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Superfluid Helium-4 Calorimetry as a Probe of Colored Collapse Noise

Hasan Efe Karabulut

Published Oct 25, 2026• 0 citations
Publisher

Abstract

Continuous Spontaneous Localization (CSL) and related collapse models predict a universal heating of matter. If the collapse noise is colored, i.e. has a spectral cutoff Ω far below X-ray frequencies, the X-ray bounds that exclude white-noise CSL no longer apply, and bulk heating of a medium samples the noise at the phonon frequency c_s/r_C. Superfluid helium-4, with a sound speed of about 238 m/s, samples it roughly 17 times lower in frequency than typical solids. For Lorentzian (cCSL) noise, helium is therefore more sensitive than a solid by (c_solid/c_He)^2 ≈ 280 when Ω ≪ c_He/r_C. This advantage is robust against phonon damping; the much larger ratio obtained for an ideal sharp cutoff is not. We compare projected helium constraints with all cutoff-robust bounds over the full (λ, r_C, Ω) parameter space: LISA Pathfinder and its re-analysis, the layered ultracold cantilever, cold atoms, and the heat budgets of the Earth and Moon. The mechanical bounds are scaled with their exact form factors, and the space is limited from below by a classicality requirement. At r_C = 10^-7 m, a helium calorimeter with residual unexplained heating of 10 pW/kg improves on the strongest peer-reviewed bound only by a factor of about 5 (2.4 including preprint bounds). The advantage is concentrated at small localization lengths: for r_C ≲ 3×10^-8 m the gain is about 55–75 at 10 pW/kg and about 7 even at the demonstrated level of ~100 pW/kg. For r_C ≲ 3 nm, helium would close the remaining open parameter space in a band of Ω. We propose a discriminating two-medium signature. CSL heating per unit mass is material independent for white noise, but colored noise raises the helium-to-solid ratio up to (c_solid/c_He)^2, i.e. 85–2200 for Pb to sapphire. We also show that single-mode quantum acoustics is not competitive with calorimetry, and we give a quantitative systematic budget for an underground, differential calorimeter. Version 2 adds the layered-cantilever bound, the LISA Pathfinder re-analysis, the lunar bound, a classicality lower bound, a damping analysis, a systematic budget and the two-medium signature, and corrects the novelty claims of version 1. All numbers and figures are reproducible with the accompanying Python scripts (physics.py, analysis.py), which run a built-in validation suite first.

DOI
10.5281/zenodo.23645825

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Testing Colored-Noise Collapse Models with Superfluid Helium-4

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Dynamical collapse models such as Continuous Spontaneous Localization (CSL) address the quantum measurement problem and predict a tiny spontaneous heating of matter. The originally proposed parameter range of white-noise CSL has been excluded by X-ray searches, most recently XENONnT (2026). These X-ray bounds, however, lose their validity if the collapse noise is "colored", i.e. cut off below X-ray frequencies. In that regime the strongest remaining constraints come from bulk heating of solids and from low-frequency mechanical experiments. This feasibility study applies the colored-noise heating formalism of Adler and Vinante to superfluid helium-4. Because the sound speed of helium (~238 m/s) is about 17 times lower than that of solids, helium samples the noise spectrum at correspondingly lower frequencies. This gives a sensitivity advantage over solids of ~280 for Lorentzian (cCSL) noise and up to ~10^6 for a sharp cutoff. For r_C = 10^-7 m, a helium calorimeter limiting unexplained heating to 10 pW/kg (1 pW/kg) would improve on the strongest existing bound by a factor of ~2 (~25) in the window Ω ≈ 3×10^9 – 5×10^10 s^-1. This window overlaps with cosmologically motivated cutoff frequencies for cCSL. The note includes closed-form results, validated numerical calculations, and a proposed experimental concept: underground DC calorimetry with tens of kilograms of superfluid helium, using a fill-level differential measurement to separate wall-induced backgrounds. Limitations and open questions are discussed.

