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arxiv: 2604.21705 · v1 · submitted 2026-04-23 · 🪐 quant-ph

Testing Spontaneous Collapse Models with Coulomb Mediated Squeezing

Suroj Dey , Peter Barker , Animesh Datta This is my paper

Pith reviewed 2026-05-09 21:34 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Continuous Spontaneous LocalizationnanospheresCoulomb interactionthermal variancecollapse modelssqueezingmotonal modesentanglement
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The pith

Detecting Coulomb-mediated variance reduction in two nanospheres bounds the CSL collapse parameter at levels matching X-ray tests.

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

The paper shows how the steady-state squeezing induced by Coulomb repulsion between two charged nanospheres reduces the thermal variance of their differential motional mode. Measuring that reduction sets an upper limit on the Continuous Spontaneous Localization parameter λ_CSL. For achievable laboratory values of charge, mass, and separation the resulting limit is comparable to X-ray-emission bounds and stronger than bulk-heating bounds. The same signature remains informative even when collapse models are extended by plausible colored noise. A separate short-time protocol using Coulomb-induced entanglement between ground-state spheres yields limits similar to early X-ray constraints.

Core claim

Detecting steady-state Coulomb-mediated reduction in the thermal variance of the differential motional mode of two nanospheres can bound the Continuous Spontaneous Localization (CSL) parameter (λ_CSL). For realistic experimental parameters the resulting bounds are comparable to those obtained from X-ray emission experiments and surpass those set by bulk-heating ones. Unlike the latter experiments, the bounds are robust against plausible coloured-noise extensions of collapse models. In the short-time regime a weak Coulomb-induced entanglement-based test between two charged nanospheres initialized in the ground state provides constraints on λ_CSL comparable to limits set by early X-ray work.

What carries the argument

Coulomb-mediated reduction in thermal variance of the differential motional mode between two charged nanospheres.

If this is right

  • The steady-state protocol yields λ_CSL bounds at least as strong as X-ray emission limits.
  • The same protocol remains valid under colored-noise extensions of CSL that weaken other tests.
  • The short-time entanglement protocol supplies comparable bounds without requiring long integration times.
  • Both protocols surpass the sensitivity of bulk-heating measurements for the same nanosphere parameters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be combined with existing optomechanical levitation setups to reach the required charge and separation values.
  • Similar variance-reduction signatures might be searched for in other charged macroscopic objects to test the same collapse models.
  • If the bounds improve further they would begin to intersect the parameter region still allowed by interferometric collapse tests.

Load-bearing premise

Any measured reduction in thermal variance is produced only by the modeled Coulomb interaction together with CSL effects and is free of unaccounted systematic noise or deviations in sphere properties and thermal environment.

What would settle it

An experiment that records exactly the thermal variance predicted by Coulomb repulsion alone, with no further reduction, would show that the method cannot set the claimed bounds on λ_CSL.

Figures

Figures reproduced from arXiv: 2604.21705 by Animesh Datta, Peter Barker, Suroj Dey.

Figure 1
Figure 1. Figure 1: FIG. 1. Two identical, levitated charged nanospheres of mass [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Our proposed bounds for parameters in Table 1 and [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Our bounds at choices of experimental parameters in [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
read the original abstract

We show that detecting steady-state Coulomb-mediated reduction in the thermal variance of the differential motional mode of two nanospheres can bound the Continuous Spontaneous Localization (CSL) parameter ($\lambda_{{\text{CSL}}}$). For realistic experimental parameters, the resulting bounds are comparable to those obtained from X-ray emission experiments and surpass those set by bulk-heating ones. Unlike these latter experiments, our bounds are robust against plausible coloured-noise extensions of collapse models. In the short-time regime, we find that a weak Coulomb-induced entanglement-based test between two charged nanospheres initialized in ground state can provide constraints on $\lambda_{\text{CSL}}$ comparable to limits set by early X-ray experiments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes using steady-state Coulomb-mediated squeezing to reduce the thermal variance of the differential motional mode of two charged nanospheres, thereby bounding the CSL parameter λ_CSL. For realistic experimental parameters the resulting limits are stated to be comparable to X-ray emission bounds and stronger than bulk-heating bounds; the limits are claimed to remain valid under colored-noise extensions of CSL. A short-time entanglement-based test with ground-state spheres is also shown to yield constraints comparable to early X-ray experiments.

Significance. If the central assumptions can be validated experimentally, the proposal supplies a new, tabletop route to constraining spontaneous-collapse models that exploits Coulomb interactions and levitated-optomechanics techniques. The explicit robustness to colored-noise CSL extensions is a genuine strength, addressing a frequent objection to other collapse-model tests. The work therefore has the potential to stimulate targeted experiments in the levitated-nanoparticle community.

major comments (2)
  1. [§4] §4 (steady-state variance derivation): the central bound on λ_CSL is obtained by attributing any observed reduction in differential-mode thermal variance exclusively to the modeled Coulomb force plus the CSL term in the master equation. No quantitative propagation of uncertainties arising from charge fluctuations, residual-gas damping variations, or small trap anharmonicities is provided; any of these effects produces an identical signature in the variance and would therefore invalidate the extracted limit without additional experimental controls.
  2. [§5] §5 (experimental feasibility): while robustness to colored-noise CSL extensions is demonstrated, the text does not supply a systematic-error budget or Monte-Carlo study showing how deviations from the assumed sphere charge, mass, or environmental temperature affect the predicted variance reduction. This omission is load-bearing because the claimed competitiveness with X-ray bounds rests on the variance reduction being CSL-dominated.
minor comments (2)
  1. [Abstract] The abstract states numerical bounds without referencing the equations or parameter table from which they are derived; a parenthetical citation to the relevant result (e.g., Eq. (XX) or Table I) would improve readability.
  2. [§2] Notation for the differential motional mode and the CSL noise operator is introduced without a compact summary table; a short glossary or equation list would aid readers unfamiliar with the optomechanics conventions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript's significance and for the detailed comments, which help strengthen the presentation of the proposal. We address each major comment below and have revised the manuscript to incorporate additional discussion and estimates on uncertainties and experimental feasibility.

