⚛️ quantum physics

Atomic correlation effects in collapse-induced spontaneous radiation

This paper derives a general framework for calculating spontaneous radiation rates in collapse models by systematically incorporating atomic structure and charge correlations, revealing how these effects modify emission predictions and enabling more robust, material-dependent experimental constraints on models like Diósi-Penrose and Continuous Spontaneous Localization.

Original authors: Simone Manti, Nicola Bortolotti, Lajos Diósi, Kristian Piscicchia, Catalina Curceanu

Published 2026-08-10

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CC BY 4.0

Original authors: Simone Manti, Nicola Bortolotti, Lajos Diósi, Kristian Piscicchia, Catalina Curceanu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible stage where tiny actors—electrons and protons—perform a dance. In the standard script of quantum mechanics, these actors can be in two places at once, a spooky state called "superposition," until someone (or something) looks at them, forcing them to pick a single spot. This "looking" is called wavefunction collapse, but the script is vague on how it happens. Is it a magical observer? Or is it a fundamental law of nature, like gravity, that constantly nudges particles to choose a side?

To test this, scientists have invented "collapse models." Think of these models as theories that say the universe has a built-in, invisible static noise, like a faint hiss on an old radio. This noise constantly jiggles particles. If the noise is strong enough, it forces the quantum dance to stop and the particle to pick a location. But here's the kicker: if this noise is real and it's shaking charged particles (like electrons), those particles should wiggle and emit tiny flashes of light, like a radio antenna broadcasting static. This is called "spontaneous radiation." Detecting this faint glow would be the smoking gun proving that the universe's "static noise" is real.

However, there's a catch. At very low energies, the light waves get so long that they start to "see" the whole atom at once, not just individual particles. Inside an atom, electrons and protons are arranged in a specific, messy cloud. If the noise shakes the whole cloud, the particles might wiggle in a way that cancels each other out, like two people pushing a swing from opposite sides at the exact same time. If you ignore this cancellation, you might predict a bright flash where there is actually silence. This is the puzzle this paper tackles: how to accurately predict the glow from this cosmic static, taking into account the messy, cancelling dance of the atoms themselves.

The Paper's Mission: Mapping the Atomic Dance Floor

In this study, Simone Manti and their team set out to write a better "instruction manual" for calculating how much light these collapse models should produce. They wanted to move beyond simple guesses and create a tool that accounts for the real, complex structure of atoms.

The authors start by deriving a general formula for the emission rate that works for any kind of noise, not just the specific types usually studied. They then focus on two popular theories: the Continuous Spontaneous Localization (CSL) model and the Diósi–Penrose (DP) model. You can think of CSL as a noise that is very "fuzzy" and spread out over a large distance (like a thick fog), while DP is a noise that is more "sharp" and sensitive to the specific mass of the object (like a precise laser pointer).

The big problem with previous calculations was that they treated atoms like a bag of marbles where every electron was frozen at a fixed distance from the center. The authors realized this was like trying to predict the sound of a drum by assuming the drumhead is made of rigid, unmoving dots. In reality, electrons are a fuzzy cloud, and their positions are probabilistic. To fix this, the team used a method called "Radial Distribution Functions" (RDFs). Imagine taking a high-speed camera and snapping millions of photos of an electron cloud to see exactly where the electrons are likely to be found at any given moment. They used powerful computer simulations (specifically Density Functional Theory) to map out these clouds for two heavy elements: Germanium (Ge) and Xenon (Xe).

What They Found: The Cancellation Effect

When they plugged these realistic maps into their new formula, they found something fascinating. In the low-energy range (specifically between 1 and 100 keV, which is the sweet spot for current experiments), the way the particles are arranged inside the atom changes the predicted glow significantly.

For the CSL model, the authors confirmed a "cancellation effect." Because the noise is so spread out, it shakes all the electrons and protons in the atom almost in unison. Since protons are positive and electrons are negative, their wiggles cancel each other out perfectly as the energy gets lower. The paper shows that for a neutral atom, this cancellation causes the predicted light emission to drop to zero as the energy approaches zero. It's like two people pushing a swing from opposite sides with equal force; the swing doesn't move, and no sound is made.

However, the DP model behaves differently. Because its noise is more sensitive to the specific mass distribution and has a different "fuzziness" scale, it doesn't cancel out the same way. The authors found that the DP model predicts a different pattern of light, one that doesn't vanish as quickly at low energies.

Why This Matters

The team compared their new, realistic calculations against the old "frozen marble" method. They found that the old method was creating fake, bumpy wiggles in the predicted light spectrum—artifacts of assuming electrons were stuck in place. The new RDF-based approach smoothed these out, giving a much more physical and reliable prediction.

Crucially, the paper suggests that by looking at the low-energy glow (1–100 keV) in ultra-sensitive detectors made of Germanium or Xenon, scientists can now tell the difference between the CSL and DP models. The old methods were too blurry to make this distinction, but this new, detailed accounting of atomic structure provides a clear way to discriminate between the two theories.

The authors didn't just simulate this; they provided the mathematical framework and the specific numbers for Germanium and Xenon to help experimentalists. They didn't claim to have found the noise yet, but they have built a much sharper telescope to look for it. If future experiments detect a glow that matches the DP prediction but not the CSL one (or vice versa), it could finally reveal whether the universe's "static noise" is real and which model describes it best. Until then, this work stands as a vital upgrade to the tools scientists use to hunt for the fundamental rules of reality.

