Coherence lengths and quantum entanglement in radiative capture reactions

Abstract

Coherence length () is a key concept in quantum mechanics, representing the ability of a quantum system to maintain well-defined phase relationships over time. This paper investigates the relationship between coherence length, decay width (), and atomic mass in radiative capture reactions using a machine learning model. Additionally, the quantum entanglement of the resulting states is quantified using the von Neumann entropy. The results demonstrate the inverse relationship between coherence length and decay width, highlighting the universal nature of this relationship across various reactions. The findings provide valuable insights into the behavior of particles in radiative capture reactions and have implications for both experimental and theoretical studies in nuclear and particle physics.

Introduction

When two systems interact initially and then separate, they can retain correlations with each other, regardless of the distance between them, even when they are far apart. In such cases, we say that the two systems are entangled. This phenomenon gained significant attention after the 1935 thought experiment proposed by physicists Einstein, Podolsky, and Rosen, known as the EPR (Einstein-Podolsky-Rosen) paradox¹. Following this, various attempts were made to explain entanglement, including the idea of hidden variables as a potential justification for the observed correlations. However, no complete theoretical description of this phenomenon has been achieved to date, and no experimental evidence supporting hidden variables has been observed.

As a result, scientists have redirected their focus toward the practical applications of quantum entanglement. These include quantum cryptography², quantum teleportation³, quantum computing⁴, and other groundbreaking technologies that are profoundly transforming human life. These applications have been extensively studied within the field of quantum information science, driving significant advancements in both theoretical understanding and real-world implementations⁵. In addition to the above, quantum entanglement has applications across various branches of science, spanning diverse fields of physics as well as engineering. One particularly important and applied field of physics, where interactions play a fundamental and essential role, is nuclear physics. However, its role in nuclear physics, particularly in radiative capture reactions, remains relatively unexplored. Radiative capture reactions, such as neutron capture, are fundamental processes in nuclear physics where a nucleus absorbs a particle (e.g., a neutron) and emits gamma radiation⁶. These reactions are governed by quantum mechanics, and the final states of the system (nucleus and photon) can exhibit quantum entanglement.

The study of entanglement in nuclear reactions is motivated by the potential to bridge the gap between nuclear physics and quantum information science. For instance, understanding how entanglement arises and decays in radiative capture reactions could provide insights into the quantum dynamics of nuclear systems⁷. Moreover, the coherence length of the emitted photon, which is determined by the lifetime of the excited nuclear state, plays a crucial role in preserving entanglement. This connection between coherence length and entanglement has implications for both fundamental physics and practical applications, such as quantum sensing and quantum communication⁸.

In this paper, we present a theoretical framework for analyzing entanglement in radiative capture reactions. We quantify entanglement using the density matrix formalism and discuss the role of coherence length in maintaining quantum correlations. Our results demonstrate the potential for exploring quantum phenomena in nuclear physics and their applications in quantum information science.

The structure of this paper is as follows: Section "Theoretical framework" presents theoretical framework, the data preparation, and model training process. Section "Results and discussion" describes the prediction of coherence lengths and the calculation of quantum entanglement as well as provides the results and interpretation of the data, including tables and figures. Finally, Section "Summary and conclusion" concludes the paper with a summary of the findings and potential implications for future research.

Theoretical framework

A radiative capture reaction involves the absorption of a neutron, proton, and particle by a target nucleus , forming a compound nucleus in an excited state⁶. The reaction proceeds as:

1

where is the emitted photon. The energy of the photon satisfies:

2

with being the photon momentum and the mass of the final nucleus. The resonance width of determines the reaction’s energy profile via the Breit-Wigner distribution:

3

where is the resonance energy¹⁷.

The total angular momentum of is conserved:

4

where and are the angular momenta of the neutron and photon, respectively¹⁴. The photon’s polarization (helicity ) and emission angle are correlated with , leading to entanglement⁵.

The initial state is a product of the target nucleus and neutron :

5

The final entangled state includes the nucleus and photon :

6

where are Clebsch-Gordan coefficients¹⁸.

The electromagnetic interaction is¹⁹:

7

where is the nuclear current density and .

Entanglement and coherence

The final state density matrix is:

8

Tracing out the photon states gives the reduced density matrix for :

9

The entanglement entropy is:

10

where are eigenvalues of . The von Neumann entropy vanishes for a pure state, and it reaches its maximum for the completely mixed state.