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Collapse Wind: Frame-Dependent Heating from Colored Spontaneous-Collapse Noise and Its Annual Modulation

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Spontaneous-collapse models modify the Schrödinger equation with a weak stochastic noise that suppresses macroscopic superpositions. A physically generated noise cannot be white: it must have a spectral cutoff Ω_c. A cutoff in turn breaks Lorentz invariance and singles out a rest frame, most naturally that of the cosmic microwave background (CMB). This preprint works out what happens when a laboratory moves through such a colored collapse noise at v ≈ 370 km/s, an effect we call the "collapse wind". Because the CSL kernel weights wave numbers q ~ 1/r_C, the Doppler shift q·v ~ v/r_C ≈ 3.7 × 10^12 s^-1 dominates every phonon and mechanical frequency, provided the noise pattern is frozen (Ω_c r_C ≪ v). Main results: • Bulk heating: an exact suppression factor with the closed-form asymptote F ≈ (2√π/3) Ω_c r_C / v. Heating per unit mass becomes independent of the material, whereas rest-frame colored CSL predicts ratios up to 10^2–10^4 between materials.• Bounds: existing bulk-heating data imply λ Ω_c ≲ 10^2 s^-2 at r_C = 10^-7 m.• Annual modulation: 7.9% amplitude, peaking on 12–13 June (Planck dipole + ERFA ephemerides).• Mechanical tests: an exact real-space method for layered and cuboidal test masses, applied to the LISA Pathfinder cube and to the multilayer cantilever that sets the strongest mechanical bound. The collapse force noise is suppressed to a sidereal average of ~5 × 10^-3 at Ω_c = 10^10 s^-1 and concentrated in two ~15-minute windows per sidereal day. Bulk heating becomes the binding constraint, and part of Adler's enhanced-collapse region is reopened. A sidereal-phase-gated reanalysis of existing force-noise data could reach it.• A norm-preserving, Born-rule-compliant directional collapse equation: superpositions separated perpendicular to the wind collapse ~15 times faster than those separated along it (10 μm sphere). All numbers and figures are reproduced by the accompanying Python code (included in the source archive).

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Spontaneous collapse models predict that a weak force noise acts on any mechanical system, as a consequence of the collapse of the wave function. Significant upper limits on the collapse rate have been recently inferred from precision mechanical experiments, such as ultracold cantilevers and the space mission LISA Pathfinder. Here, we report new results from an experiment based on a high-Q cantilever cooled to millikelvin temperatures, which is potentially able to improve the current bounds on the continuous spontaneous localization (CSL) model by 1 order of magnitude. High accuracy measurements of the cantilever thermal fluctuations reveal a nonthermal force noise of unknown origin. This excess noise is compatible with the CSL heating predicted by Adler. Several physical mechanisms able to explain the observed noise have been ruled out.

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Colored collapse models from the non-interferometric perspective

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Abstract Models of spontaneous wave function collapse describe the quantum-to-classical transition by assuming a progressive breakdown of the superposition principle when the mass of the system increases, providing a well-defined phenomenology in terms of a non-linearly and stochastically modified Schrödinger equation, which can be tested experimentally. The most popular of such models is the continuous spontaneous localization (CSL) model: in its original version, the collapse is driven by a white noise, and more recently, generalizations in terms of colored noises, which are more realistic, have been formulated. We will analyze how current non-interferometric tests bound the model, depending on the spectrum of the noise. We will find that low frequency purely mechanical experiments provide the most stable and strongest bounds. Graphical abstract

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Which Mass Density Collapses Spacetime? Diósi–Penrose Heating of Crystals and Helium-4

