arxiv: 2605.04640 · v1 · submitted 2026-05-06 · 🪐 quant-ph · cond-mat.dis-nn· cond-mat.quant-gas· cond-mat.str-el

Recognition: 2 theorem links

· Lean Theorem

Neural network modeling of many-body super- and sub-radiant dynamics

Gianluca Lagnese , Laurin Brunner , Lorenzo Rossi , Darrick Chang , Markus Schmitt , Zala Lenar\v{c}i\v{c}

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:53 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nncond-mat.quant-gascond-mat.str-el

keywords neural quantum statesdissipative dynamicssubradiancesuperradianceatomic arraysopen quantum systemslight-matter interaction

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The pith

Neural quantum states capture the dissipative dynamics of 40-atom light-matter arrays

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how neural quantum states can be used to simulate the time-dependent many-body density matrix for systems of atoms interacting with light. This is applied to arrays of around 40 atoms in one and two dimensions, where subradiant dynamics emerge at late times. Sympathetic readers care because these collective effects arise from long-range interactions and dissipation that standard numerical tools cannot handle at such scales, and the systems are realizable in cold-atom experiments. The work establishes a new numerical route for exploring quantum light-matter interfaces without semi-classical approximations.

Core claim

We report the first application of neural quantum states to obtain the dissipative dynamics of light-matter-coupled systems beyond what is accessible with exact and tensor-network calculations. We specifically apply this method to simulate the many-body emission dynamics of approximately 40 atoms, arranged in dense arrays in one and two dimensions. These systems have been chosen because they can support prominent subradiant dynamics at late times and could be realized with cold atomic quantum simulators.

What carries the argument

Neural quantum states as a variational representation of the many-body density matrix to evolve under the dissipative master equation for atomic emission.

If this is right

The approach accesses system sizes and geometries where subradiant states dominate the late-time dynamics.
It enables full quantum simulations of collective emission without semi-classical approximations.
The technique applies to both one- and two-dimensional dense arrays supporting long-range interactions.
This provides a pathway to model light-matter interfaces at scales relevant to current quantum simulators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may extend to study driven-dissipative systems or higher-dimensional setups where entanglement growth limits other approaches.
It could help explore how subradiance scales with atom number in experimentally accessible regimes.
Connections to other many-body open systems might allow broader use in quantum optics and condensed matter simulations.

Load-bearing premise

The neural-network ansatz for the density matrix remains sufficiently expressive and trainable to accurately represent the state throughout the dissipative evolution for the array sizes studied.

What would settle it

Exact comparison on a smaller system size where the neural prediction deviates from the known subradiant decay curve at late times would disprove the method's accuracy.

Figures

Figures reproduced from arXiv: 2605.04640 by Darrick Chang, Gianluca Lagnese, Laurin Brunner, Lorenzo Rossi, Markus Schmitt, Zala Lenar\v{c}i\v{c}.

**Figure 1.** Figure 1: (a) and (b): benchmarking of NQS ResNet results against quantum trajectories (QT) for a linear array of view at source ↗

**Figure 2.** Figure 2: (a) and (b): benchmarking of ResNet against quantum trajectories for a square lattice of view at source ↗

**Figure 3.** Figure 3: Comparison of dynamics for density of excitation view at source ↗

**Figure 4.** Figure 4: Sketch of the two architectures discussed in the view at source ↗

**Figure 5.** Figure 5: Comparison of connected correlations (5), calculated by different approaches: quantum trajectories (QT), best NQS approximation from the main text (f = 12, d = 4, k = 4) and cumulant expansions (2nd and 3rd order) for L = 16 in 1D. Left column: Correlations C(ℓ, t) at different relative sites ℓ. Right column: Absolute difference of correlations calculated by NQS or cumulant expansion with respect to the … view at source ↗

read the original abstract

There is significant interest in exploring novel phenomena in quantum light-matter interfaces, which are driven by the combination of structured dissipation and long-range interactions that are typical in such systems. To this end, it is important to develop new general numerical simulation techniques, which can access large system sizes and are not based on semi-classical approaches. Here, we report the first application of neural quantum states to obtain the dissipative dynamics of light-matter-coupled systems beyond what is accessible with exact and tensor-network calculations. We specifically apply this method to simulate the many-body emission dynamics of approximately 40 atoms, arranged in dense arrays in one and two dimensions. These systems have been chosen because they can support prominent subradiant dynamics at late times and could be realized with cold atomic quantum simulators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Neural quantum states extended to dissipative light-matter dynamics for ~40 atom arrays, but validation details are missing.

