K \lesssim 0.7\times 10^{-13}\ \mathrm{T\,m^{-1}\,A^{-1}}$, and both translate through $\eta_0 = \mu_0 I/(2\pi d^2)$ into the same relative current stability $\delta I/I \lesssim 1.3\times 10^{-8}$ for $\Gamma \sim 100\ \mathrm{Hz}$. Solving the noise-perturbed equations of motion in the Fourier domain and evaluating the Gaussian contrast formula then gives trajectory deviations $\sim 10^{-19}$ m and contrast $C \approx 1$ for both noise types, so the paper concludes that the Humpty-Dumpty problem does not degrade the interference at the permitted noise level.

Load-bearing premise

The load-bearing assumption is that the dephasing rate can be computed from noise-induced changes in the Lagrangian coefficients while the arm trajectories are held fixed; the paper's own Appendix D order-of-magnitude estimate ($\delta x_j\,\eta_0 \sim 10^{-11}$ T versus $x_j\,\delta\eta \sim 10^{-13}$ T) indicates that trajectory-deviation terms are not negligible, so this assumption carries the quoted $\delta I/I$ bound.

Editorial extensions

If this is right

  • A superconducting chip current supply with relative stability near $10^{-8}$ is sufficient to hold magnetic-gradient dephasing at roughly $10^{2}$ Hz in a nanodiamond Stern-Gerlach interferometer.
  • Humpty-Dumpty trajectory mismatch is not a bottleneck at these noise levels: the expected contrast stays close to one, so closing the loop and performing spin readout is feasible.
  • The same $\delta I/I$ budget holds for white noise, flicker noise, and by the paper's comparison any spectral exponent $\alpha$ between 0 and 1.5, so exact knowledge of the noise colour is not required to set tolerances.
  • Increasing the particle-wire distance $d$ weakens the gradient and tightens the required current stability while also enlarging the superposition size; the choice of $d$ is therefore a trade-off.
  • Repeating the experiment many times tightens the bounds on the noise amplitudes $A$ and $K$ but leaves $\delta I/I$ unchanged, so signal averaging does not relax the fundamental current-stability requirement.

Reading between the lines

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

  • Because Appendix D reports $\delta x_j\,\eta_0 \sim 10^{-11}$ T versus $x_j\,\delta\eta \sim 10^{-13}$ T, I infer that the headline $\delta I/I \le 10^{-8}$ bound could be optimistic; recomputing $\Gamma$ with the $F_{\rm dev}(\xi)$ term is a direct check.
  • The apparent near-universality of $\delta I/I$ across noise spectra suggests that chip geometry, not noise colour, is the main engineering lever; pushing $d$ down relaxes the current-stability requirement at the cost of other near-field noise sources.
  • The same harmonic-oscillator transfer-function method could be applied to other levitated-particle superposition proposals with different spin or mass susceptibilities, producing a general magnetic-noise budget for tabletop tests of gravity.
  • The contrast estimate assumes a Gaussian ground-state wavefunction; I infer that thermal or non-Gaussian motional states, or finite NV spin coherence, could lower the achievable contrast even when trajectory deviations remain small.
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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 / 4 minor

Summary. The paper analyzes magnetic-field-gradient noise in a Stern-Gerlach-type matter-wave interferometer with a levitated NV-center nanodiamond. It models the two interferometer arms as harmonic oscillators with noisy Lagrangian coefficients, derives the phase-noise variance for white and 1/f (flicker) noise, and obtains a bound on the current-noise-to-signal ratio, δI/I ≲ 1.3×10⁻⁸, for a dephasing rate Γ ≲ 100 Hz. It then addresses the Humpty-Dumpty problem by computing trajectory deviations and estimating the interference contrast, concluding that the contrast remains close to unity for the allowed noise levels. Appendix D explicitly treats the trajectory-deviation terms that were dropped in the main dephasing calculation and reports that these contributions cannot be neglected.

Significance. The topic is timely and relevant for proposals aimed at macroscopic spatial superpositions and quantum-gravity-mediated entanglement tests. The manuscript has several strengths: the stochastic Lagrangian framework is essentially self-contained, the noise amplitudes A and K are constrained rather than fitted to the target result, the transfer-function formalism in Eqs. (6)-(10) and (29)-(30) is explicit, and the paper makes a falsifiable prediction about the allowed current-noise level. The comparison between white and flicker noise and the claimed robustness of the δI/I bound across spectral exponents is also a useful extension. However, the central quantitative claims are undermined by an internal contradiction with Appendix D and by numerical and dimensional errors in the contrast section. As submitted, the paper does not establish its headline constraint or its Humpty-Dumpty conclusion.