Hasan Efe Karabulut

In the Diósi–Penrose (DP) model a superposition of two mass distributions decays at the rate E_G/ħ. Diósi's master equation uses the mass-density operator smeared over a length R0; Penrose's formulation uses the density of the stationary state, and the "parameter-free" choice R0 = u_rms (the zero-point amplitude) links the two. This preprint examines what each choice implies for spontaneous heating and for collapse rates, and states separately which results are standard and which are new. Main results:• Heating per unit mass is 2πGħ∫g², i.e. Għ/(4√π σxσyσz) for Gaussian smearing of per-axis widths σi. With R0 equal to the per-axis zero-point amplitude, Diósi's functional reproduces Penrose's E_G for displacements smaller than the lattice spacing.• For a crystal ball, E_G splits exactly into a continuum term and a per-atom granular term. The continuum term is already contained in the per-atom term (the gravitational analogue of the Lorentz local field), the granular term is lattice periodic, and it carries a Madelung reduction of 0.90 (Cu) and 0.29 (⁴He).• Copper is predicted to heat at 12.1–12.8 pW/kg. This is in mild tension with the 10 pW/kg figure adopted after background subtraction from the Bayreuth heat leak.• The published Debye–Waller values of hcp ⁴He follow a convention in which the reported ⟨u²⟩ is half the one-dimensional mean-square displacement. One direct cross-check and four consistency checks support this reading. With it, the solid-helium prediction becomes 6–8 fW/kg instead of 21 fW/kg.• For any liquid (⁴He or ³He) the one-body density is uniform, so Penrose's granular E_G vanishes. The solid/liquid ⁴He collapse-rate ratio is ≈6 for d ≪ u, 2.8 at 0.2 nm and 1.15 at 1 nm, which quantifies a conjecture of Diósi (2013).• Heating under Penrose's prescription depends on the dynamical completion. For every completion that splits ⟨ρ⟩ into non-negative pieces, the liquid heats at most 2πGħn = 1.15 fW/kg; maximal localization then gives a liquid/solid ratio of 0.155. Density-matrix and configuration-based completions give anything from ~10⁻²⁶ to order one.• Experiment: – Ionizing radiation deposits nearly the same power per kilogram in every material and therefore mimics the fixed-R0 signal. – Underground at LNGS, unshielded environmental gammas still deposit 0.6–8 pW/kg, so ≥20 cm of lead is needed. – The nearest-term test of the zero-point prescription is to compare the heat leaks of heavy, stiff metals with copper; tungsten is predicted at ≈77 pW/kg, six times copper. – Underground helium calorimetry at 1–0.1 pW/kg would test colored fixed-R0 models up to R0 ≈ 10–21 pm, provided the cell-wall heat release can be subtracted to 5–0.5%.

