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arXiv:0707.3134 · Journal of Physics A Mathematical and Theoretical

Photon-emission rate from atomic systems in the CSL model

Stephen L Adler, Fethi M. Ramazanoğlu

Published Oct 16, 2007• 72 citations
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Abstract

We calculate the photon emission rate from a general atomic system in the mass- proportional CSL model. For an isolated charged particle emitting kilovolt gamma rays, our results agree with those obtained by Fu. For a neutral atomic system, photon emission is strongly suppressed for photon wavelengths much larger than the atomic radius. However, for kilovolt gamma rays, Fu's result is modified by a structure factor that is of order unity, giving no rate suppression. Our calculation is readily generalized to the case of non-white noise, noise couplings that are not mass-proportional, and general (non-Gaussian) spatial correlation functions, and corresponding results are given. We briefly discuss the implications of our calculation for upper bounds on the CSL model parameters.

DOI
10.1088/1751-8113/40/44/017

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•Sep 5, 2026•0 citations
We calculate the photon emission rate from a general atomic system in the mass- proportional CSL model. For an isolated charged particle emitting kilovolt gamma rays, our results agree with those obtained by Fu. For a neutral atomic system, photon emission is strongly suppressed for photon wavelengths much larger than the atomic radius. However, for kilovolt gamma rays, Fu's result is modified by a structure factor that is of order unity, giving no rate suppression. Our calculation is readily generalized to the case of non-white noise, noise couplings that are not mass-proportional, and general (non-Gaussian) spatial correlation functions, and corresponding results are given. We briefly discuss the implications of our calculation for upper bounds on the CSL model parameters.
We calculate the photon emission rate from a general atomic system in the mass-proportional CSL model. For an isolated charged particle emitting kilovolt gamma rays, our results agree with those obtained by Fu. For a neutral atomic system, photon emission is strongly suppressed for photon wavelengths much larger than the atomic radius. However, for kilovolt gamma rays, Fu’s result is modified by a structure factor that is of order unity, giving no rate suppression. Our calculation is readily generalized to the case of non-white noise, noise couplings that are not mass-proportional, and general (non-Gaussian) spatial correlation functions, and corresponding results are given. We briefly discuss the implications of our calculation for upper bounds on the CSL model parameters. 1
Collapse models introduce stochastic and nonlinear modifications in the quantum dynamics, predicting observable effects, such as spontaneous radiation from charged particles, which can be used to constrain their parameters. Recently, attention has focused on the 1-100 keV energy range, where the wavelength of the emitted photons becomes comparable to atomic dimensions, making the emission sensitive to atomic structure and leading to model-dependent behaviors that enable their discrimination. Here, we derive a general expression for the spontaneous emission rate for arbitrary noise, providing a framework that systematically incorporates the atomic structure through the radial distribution of the emitters, modulated by the specific collapse model. The formalism recovers previous results in the appropriate limits and naturally includes new low-energy effects, such as cancellation mechanisms arising from charge correlations. We evaluate the rates for germanium and xenon within the Diósi-Penrose and Continuous Spontaneous Localization models, showing how these correlations modify the predicted emission rates. This approach provides a unified framework to account for atomic effects and enables more robust, material-dependent experimental constraints on collapse model parameters.
Abstract We study spontaneous radiation emission from matter, as predicted by the Continuous Spontaneous Localization (CSL) collapse model. We show that, in an appropriate range of energies of the emitted radiation, the largest contribution comes from the atomic nuclei. Specifically, we show that in the energy range E∼10 − 105E\sim 10\,-\,10^{5}E∼10−105 E ∼ 10 - 10 5 keV the contribution to the radiation emission from the atomic nuclei grows quadratically with the atomic number of the atom, overtaking the contribution from the electrons, which grows only linearly. This theoretical prediction is then compared with the data from a dedicated experiment performed at the extremely low background environment of the Gran Sasso underground National Laboratory, where the radiation emitted form a sample of Germanium was measured.As a result, we obtain the strongest bounds on the CSL parameters for rC≤10−6r_C\le 10^{-6}rC​≤10−6 r C ≤ 10 - 6 m, improving the previous ones by more than an order of magnitude.
In this paper new upper limits on the parameters of the Continuous Spontaneous Localization (CSL) collapse model are extracted. To this end the X-ray emission data collected by the IGEX collaboration are analyzed and compared with the spectrum of the spontaneous photon emission process predicted by collapse models. This study allows to obtain the most stringent limits within a relevant range of the CSL model parameters, with respect to any other method. The collapse rate λ\lambdaλ and the correlation length rCr_CrC​ are mapped, thus allowing to exclude a broad range of the parameter space.
The strongest experimental bound on white, mass-proportional continuous spontaneous localization (CSL) at the canonical correlation length rC = 10-7 m is the Majorana Demonstrator's 2022 search for collapse-induced X-ray emission from germanium (Phys Rev Lett 129, 080401). That bound was derived assuming the emission rate scales as 1/E with the coherent-nucleus-plus-electron amplitude S = Np2 + Ne. In 2024, Piscicchia and colleagues showed the true atomic-system emission spectrum departs from 1/E in exactly the band the Majorana search used, and called for the Majorana analysis to be reconsidered. This paper bounds that departure two ways. First, an assumption-free bound: the two k-dependent terms of the emission strength are bounded separately by the true extrema of sinc(x) (minimum -0.2172, not -1), giving 0 ≤ S(k) ≤ Np2+Ne2+2|sincmin|NpNe for every photon momentum and every atomic configuration. For germanium (Np = Ne = 32) this caps any atomic-structure correction at a factor 2.361 (not the looser 3.879 obtained from the naive triangle inequality, which is not attainable): the tightest conceivable corrected ceiling is 2.09 × 10-15 s-1, and the GRW parameter point survives it by a further factor 20.9. Second, a Hartree-Fock argument, given hypotheses stated explicitly in the paper, that within that structure the correction can only weaken the bound, not tighten it. Under the best currently published spectrum (Manti, Bortolotti, Diósi, Piscicchia and Curceanu 2026, arXiv:2608.07205), GRW survives by 79-87 times, against a published margin of 49.4 times. All arithmetic is executed in VERIFY.py and reproduced in VERIFY-OUTPUT.txt. Combined with a scoped 34-source census of the field's other constraint classes, the result is that the white-CSL window at rC = 10-7 m remains open and is bounded against future spectrum corrections by a proved ceiling, strengthened by a hypothesis-bearing argument for its sign.