major comments (4)
  1. [Sec. IV, Eq. (29) and Appendix D (Eqs. D5-D6)] The headline bound on δI/I is not supported by the manuscript's own analysis. The main dephasing calculation, Eq. (29), is based on Eq. (26), which explicitly drops all δx_j trajectory-deviation terms from Eq. (25). Appendix D reintroduces those terms and estimates x_j δη ∼ 10⁻¹³ T and δx_j η0 ∼ 10⁻¹¹ T, then states that the dephasing due to deviations 'cannot be neglected' and 'has to be accounted for while obtaining precise bounds.' Since Γ is a variance of the accumulated phase, an order-of-magnitude factor of about 100 in the dominant phase amplitude implies that the omitted Γ_dev is roughly 10⁴ times the retained contribution at the same noise level, unless there is a coherent cancellation between the two contributions. No such cancellation is demonstrated. Consequently, the bounds δI/I ≤ 1.3×10⁻⁸ in Eqs. (41) and (46) are not established; a proper inclusion of Γ_dev could tighten them by about two orders of magnitude.
  2. [Sec. V B, Eq. (62)] The numerical value in Eq. (62) is inconsistent with the stated parameters. Using the parameter values of Sec. IV A and the mass implied by Eq. (36), the expression (2ℏγeA/(mω0))² (2π/ω0) evaluates to approximately 10⁻³³ m², not 4×10⁻⁷⁰ m². This discrepancy of many orders of magnitude means that the reported trajectory-deviation variance cannot be used to support the claim that the contrast is near unity.
  3. [Sec. V B, Eqs. (58) and (63)] Eq. (63) evaluates the momentum mismatch at the wrong point in the trajectory. At the closing time T_exp = 2π/ω0, the unperturbed momentum difference in Eq. (58) vanishes because sin(ω0 t) = 0, yet Eq. (63) uses the amplitude 2ℏγeη0/ω0 and reports ⟨Δp²⟩ = 5.25×10⁻²². The noise-induced momentum deviations δp_j from Eq. (52) are also omitted. The resulting contrast, C(T) ≈ 1, therefore does not address the actual phase-space mismatch at the time of recombination.
  4. [Sec. V B, Eqs. (54)-(61)] The contrast calculation mixes unit conventions inconsistently. The text states 'We consider ℏ = 1' before Eq. (59), but the same section uses SI values of ℏ, γe, m, and σx, for example in Eqs. (55), (58), and (61). With ℏ = 1, σp = 1/(2σx), but with SI units the correct relation is σp = ℏ/(2σx). This dimensional inconsistency makes the quantitative contrast predictions unreliable and must be repaired before the Humpty-Dumpty conclusion can be assessed.
minor comments (4)
  1. [Table I and Sec. IV A] The nanodiamond mass m is not listed in Table I even though it enters Eqs. (36), (55), and (62); the authors should state the assumed mass or radius explicitly (the numerical value m ≈ 10⁻¹⁵ kg is implied by Eq. (36)).
  2. [Eqs. (38) and (43)] The dimensionless integrals are labeled with units of s⁻¹ (e.g., '× 1.8 s⁻¹'); the integral values themselves should be dimensionless, and the units should be carried by the prefactor only.
  3. [Sec. IV A, Eq. (35)] The text writes Γ ≈ 155 Hz⁻¹ for a decoherence rate; the units should be Hz or s⁻¹, not Hz⁻¹.
  4. [Appendix D] The discussion around Eq. (26) says the trajectory fluctuations are ignored and later says they cannot be neglected; a forward reference to Appendix D and a reconciliation of these statements are needed for the reader to follow the status of the main result.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the δI/I bound is an inverted constraint from a hand-set Γmax, and the Appendix D trajectory-deviation caveat is a correctness limitation, not a circular step.

full rationale

The derivation chain is self-contained. The central bound δI/I ≤ 1.3×10^-8 is obtained by inverting the algebraic expression for the dephasing rate Γ (Eqs. (38)-(41) for white noise and Eqs. (43)-(46) for flicker noise) after imposing an externally chosen target Γmax ≈ 100 Hz. That target is a design requirement, not the conclusion being derived, so no step assumes the result. Eq. (47) is a rewriting of the same algebraic relation, not a separate prediction. The Humpty-Dumpty contrast calculation in Sec. V takes the noise bounds from Sec. IV as inputs and checks that the resulting trajectory deviations are small; the contrast calculation is not used to fix any parameter in the dephasing budget, so there is no feedback loop of the kind that would constitute fitted-input-called-prediction or self-definitional circularity. The cited prior work supplies the Stern-Gerlach Lagrangian ([54,79], the latter being external) and chip parameters such as I = 12 A and d = 20 µm ([68], an overlapping-author preprint); these are premises for the model, not conclusions that presuppose the paper's target result. Appendix D does contain an important limitation statement: it estimates x_j δη ∼ 10^-13 T versus δx_j η0 ∼ 10^-11 T and says the trajectory-deviation dephasing 'cannot be neglected' and 'has to be accounted for while obtaining precise bounds on experimental parameters.' That is a correctness or completeness problem with the main analysis, but it is not circularity: the paper does not redefine Γ to include Γdev and then claim to predict it as an independent output. No circular step can be exhibited from the paper's own equations, so the circularity score is 0.