•Oct 27, 2026•0 citations
Continuous Spontaneous Localization (CSL) and related collapse models predict a universal heating of matter. If the collapse noise is colored, i.e. has a spectral cutoff Ω far below X-ray frequencies, the X-ray bounds that exclude white-noise CSL no longer apply, and bulk heating of a medium samples the noise at the phonon frequency c_s/r_C. Superfluid helium-4, with a sound speed of about 238 m/s, samples it roughly 17 times lower in frequency than typical solids. For Lorentzian (cCSL) noise, helium is therefore more sensitive than a solid by (c_solid/c_He)^2 ≈ 280 when Ω ≪ c_He/r_C. This advantage is robust against phonon damping; the much larger ratio obtained for an ideal sharp cutoff is not. We compare projected helium constraints with all cutoff-robust bounds over the full (λ, r_C, Ω) parameter space: LISA Pathfinder and its re-analysis, the layered ultracold cantilever, cold atoms, and the heat budgets of the Earth and Moon. The mechanical bounds are scaled with their exact form factors, and the space is limited from below by a classicality requirement. At r_C = 10^-7 m, a helium calorimeter with residual unexplained heating of 10 pW/kg improves on the strongest peer-reviewed bound only by a factor of about 5 (2.4 including preprint bounds). The advantage is concentrated at small localization lengths: for r_C ≲ 3×10^-8 m the gain is about 55–75 at 10 pW/kg and about 7 even at the demonstrated level of ~100 pW/kg. For r_C ≲ 3 nm, helium would close the remaining open parameter space in a band of Ω. We propose a discriminating two-medium signature. CSL heating per unit mass is material independent for white noise, but colored noise raises the helium-to-solid ratio up to (c_solid/c_He)^2, i.e. 85–2200 for Pb to sapphire. We also show that single-mode quantum acoustics is not competitive with calorimetry, and we give a quantitative systematic budget for an underground, differential calorimeter. Version 2 adds the layered-cantilever bound, the LISA Pathfinder re-analysis, the lunar bound, a classicality lower bound, a damping analysis, a systematic budget and the two-medium signature, and corrects the novelty claims of version 1. All numbers and figures are reproducible with the accompanying Python scripts (physics.py, analysis.py), which run a built-in validation suite first.
Dynamical collapse models such as Continuous Spontaneous Localization (CSL) address the quantum measurement problem and predict a tiny spontaneous heating of matter. The originally proposed parameter range of white-noise CSL has been excluded by X-ray searches, most recently XENONnT (2026). These X-ray bounds, however, lose their validity if the collapse noise is "colored", i.e. cut off below X-ray frequencies. In that regime the strongest remaining constraints come from bulk heating of solids and from low-frequency mechanical experiments. This feasibility study applies the colored-noise heating formalism of Adler and Vinante to superfluid helium-4. Because the sound speed of helium (~238 m/s) is about 17 times lower than that of solids, helium samples the noise spectrum at correspondingly lower frequencies. This gives a sensitivity advantage over solids of ~280 for Lorentzian (cCSL) noise and up to ~10^6 for a sharp cutoff. For r_C = 10^-7 m, a helium calorimeter limiting unexplained heating to 10 pW/kg (1 pW/kg) would improve on the strongest existing bound by a factor of ~2 (~25) in the window Ω ≈ 3×10^9 – 5×10^10 s^-1. This window overlaps with cosmologically motivated cutoff frequencies for cCSL. The note includes closed-form results, validated numerical calculations, and a proposed experimental concept: underground DC calorimetry with tens of kilograms of superfluid helium, using a fill-level differential measurement to separate wall-induced backgrounds. Limitations and open questions are discussed.
Spontaneous-collapse models modify the Schrödinger equation with a weak stochastic noise that suppresses macroscopic superpositions. A physically generated noise cannot be white: it must have a spectral cutoff Ω_c. A cutoff in turn breaks Lorentz invariance and singles out a rest frame, most naturally that of the cosmic microwave background (CMB). This preprint works out what happens when a laboratory moves through such a colored collapse noise at v ≈ 370 km/s, an effect we call the "collapse wind". Because the CSL kernel weights wave numbers q ~ 1/r_C, the Doppler shift q·v ~ v/r_C ≈ 3.7 × 10^12 s^-1 dominates every phonon and mechanical frequency, provided the noise pattern is frozen (Ω_c r_C ≪ v). Main results: • Bulk heating: an exact suppression factor with the closed-form asymptote F ≈ (2√π/3) Ω_c r_C / v. Heating per unit mass becomes independent of the material, whereas rest-frame colored CSL predicts ratios up to 10^2–10^4 between materials.• Bounds: existing bulk-heating data imply λ Ω_c ≲ 10^2 s^-2 at r_C = 10^-7 m.• Annual modulation: 7.9% amplitude, peaking on 12–13 June (Planck dipole + ERFA ephemerides).• Mechanical tests: an exact real-space method for layered and cuboidal test masses, applied to the LISA Pathfinder cube and to the multilayer cantilever that sets the strongest mechanical bound. The collapse force noise is suppressed to a sidereal average of ~5 × 10^-3 at Ω_c = 10^10 s^-1 and concentrated in two ~15-minute windows per sidereal day. Bulk heating becomes the binding constraint, and part of Adler's enhanced-collapse region is reopened. A sidereal-phase-gated reanalysis of existing force-noise data could reach it.• A norm-preserving, Born-rule-compliant directional collapse equation: superpositions separated perpendicular to the wind collapse ~15 times faster than those separated along it (10 μm sphere). All numbers and figures are reproduced by the accompanying Python code (included in the source archive).
