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arXiv:1801.00509 · Physical Review A

Bulk heating effects as tests for collapse models

Stephen L. Adler, Andrea Vinante

Published May 18, 2018• 78 citations
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Abstract

We discuss limits on the noise strength parameter in mass-proportional-coupled wave-function collapse models implied by bulk heating effects and examine the role of the noise power spectrum in comparing experiments of different types. This comparison utilizes a calculation of the rate of heating through phonon excitation implied by a general noise power spectrum $\ensuremath{\lambda}(\ensuremath{\omega})$. We find that, in the standard heating formula, the reduction rate $\ensuremath{\lambda}$ is replaced by ${\ensuremath{\lambda}}_{\mathrm{eff}}=\frac{2}{3{\ensuremath{\pi}}^{3/2}}\ensuremath{\int}{d}^{3}w{e}^{\ensuremath{-}{\stackrel{P\vec}{w}}^{2}}{\stackrel{P\vec}{w}}^{2}\ensuremath{\lambda}({\ensuremath{\omega}}_{L}(\stackrel{P\vec}{w}/{r}_{c}))$, with ${\ensuremath{\omega}}_{L}(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}})$ being the longitudinal acoustic-phonon frequency as a function of wave number $\stackrel{P\vec}{q}$, and with ${r}_{C}$ being the noise correlation length. Hence if the noise power spectrum is cut off below ${\ensuremath{\omega}}_{L}(|\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}|\ensuremath{\sim}{r}_{c}^{\ensuremath{-}1})$, the bulk heating rate is sharply reduced, allowing compatibility with current experimental results. We suggest possible new bulk heating experiments that can be performed subject to limits placed by natural heating from radioactivity and cosmic rays. The proposed experiments exploit the vanishing of thermal transport in the low-temperature limit.

DOI
10.1103/physreva.97.052119

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We discuss limits on the noise strength parameter in mass-proportional-coupled wave-function collapse models implied by bulk heating effects and examine the role of the noise power spectrum in comparing experiments of different types. This comparison utilizes a calculation of the rate of heating through phonon excitation implied by a general noise power spectrum \ensuremathλ(\ensuremathω)\ensuremath{\lambda}(\ensuremath{\omega})\ensuremathλ(\ensuremathω). We find that, in the standard heating formula, the reduction rate \ensuremathλ\ensuremath{\lambda}\ensuremathλ is replaced by {\ensuremath{\lambda}}_{\mathrm{eff}}=\frac{2}{3{\ensuremath{\pi}}^{3/2}}\ensuremath{\int}{d}^{3}w{e}^{\ensuremath{-}{\stackrel{P\vec}{w}}^{2}}{\stackrel{P\vec}{w}}^{2}\ensuremath{\lambda}({\ensuremath{\omega}}_{L}(\stackrel{P\vec}{w}/{r}_{c})), with {\ensuremath{\omega}}_{L}(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}) being the longitudinal acoustic-phonon frequency as a function of wave number \stackrel{P\vec}{q}, and with rC{r}_{C}rC​ being the noise correlation length. Hence if the noise power spectrum is cut off below {\ensuremath{\omega}}_{L}(|\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}|\ensuremath{\sim}{r}_{c}^{\ensuremath{-}1}), the bulk heating rate is sharply reduced, allowing compatibility with current experimental results. We suggest possible new bulk heating experiments that can be performed subject to limits placed by natural heating from radioactivity and cosmic rays. The proposed experiments exploit the vanishing of thermal transport in the low-temperature limit.
Models of spontaneous wave function collapse predict a small heating rate for a bulk solid, as a result of coupling to the noise field that causes collapse. This rate is small enough that ambient radioactivity and cosmic ray flux on the surface of the earth can mask the heating due to spontaneous collapse. In this paper we estimate the background noise due to $\ensuremath{\gamma}$ radiation and cosmic ray muon flux, at different depths. We demonstrate that a low-temperature underground experiment at a depth of about 6.5 kilometer water equivalent would have a low enough background to allow detection of bulk heating for a collapse rate $\ensuremath{\lambda}$ of ${10}^{\ensuremath{-}16}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$ using presently available technology.
Heating induced by the noise postulated in wave function collapse models leads to a lower bound to the temperature of solid objects. For the noise parameter values λ=coupling strength∼10−8s−1λ={\rm coupling~strength}\sim 10^{-8} {\rm s}^{-1}λ=coupling strength∼10−8s−1 and rC=correlation length∼10−5cmr_C ={\rm correlation~length} \sim 10^{-5} {\rm cm}rC​=correlation length∼10−5cm, which were suggested \cite{adler1} to make latent image formation an indicator of wave function collapse and which are consistent with the recent experiment of Vinante et al. \cite{vin}, the effect may be observable. For metals, where the heat conductivity is proportional to the temperature at low temperatures, the lower bound (specifically for RRR=30 copper) is $\sim 5\times 10^{-11} (L/r_C) $K, with L the size of the object. For the thermal insulator Torlon 4203, the comparable lower bound is ∼3×10−6(L/rc)0.63\sim 3 \times 10^{-6} (L/r_c)^{0.63}∼3×10−6(L/rc​)0.63 K. We first give a rough estimate for a cubical metal solid, and then give an exact solution of the heat transfer problem for a sphere.
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).
Collapse models postulate that space is filled with a collapse noise field, inducing quantum Brownian motions, which are dominant during the measurement, thus causing collapse of the wave function. An important manifestation of the collapse noise field, if any, is thermal energy generation, thus disturbing the temperature profile of a system. The experimental investigation of a collapse-driven heating effect has provided, so far, the most promising test of collapse models against standard quantum theory. In this paper, we calculate the collapse-driven heat generation for a three-dimensional multi-atomic Bravais lattice by solving stochastic Heisenberg equations. We perform our calculation for the mass-proportional continuous spontaneous localization collapse model with nonwhite noise. We obtain the temperature distribution of a sphere under stationary-state and insulated surface conditions. However, the exact quantification of the collapse-driven heat-generation effect highly depends on the actual value of cutoff in the collapse noise spectrum.
The possibility that the collapse of the wave function in quantum mechanics is real and ultimately connected to (classical) gravity has been debated for decades with main contributions by Diósi and Penrose (DP). In particular, Diósi proposed a noise-based dynamical reduction model, which captures the same order of magnitudes for the collapse time suggested by Penrose based on heuristic arguments. This is known in the literature as the DP model (Diósi–Penrose). A peculiarity of the DP model is the prediction of spontaneous heating of matter, which can be tested without the need for massive quantum superpositions. Notably, a very similar effect is predicted by recent theoretical approaches to gravity as a classical-only information channel. Here, we reconsider the current constraints on the DP model from spontaneous heating by analyzing experimental situations not properly considered before. We argue that the parameter-free version of the DP model is close to be ruled out by standard heat leak measurements at ultralow temperatures with a conclusive exclusion likely within reach with existing technology. This result would strengthen a recent claim of exclusion inferred by spontaneous x-ray emission experiments, which relies on the somewhat stronger assumption that the DP noise field is white up to x-ray frequencies.