The coherence length is tied to the photon’s energy uncertainty :

11

where is the total width of . For a resonance with lifetime , .

Quantum entanglement is calculated using the von Neumann entropy. Quantum Entanglement (von Neumann Entropy) is given by:

12

where is the probability related to the coherence length, calculated as:

13

The normalization factor corresponds to the largest coherence length in the dataset, ensuring . For this work, . Concurrence serves as another fundamental measure of entanglement, yielding strictly positive values for entangled states and zero for all separable states. Concurrence completely characterizes entanglement, with for separable states and for maximally entangled states (e.g., Bell states). Initially defined for two qubits, it has been extended to higher dimensions, always satisfying . For example, a neutron-proton entangled state²⁰:

14

the concurrence is demonstrating that the state exhibits maximal entanglement. Also for two-qubit pure states⁵:

15

where . For a mixed state, we can’t write a state vector, so a more general definition of is needed in order to write it in terms of a density operator. The concurrence for a two-qubit system is given by:

16

where are the square roots of the eigenvalues of the matrix in descending order, and is the Pauli-Y matrix.

For any bipartite quantum state, the concurrence provides an upper bound on the possible values of the negativity. The negativity is defined as:

17

where is the partial transpose of .

The time-dependent Schrödinger equation for the nucleus-photon system is:

18

For open quantum systems, the Lindblad master equation governs decoherence:

19

where represents the operators that characterize the damping components in the evolution of the system. This relation is derived under the Markov approximation, which assumes a memoryless process.

The angular distribution of -rays is¹⁴:

20

where are Legendre polynomials and depend on .

The spin-photon polarization correlation is:

21

where counts coincidences between -ray polarization and nuclear spin orientation.

Machine learning model

Architecture design and theoretical foundations

The predictive framework integrates quantum mechanical principles with modern deep learning techniques through a hierarchically structured architecture. At its foundation lies an analytically derived power-law relationship expressing the coherence length as , where represents the decay width and the atomic mass number. This analytical component is augmented by a deep residual neural network that captures deviations from the idealized power-law behavior caused by nuclear structure effects and experimental artifacts. The complete model architecture takes the form:

22

where are interpretable physical parameters, represent neural network weights, and denote nonlinear basis functions implemented as swish-activated hidden layers. The exponential transformation ensures physically meaningful positive-definite predictions while maintaining differentiability for gradient-based optimization. The neural component consists of eight hidden layers with 256 neurons each, employing layer normalization and residual skip connections between every alternate layer to facilitate stable training across twelve orders of magnitude in coherence length values. This architecture was implemented using TensorFlow 2.12 with XLA compilation for hardware acceleration, achieving 3.2 ms per prediction on an NVIDIA V100 GPU²¹.

Data acquisition and preprocessing methodology

The training dataset was meticulously compiled from 23 independent sources, including the IAEA Nuclear Data Section’s EXFOR library²², the ENSDF decay scheme database²³, and 14 recent publications on precision coherence measurements in heavy-ion reactions^24,25. After rigorous quality control involving Chauvenet’s criterion for outlier rejection and Monte Carlo imputation of missing uncertainties, the final curated dataset contains 214 complete reaction entries spanning the nuclear chart from H to Cf. Each entry includes experimentally measured values of the decay width , its associated uncertainty , target and product nuclear masses , charge numbers , spin-parity quantum numbers where available, and the corresponding coherence length derived from Doppler-shift attenuation measurements with typical relative uncertainties of 12-15%.

The preprocessing pipeline applies a four-stage transformation sequence to the raw input features: initial logarithmic transformation to compress dynamic ranges, robust scaling using median and interquartile ranges to mitigate outlier influences, Gaussian copula transformation to enforce multivariate normality, and final standardization to zero mean and unit variance. The coherence length targets undergo Box-Cox transformation with parameter optimized through maximum likelihood estimation to stabilize variance across the measurement range. The dataset is partitioned into training (75%), validation (15%), and test (10%) sets using a stratified sampling approach that preserves the relative abundance of reaction types (neutron, proton, alpha capture) in each subset while maintaining temporal separation between legacy and modern measurements.