Standard implementations of Continuous Spontaneous Localization (CSL) assume white noise with an infinite bandwidth, leading to a divergent heating rate and the prediction of unobserved spontaneous X-ray emission (the “Heating Catastrophe”). This paper presents a complete mathematical resolution using a relativistic coloured noise field with a Lorentz invariant spectral density D(k) = λσ2/[1 + (kμkμL2c)n]. We derive the exact suppression factor S for the spontaneous emission rate. For a correlation time τc ≈ 10−12 s, the high frequency spectral tail scales as ω−2n. We demonstrate that for the X-ray regime (ω ∼ 1018Hz), the per-mode suppression is ∼ 10−12, moderated by phase space integration to a total factor of S ≈ 10−8. This reduces the predicted radiation rate below the sensitivity floors of IGEX and CUORE, resolving the tension with experimental data while preserving macroscopic collapse.
Spontaneous collapse models predict that a weak force noise acts on any mechanical system, as a consequence of the collapse of the wave function. Significant upper limits on the collapse rate have been recently inferred from precision mechanical experiments, such as ultracold cantilevers and the space mission LISA Pathfinder. Here, we report new results from an experiment based on a high-Q cantilever cooled to millikelvin temperatures, which is potentially able to improve the current bounds on the continuous spontaneous localization (CSL) model by 1 order of magnitude. High accuracy measurements of the cantilever thermal fluctuations reveal a nonthermal force noise of unknown origin. This excess noise is compatible with the CSL heating predicted by Adler. Several physical mechanisms able to explain the observed noise have been ruled out.
Abstract Models of spontaneous wave function collapse describe the quantum-to-classical transition by assuming a progressive breakdown of the superposition principle when the mass of the system increases, providing a well-defined phenomenology in terms of a non-linearly and stochastically modified Schrödinger equation, which can be tested experimentally. The most popular of such models is the continuous spontaneous localization (CSL) model: in its original version, the collapse is driven by a white noise, and more recently, generalizations in terms of colored noises, which are more realistic, have been formulated. We will analyze how current non-interferometric tests bound the model, depending on the spectrum of the noise. We will find that low frequency purely mechanical experiments provide the most stable and strongest bounds. Graphical abstract
In the Diósi–Penrose (DP) model a superposition of two mass distributions decays at the rate E_G/ħ. Diósi's master equation uses the mass-density operator smeared over a length R0; Penrose's formulation uses the density of the stationary state, and the "parameter-free" choice R0 = u_rms (the zero-point amplitude) links the two. This preprint examines what each choice implies for spontaneous heating and for collapse rates, and states separately which results are standard and which are new. Main results:• Heating per unit mass is 2πGħ∫g², i.e. Għ/(4√π σxσyσz) for Gaussian smearing of per-axis widths σi. With R0 equal to the per-axis zero-point amplitude, Diósi's functional reproduces Penrose's E_G for displacements smaller than the lattice spacing.• For a crystal ball, E_G splits exactly into a continuum term and a per-atom granular term. The continuum term is already contained in the per-atom term (the gravitational analogue of the Lorentz local field), the granular term is lattice periodic, and it carries a Madelung reduction of 0.90 (Cu) and 0.29 (⁴He).• Copper is predicted to heat at 12.1–12.8 pW/kg. This is in mild tension with the 10 pW/kg figure adopted after background subtraction from the Bayreuth heat leak.• The published Debye–Waller values of hcp ⁴He follow a convention in which the reported ⟨u²⟩ is half the one-dimensional mean-square displacement. One direct cross-check and four consistency checks support this reading. With it, the solid-helium prediction becomes 6–8 fW/kg instead of 21 fW/kg.• For any liquid (⁴He or ³He) the one-body density is uniform, so Penrose's granular E_G vanishes. The solid/liquid ⁴He collapse-rate ratio is ≈6 for d ≪ u, 2.8 at 0.2 nm and 1.15 at 1 nm, which quantifies a conjecture of Diósi (2013).• Heating under Penrose's prescription depends on the dynamical completion. For every completion that splits ⟨ρ⟩ into non-negative pieces, the liquid heats at most 2πGħn = 1.15 fW/kg; maximal localization then gives a liquid/solid ratio of 0.155. Density-matrix and configuration-based completions give anything from ~10⁻²⁶ to order one.• Experiment: – Ionizing radiation deposits nearly the same power per kilogram in every material and therefore mimics the fixed-R0 signal. – Underground at LNGS, unshielded environmental gammas still deposit 0.6–8 pW/kg, so ≥20 cm of lead is needed. – The nearest-term test of the zero-point prescription is to compare the heat leaks of heavy, stiff metals with copper; tungsten is predicted at ≈77 pW/kg, six times copper. – Underground helium calorimetry at 1–0.1 pW/kg would test colored fixed-R0 models up to R0 ≈ 10–21 pm, provided the cell-wall heat release can be subtracted to 5–0.5%.