read point-by-point responses

  1. Referee: §4 (steady-state variance derivation): the central bound on λ_CSL is obtained by attributing any observed reduction in differential-mode thermal variance exclusively to the modeled Coulomb force plus the CSL term in the master equation. No quantitative propagation of uncertainties arising from charge fluctuations, residual-gas damping variations, or small trap anharmonicities is provided; any of these effects produces an identical signature in the variance and would therefore invalidate the extracted limit without additional experimental controls.

    Authors: We agree that experimental uncertainties could in principle mimic the predicted variance reduction and that a discussion of controls is warranted. In the revised manuscript we expand §4 with a new paragraph that (i) identifies the dominant confounding mechanisms, (ii) outlines standard experimental techniques (independent damping-rate measurements, charge-stability monitoring via ionisation or Kelvin-probe methods, and anharmonicity calibration via higher-order sideband spectroscopy) already routine in levitated-optomechanics setups, and (iii) supplies order-of-magnitude estimates showing that, with present-day parameter control, these effects remain sub-dominant to the CSL-induced squeezing for the quoted experimental values. This addition clarifies the conditions under which the bound remains valid without changing the central derivation. revision: yes

  2. Referee: §5 (experimental feasibility): while robustness to colored-noise CSL extensions is demonstrated, the text does not supply a systematic-error budget or Monte-Carlo study showing how deviations from the assumed sphere charge, mass, or environmental temperature affect the predicted variance reduction. This omission is load-bearing because the claimed competitiveness with X-ray bounds rests on the variance reduction being CSL-dominated.

    Authors: We accept that a more explicit error budget strengthens the feasibility claim. The revised §5 now includes a dedicated paragraph presenting analytical sensitivity estimates for ±10 % variations in charge, mass, and temperature around the nominal values used in the figures. These estimates demonstrate that the differential-mode variance reduction remains observable and yields CSL bounds competitive with X-ray limits within the parameter ranges accessible in current levitated-nanoparticle experiments. While a full Monte-Carlo propagation is beyond the scope of this theoretical proposal, the added estimates directly address the load-bearing nature of the competitiveness claim. revision: yes

Circularity Check

0 steps flagged

Forward calculation of CSL bounds from model parameters shows no circularity

full rationale

The paper proposes an experimental test and performs forward calculations of steady-state variance reduction and resulting λ_CSL bounds using the CSL master equation plus realistic nanosphere parameters. No derivation step reduces by construction to a fitted input, self-defined quantity, or load-bearing self-citation. The central result is a predicted experimental sensitivity, independent of the data it would later constrain. Assumptions about systematics are acknowledged as experimental limitations rather than hidden in the math.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard CSL model, thermal equilibrium assumptions for the nanospheres, and the validity of the Coulomb interaction in the motional regime; no free parameters are introduced beyond the CSL rate itself being bounded.

axioms (2)
  • domain assumption Standard quantum mechanics plus the CSL collapse model governs the dynamics of the charged nanospheres.
    Invoked throughout the abstract as the framework for calculating variance reduction.
  • domain assumption The nanospheres remain in thermal equilibrium with the environment except for the CSL and Coulomb effects.
    Required for the steady-state variance calculation to hold.

pith-pipeline@v0.9.0 · 5406 in / 1215 out tokens · 33128 ms · 2026-05-09T21:34:11.013311+00:00 · methodology

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Reference graph

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    Continuous Spontaneous Localization (CSL) Model The CSL master equation reads [2, 19]: ∂ˆρ(t) ∂t =− i ℏ[ ˆH,ˆρ(t)]− λCSL(4πr2 CSL)3/2 2m2 0 ˆ d3x[ ˆM(x),[ ˆM(x),ˆρ(t)]].(A15) With: ˆM(x) = X α∈{1,2} NX i=1 m(α) i (2πr2 CSL)3/2 exp − |r(α) i −x| 2 2r2 CSL ! ,(A16) The DP master equation and the CSL master equation has very similar structures, here the coll...

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    Bounds on The CSL Model Using the diffusion coefficients derived for CSL, Eq. (A20) and Eq. (A21), we can boundλ CSL from the above entanglement observation criteria (D8). To gain some intuition, we look at the case whered≫r c. In this regime, we can ignoreD 12 CSL and obtain a simplified expression for the bound: D11 CSL < δ2mω2ℏ(1−f) 2 , f= 2Dth δ2mω2ℏ ...

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    Proceeding as before, we have: D11 DP < δ2mω2ℏ(1−f) 2 , f= 2Dth δ2mω2ℏ .(D11) Using Eq

    Bounds on DP Model Unlike the CSL correlating noise, the DP correlating noise is long-ranged, in the regime,d≫ R D,D 12 DP → − Gℏm2 d3 ; which exactly the Newtonian-potential, upto a factor ofℏ.Nevertheless, in this regime, we can ignore the correlating noise term compared to the local noise, which is unaffected by the largedlimit. Proceeding as before, w...

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