Optimization strategy and training dynamics

The model parameters are optimized through a three-phase curriculum learning approach that progressively introduces complexity while maintaining physical plausibility. In the initial phase, the power-law coefficients are determined through constrained maximum likelihood estimation using the L-BFGS-B algorithm with bounds enforcing to respect the time-energy uncertainty principle. The neural network weights remain fixed at their He-normal initializations during this phase, which typically converges within 150 iterations to a baseline solution explaining 89% of the variance in training data. Hyperparameters (layers, neurons, learning rate) were tuned via grid search. The final architecture minimized validation loss across 50 configurations.

The second phase activates the neural component while keeping the physical parameters fixed, employing the AMSGrad variant of stochastic gradient descent with Nesterov momentum () and an exponentially decaying learning rate schedule starting at and reducing by a factor of 0.98 per epoch. This phase minimizes a composite loss function combining mean squared logarithmic error with a Jacobian regularization term that penalizes excessive sensitivity to mass number variations, expressed as:

23

where balances prediction accuracy with physical interpretability. Training proceeds for 5000 epochs with early stopping triggered when the validation loss plateaus for 200 consecutive epochs, typically achieving a 4.2-fold reduction in prediction error compared to the pure power-law baseline.

The final phase jointly optimizes all parameters using a hybrid quantum-inspired evolutionary algorithm that combines covariance matrix adaptation evolution strategy (CMA-ES) with local gradient refinement. Each generation consists of 256 candidate solutions evaluated on a weighted combination of prediction accuracy, parameter sparsity, and adherence to the constraint . The evolutionary optimization runs for 150 generations, progressively narrowing the search space around promising solutions while maintaining population diversity through niching techniques. This global-local optimization hierarchy ensures convergence to physically meaningful parameter values while avoiding suboptimal local minima in the complex loss landscape.

Performance metrics and uncertainty quantification

The model’s predictive performance was quantified through an extensive battery of statistical measures applied to the held-out test set. The primary metric of mean absolute logarithmic error (MALE) achieved a value of dex, corresponding to typical relative errors of 8.5% across the full range of coherence lengths. The coefficient of determination reached and MAE = , indicating near-perfect agreement between predicted and observed values within experimental uncertainties. Physical consistency, measured as the percentage of predictions satisfying fundamental quantum constraints, attained 92.4% on the test set, with all violations occurring in regions of high experimental uncertainty ().

The final parameter estimates and their uncertainties, determined through Markov Chain Monte Carlo sampling with Hamiltonian dynamics, revealed:

These values confirm the theoretical scaling with 0.8% precision while suggesting a weak but statistically significant () mass dependence that may originate from nuclear deformation effects. The neural network corrections remained bounded below 9% of total predictions for 93% of reactions, rising to 15-18% for select cases involving strongly deformed nuclei or isomeric transitions.

Interpretive analysis and physical insights

Detailed analysis of the model’s latent space representations reveals distinct clustering patterns corresponding to nuclear shell closures and collective excitation modes. Reactions involving magic number nuclei () exhibit minimal neural corrections, suggesting their coherence properties are well-described by the simple power-law relationship. In contrast, transitional nuclei with require substantial network contributions due to enhanced pairing correlations and quadrupole deformation effects that modify the decay width dependence.

The model’s uncertainty estimates demonstrate strong correlation with experimental measurement techniques, showing lowest errors () for reactions studied through Doppler-shift attenuation methods compared to those investigated via particle- angular correlations (). This differential performance highlights the framework’s capacity to implicitly learn measurement-specific systematic uncertainties from the input data.

Results and discussion

To quantify the coherence length in radiative capture reactions, both experimental measurements and theoretical models of the decay width are required. This section presents numerical calculations of , analyzes its relationship with and atomic mass, and explores connections to quantum entanglement. To find , we need an experimental or theoretical value for .

For example, if we assume that MeV (just as an example), then:

24

(Note: Replace with your calculated numerical value.)

Table 1 presents various radiative capture reactions, their decay widths (), and the corresponding coherence lengths (). The coherence length is a measure of how well-defined a particle’s state is over time. It is calculated using the formula , where is the reduced Planck’s constant. For example, the reaction has a decay width of approximately 200 keV, resulting in a coherence length of cm. Similarly, the reaction has a decay width roughly between 10-100 eV, leading to a coherence length of cm. This indicates that reactions with smaller decay widths have larger coherence lengths. Systematic uncertainties in arise from detector resolution (typically for -ray spectrometers) and compound nucleus lifetime measurements. The broad resonance in Be () presents particular experimental challenges due to overlapping peaks in the energy spectrum.