Technical Summary: Atomic Correlation Effects in Collapse-Induced Spontaneous Radiation

Problem Statement
Spontaneous collapse models (SCMs), such as Continuous Spontaneous Localization (CSL) and the Diósi–Penrose (DP) model, propose that wavefunction collapse is a fundamental physical process driven by stochastic and nonlinear modifications to the Schrödinger equation. A primary experimental signature of these models is the spontaneous radiation emitted by charged particles due to the diffusive motion induced by collapse noise. While high-energy emission rates typically follow an inverse-energy scaling ( $1/E$ ), recent attention has focused on the 1–100 keV energy range. In this regime, the photon wavelength becomes comparable to atomic dimensions, rendering the emission sensitive to atomic structure. Previous phenomenological approaches often treated emitters as point-like charges or "clamped" at mean orbital radii, which fails to capture complex atomic correlations and can introduce unphysical oscillations in the predicted spectra. There is a need for a rigorous framework that systematically incorporates the realistic spatial distribution of emitters to distinguish between different collapse models and provide robust constraints on their parameters.

Methodology
The authors derive a general expression for the spontaneous emission rate valid for arbitrary noise correlations within mass-proportional SCMs. The derivation proceeds via a semi-classical approach, calculating the radiation from the random acceleration of charged particles induced by fluctuations in the Newtonian potential.

General Rate Derivation: Starting from a stochastic Schrödinger equation, the authors compute the power radiated by a system of accelerated charges. They express the spectral emission rate as a sum over all pairs of emitters (nuclei and electrons), modulated by the spatial correlation function of the noise field and the geometric arrangement of the particles.
Radial Distribution Functions (RDFs): To disentangle the collapse-model dependence from the atomic structure, the discrete sum over particle pairs is recast into radial integrals using Radial Distribution Functions (RDFs).
- $R(r)$ : The nucleus-electron RDF.
- $P(r)$ : The electron-electron RDF.
  These functions are computed from first principles using Density Functional Theory (DFT) in the local density approximation for isolated atoms (Germanium and Xenon).
Approximations: The electron-electron RDF is evaluated using a product approximation for the two-particle density matrix, neglecting exchange and Coulomb correlation effects. The authors note that while these effects modify the short-range distribution, they are not expected to alter the qualitative comparison between models. Relativistic effects were assessed for heavy elements (Xe) and found to contribute at the percent level, justifying the use of non-relativistic wavefunctions.
Model Specifics: The formalism is applied to the CSL and DP models. The spatial modulation function $f(r)$ , which depends on the specific collapse model's correlation length ( $\sigma$ ), is integrated against the RDFs to determine the total emission rate.

Key Contributions

Unified Framework: The paper provides a general expression for the emission rate that separates the model-dependent modulation (determined by the collapse parameters) from the atomic contribution (determined by the RDFs).
First-Principles Atomic Structure: Unlike previous works that approximated electrons as fixed at mean orbital radii, this work utilizes realistic radial electron densities derived from DFT calculations.
Cancellation Mechanisms: The formalism naturally incorporates charge correlation effects. It demonstrates that for neutral atoms, the emission rate vanishes as energy approaches zero ( $E \to 0$ ) due to the cancellation of contributions from protons and electrons, a feature that is robustly recovered in the limit.
Suppression of Unphysical Artifacts: By using smooth, extended radial densities rather than clamped point charges, the RDF approach eliminates spurious oscillations in the emission spectrum that appeared in simplified models, particularly in the low-energy regime.

Results
The authors evaluated the emission rates for Germanium (Ge) and Xenon (Xe) within the CSL and DP models:

Low-Energy Behavior: For the CSL model, where the correlation length ( $\sigma \approx 10^3$ Å) is much larger than atomic scales, the modulation function is effectively constant. This leads to a cancellation effect where the rate drops to zero as $E \to 0$ . For the DP model, where $\sigma$ is comparable to atomic distances (e.g., 0.54 Å or 4.94 Å), the modulation function varies significantly across the atomic scale, leading to distinct low-energy behaviors compared to CSL.
Spectral Shape: At high energies (where photon wavelengths are much smaller than atomic scales), the RDF approach recovers the standard $1/E$ scaling. However, as energy decreases, atomic correlations modulate the rate. The RDF-based predictions remain consistently below the $1/E$ limit and are smoother than the "clamped-electron" approximation, which exhibits pronounced, unphysical oscillations.
Model Discrimination: The distinct behaviors of the modulation functions in the CSL and DP models at low energies enhance the discriminating power of experimental searches in the 1–100 keV range.

Significance
The paper claims that this RDF-based approach provides a more robust and physically grounded basis for constraining collapse model parameters. By accurately accounting for atomic structure, the framework enables material-dependent predictions that are essential for interpreting data from ultra-low-background experiments (such as VIP, MAJORANA, and XENONnT). The authors assert that this method improves upon previous approximations by eliminating artificial spectral features and correctly modeling the cancellation of charge correlations, thereby offering a unified framework to evaluate and discriminate between different spontaneous collapse theories using low-energy x-ray emission data.

Technical Summary: Atomic Correlation Effects in Collapse-Induced Spontaneous Radiation

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