Assumptions & free parameters 5 free parameters · 8 assumptions · 0 invented entities

The central dephasing bound rests on a standard stochastic-noise formalism plus a set of assumed chip parameters from ref. 68, a preprint by overlapping authors. The main free choice is the target Γmax~100 Hz; the most fragile assumption is that trajectory deviations can be neglected in Eq. (29), which Appendix D later questions. No new physical entities are introduced.

free parameters (5)
  • Maximum tolerable dephasing rate Γmax = ~100 Hz (chosen; 10% coherence would be Γ≈155 s⁻¹)
    Hand-chosen target used to invert the dephasing integrals into bounds on A and K; the abstract suggests O(ω0/2π)≈67 Hz instead.
  • Nanoparticle mass m = ~10⁻¹⁵ kg
    Used for σx, Δx_max, and the contrast; no particle radius or independent source is given in this paper.
  • Bias magnetic field B0 = 0.2 T
    Taken from Table I and ref. 68; enters H and the zero-point wavefunction width.
  • Wire current I = 12 A
    Taken from Table I and ref. 68; sets η0 and ω0, hence the central δI/I bound.
  • Wire-particle distance d = 20 µm
    Taken from Table I and ref. 68; sets η0 and the flicker-noise PSD coefficient.
assumptions (8)
  • standard math The gradient noise is a stationary Gaussian classical process, so the Wiener-Khinchin theorem gives Γ from the PSD, Eq. (6).
    Used in Section II and Appendix A to convert ensemble-averaged phase variance into a frequency integral.
  • domain assumption The nanoparticle motion is one-dimensional along x, with x(0)=0, x-dot(0)=0, y=0, and gravity ignored.
    Stated after Eq. (18) and around Eq. (22); a real Stern-Gerlach interferometer has 3D motion and rotations.
  • domain assumption The magnetic field is that of an infinite thin wire, B=μ0I/(2πd).
    Eqs. (15)-(17); finite chip geometry and bias-field noise are not modeled.
  • domain assumption The NV-diamond dynamics follow the Stern-Gerlach Lagrangian with Sx=±1 and zero-field splitting D from refs. 53-55 and 79.
    Eq. (18) is assumed without derivation; it is the whole dynamical model.
  • domain assumption Superconducting Nb wires exhibit flicker noise with α≈1 and PSD S_II=K I²/|ω|^α.
    From refs. 76 and 77 and used in Eq. (42) for the flicker bound.
  • ad hoc to paper In the main dephasing calculation, the noise changes the Lagrangian coefficients but not the trajectories xR and xL.
    Explicitly assumed before Eq. (4) and used for Eq. (29); Appendix D says trajectory-deviation dephasing cannot be neglected, making this load-bearing.
  • ad hoc to paper A 10% coherence target sets Γmax ~100 Hz.
    Introduced in Sec. IV A; it is a design choice, not derived.
  • domain assumption The wavefunction remains a Gaussian ground state of a harmonic oscillator with σx=sqrt(ℏ/2mω0).
    Eqs. (54)-(61) for the contrast; ignores squeezing, rotations, and phonon modes.

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Pith. "Pith review of Magnetic noise in macroscopic quantum spatial superposition." pith.science (2026). https://pith.science/paper/QUSGUF2W

@misc{pith2026250413252,
  author       = {Pith},
  title        = {Pith review of: Magnetic noise in macroscopic quantum spatial superposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QUSGUF2W}},
  note         = {Machine review of arXiv:2504.13252}
}
abstract

In this paper, we will show how random fluctuations in the magnetic field will jitter the paths of a matter-wave interferometer randomly, hence, decohere the quantum superposition. To create a large spatial superposition with nanoparticles, we envisage embedding a spin in a nanoparticle as a defect and applying an inhomogeneous magnetic field as in a Stern-Gerlach type experiment to create a macroscopic quantum superposition. Such matter-wave interferometers are the cornerstone for many new fundamental advancements in physics; particularly, adjacent matter-wave interferometers can use entanglement features to test physics beyond the Standard Model, test the equivalence principle, improve quantum sensors, and test the quantum nature of spacetime in a lab. In particular, we will use white and flicker noise to study the decoherence and constrain the parameters keeping in mind ambient temperatures suitable for superconducting wires embedded on a chip. We will show that to obtain a tiny spatial superposition of a nanometer separation, $\Delta x \sim {\cal O} (10^{-9})$m and to minimize decoherence, $\Gamma\leq {\cal O}(\frac{\omega_0}{2\pi})$, where $\Gamma$ is the decoherence and $\omega_0$ is the frequency of the oscillator, we will need current fluctuations to be $\delta I/I\leq {\cal O}(10^{-8})$, which is not impossible to obtain in superconducting wire arrangements. For such tiny fluctuations, we demonstrate that the Humpty-Dumpty problem in a matter-wave interferometer arising from a mismatch in position and momentum does not cause a loss in contrast.

Figures

Figures reproduced from arXiv: 2504.13252 by the authors.

Figure 1
Figure 1. FIG. 1: Dependence of the dephasing rate Γ [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Contour plot showing the behaviour of the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Comparison of the upper bound on the relative [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: FIG. 5: Numerical simulations of trajectory deviations [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Plot of Γ [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The graph show the nature of the integrands in [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

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Reference graph

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Pith tools

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