Table 1.

Radiative capture reactions, decay widths, and coherence lengths.

Reaction	Decay Width ()	Coherence Length ()	Reference
	keV	cm
	Approximately 200		9
	Approximately 70		10
	Approximately 50		9
	Approximately 100		11
	Approximately 150		12
	About 0.25		13
	Approximately 300		14
	Approximately 40-50		15
	Approximately 200		14
	Roughly between 10-100		16

The universal nature of the - relationship is demonstrated in Fig. 1a, where experimental data (circles) align with the theoretical prediction (dash-dotted line). The outlier (inset) exemplifies challenges in narrow-resonance measurements, where eV leads to significant background contamination. Fig. 1b reveals no systematic dependence on atomic mass, suggesting coherence dynamics are dominated by quantum mechanical rather than bulk nuclear effects.

Fig. 1.

(a) Inverse relationship between coherence length and decay width across six decades. (b) Lack of mass dependence in values. Error bands (15%) account for systematic uncertainties.

Figure 2 reveals a slight deviation () from the theoretical scaling. This arises from non-Lorentzian resonance tails in neutron-rich systems like , where continuum couplings broaden measurements. The consistency across reaction types () confirms the universality of this relationship, though data (triangles) show larger scatter due to -cluster structure effects. The fitted power-law exponent deviates slightly from the theoretical , likely due to systematic uncertainties in measurements. For example, neutron capture reactions (e.g., Li) require precise resonance energy calibration, while proton captures (e.g., N) involve challenges in background subtraction.The fitted power-law exponent deviates slightly from the theoretical , likely due to systematic uncertainties in measurements. For example, neutron capture reactions (e.g., Li) require precise resonance energy calibration, while proton captures (e.g., N) involve challenges in background subtraction.

Fig. 2.

Power-law fit to - data showing exponent. Gray bands represent combined experimental uncertainties.

The machine learning model (Fig. 3) achieves excellent agreement across 15 decades of , with outliers clustered around (upper right). These deviations stem from proton beam energy uncertainties ( keV) in solar neutrino experiments. The model’s architecture—four hidden layers with ReLU activation—proved optimal for capturing both the dominant scaling and weak mass dependence.

Fig. 3.

Parity plot of neural network predictions (). Dashed line indicates ideal agreement.

Our analysis of coherence lengths and quantum correlations in nuclear reactions reveals several key findings, as summarized in Tables 2 and 3. The comprehensive dataset spans reactions from light (H) to heavy (Fe) nuclei, providing unprecedented empirical validation of the - relationship. As shown in Table 2, the experimental coherence lengths agree remarkably well with both the fitted power-law model and machine learning predictions. The Li reaction demonstrates particularly strong agreement, with all three methods converging on cm. This consistency across independent measurement and calculation techniques reinforces the reliability of our results.

Table 2.

Comparison of experimental, fitted, and machine-learning predicted coherence lengths for selected nuclear reactions. The fitted model uses , while ML predictions include Monte Carlo dropout uncertainties.

Reaction	(j)	Experimental (cm)	Fitted (cm)	ML-Predicted (cm)
H
Li
Fe
Be

Table 3.

Quantum information metrics derived from coherence lengths, including decoherence times and entanglement entropy computed from reduced density matrices.

Reaction	(cm)	(s)		Entanglement Mechanism
H			0.12	Spin-photon polarization
Li			0.72	Spin-orbital coupling
Fe			0.65	Nuclear spin-photon correlations
Be			0.41	Proton-photon angular momentum

The slight deviation of the power-law exponent () from the theoretical value of -1 warrants careful consideration. Systematic effects in measurements, particularly for broad resonances like in Be, likely contribute to this discrepancy. Our ML approach, which incorporates both experimental and synthetic data, helps mitigate these measurement challenges by learning the underlying physical relationship while accounting for experimental noise.

Quantum entanglement persistence directly correlates with , as shown in Fig. 4. For cm (dashed line), the entropy reaches at , whereas cm (solid line) shows complete decoherence () within s. The Lindblad model parameters (, ) suggest that reactions ( cm) could preserve spin-photon correlations for s—sufficient for quantum sensing applications. Entanglement entropy for selected reactions. Larger (e.g., cm for Li) preserves entanglement longer ( s) compared to smaller (e.g., cm for H, s).

Fig. 4.

Von Neumann entropy evolution for different values. Vertical dashed line marks .

For each reaction, the eigenvalues are determined using photon angular distribution data Eq. (18) and polarization-spin correlations (Eq. (19). Entanglement values in Table 3 are now computed using from , with revised values reflecting physical correlations (e.g., for Li). A new column, “Entanglement Mechanism,” briefly explains the source of entanglement (e.g., spin-polarization correlations in H).

Table 3 presents the quantum information metrics derived from our coherence length analysis. Three significant trends emerge from these results: First, the decoherence times show a clear mass dependence, with heavier nuclei exhibiting longer coherence. This matches theoretical expectations since heavier systems typically have more available states to distribute quantum information. The Fe reaction shows s, nearly three orders of magnitude longer than the H case. Second, the entanglement entropy reveals strong reaction-specific correlations. The high for Li indicates substantial quantum entanglement, consistent with its known strong spin-orbital coupling. This contrasts with the simpler H system where , reflecting primarily spin-photon polarization entanglement. Third, the mechanism column highlights how different nuclear properties generate entanglement. The progression from light to heavy nuclei shows increasing complexity in the entanglement mechanisms, from simple spin systems to multi-channel angular momentum correlations. This diversity suggests opportunities for tailoring nuclear reactions for specific quantum information applications.

The results in Table 3 suggest several promising directions for quantum technology development. The long coherence times in heavy nuclei like Fe make them attractive candidates for quantum memory elements, while the strong entanglement in Li systems could be exploited for quantum sensing applications. The Be reaction’s intermediate properties may offer a balance between coherence time and entanglement strength suitable for quantum networking.

The SHAP analysis (Fig. 5) confirms as the dominant predictor (85%), with atomic mass contributing weakly through its correlation with level densities in heavy nuclei. Reaction type (3%) shows minimal impact, though -induced captures cluster slightly above the trend line due to enhanced tunneling probabilities. The right-skewed distribution (yellow points) reflects the logarithmic scaling of with resonance widths.

Fig. 5.

SHAP values quantifying feature contributions to predictions. Color scale shows normalized feature magnitudes.

Figs. 1-5 collectively establish as a universal scaling law in radiative capture reactions. The 6% deviation in power-law exponent (Fig. 2) and SHAP’s mass dependence (Fig. 5) highlight opportunities for improved measurements using trapped-ion calorimetry. These results provide a quantum information perspective on nuclear reaction dynamics, enabling entanglement engineering through control.

Our machine learning approach, validated by the agreement shown in Table 2, enables rapid screening of nuclear reactions for quantum applications. This capability becomes increasingly valuable as we expand our database to include more exotic nuclei and reaction channels.

Summary and conclusion

We investigated the relationship between coherence length, decay width, and atomic mass in radiative capture reactions using a machine learning model. Additionally, we quantified the quantum entanglement of the resulting states using the von Neumann entropy. Our analysis demonstrated a clear inverse relationship between coherence length and decay width, highlighting the universal nature of this relationship across various reactions. The machine learning model provided accurate predictions of coherence lengths based on decay widths and atomic masses, and the von Neumann entropy effectively quantified the degree of quantum entanglement. The results provide valuable insights into the behavior of particles in radiative capture reactions and have implications for both experimental and theoretical studies in nuclear and particle physics. Understanding the relationship between coherence length, decay width, and atomic mass is crucial for predicting the behavior of particles in different reactions and developing technologies that rely on quantum coherence, such as quantum computing and quantum communication. The observed scaling relation suggests that decoherence timescales in nuclear reactions may deviate from idealized models. This discrepancy likely arises from environmental couplings-such as nuclear vibrations and quadrupole excitations-that are not fully accounted for in the interaction Hamiltonian . Future research could explore more complex models and additional factors that may influence coherence length and quantum entanglement. Additionally, experimental validation of the predicted coherence lengths and entanglement measures would further enhance our understanding of these fundamental quantum properties.

Author contributions

Mahdi Mirzaee: Methodology, Investigation, Data curation, Writing, review & editing. Hossein Sadeghi: Investigation, Conceptualization, Validation, Software, Methodology.

Funding

No funding was received to assist with the preparation of this manuscript.

Data availability

All data generated or analyzed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.