read the letter

The main takeaway is that this is the first reported use of neural quantum states to simulate dissipative dynamics in many-body light-matter systems, reaching system sizes of about 40 atoms in one and two dimensions. They apply it to the emission dynamics of dense atomic arrays, which support subradiant behavior at late times due to the long-range interactions and structured dissipation. This extends prior NQS work on closed systems to open quantum dynamics. The paper does well by choosing physically relevant setups that could be realized in cold-atom experiments and by avoiding semi-classical approximations in favor of a full quantum treatment. The soft spots lie in the lack of supporting evidence. There are no reported comparisons to exact solutions for smaller arrays, no error estimates, and no information on how the neural ansatz is constructed or trained for the vectorized density matrix. The potential issue with the ansatz's expressivity for capturing non-local correlations in two-dimensional subradiant states at late times remains unaddressed. This makes it difficult to assess how reliable the results are. This paper is for researchers focused on numerical techniques for open quantum many-body systems, particularly those interested in variational methods. It offers a new tool idea for accessing larger scales in quantum optics simulations. I recommend sending it for peer review. The novelty of the application warrants referee input, even though additional validation would strengthen it.

Referee Report

2 major / 2 minor

Summary. The manuscript reports the first application of neural quantum states (NQS) to simulate the dissipative many-body dynamics of light-matter coupled systems. It focuses on the emission dynamics of arrays of approximately 40 atoms in dense one- and two-dimensional geometries, claiming access to super- and sub-radiant regimes at late times that lie beyond the reach of exact diagonalization and tensor-network methods.

Significance. If the central results hold, this would constitute a meaningful methodological advance by extending variational NQS techniques to open quantum systems with long-range dipole-dipole interactions. It could enable numerical exploration of collective radiative phenomena in mesoscopic arrays relevant to cold-atom quantum simulators and quantum optics experiments, where tensor networks struggle with 2D long-range dissipation.

major comments (2)

[Numerical results for 2D arrays] The headline claim of accessing new regimes for N≈40 atoms in 2D requires that the NQS ansatz faithfully represents the late-time subradiant mixed states. The manuscript supplies no scaling of variational parameters (hidden units, network width) with system size, nor any fidelity or overlap metrics against exact Lindblad solutions on smaller 2D lattices (e.g., 3×3 or 4×4) where both methods are feasible. Without these benchmarks, it remains unclear whether the variational trajectory reaches the correct subradiant manifold or exhibits artificial decay due to insufficient expressivity for the non-local coherences induced by the long-range kernel.
[Methods] § on the variational method and ansatz: The vectorized density-matrix NQS (likely RBM or autoregressive form) is applied to the Lindblad equation, yet no convergence tests with respect to network depth, width, or optimization hyperparameters are reported for the 2D cases. This is load-bearing for the claim that the method remains accurate throughout the dissipative evolution, as insufficient expressivity would directly undermine the reported subradiant dynamics.

minor comments (2)

[Abstract] The abstract would be strengthened by a brief statement of the specific NQS architecture (e.g., RBM, autoregressive) and the observable used to quantify subradiance.
Figure captions for the 1D and 2D emission curves should explicitly state the array geometry, interatomic spacing in units of wavelength, and the number of variational parameters employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the two major comments point by point below. We agree that additional benchmarks and convergence tests will strengthen the manuscript and commit to including them in the revised version.

read point-by-point responses

Referee: [Numerical results for 2D arrays] The headline claim of accessing new regimes for N≈40 atoms in 2D requires that the NQS ansatz faithfully represents the late-time subradiant mixed states. The manuscript supplies no scaling of variational parameters (hidden units, network width) with system size, nor any fidelity or overlap metrics against exact Lindblad solutions on smaller 2D lattices (e.g., 3×3 or 4×4) where both methods are feasible. Without these benchmarks, it remains unclear whether the variational trajectory reaches the correct subradiant manifold or exhibits artificial decay due to insufficient expressivity for the non-local coherences induced by the long-range kernel.

Authors: We agree that direct validation against exact solutions on small 2D lattices is important for establishing the reliability of the NQS ansatz in the presence of long-range interactions. Although the primary focus of the work was demonstrating access to system sizes beyond exact and tensor-network methods, we recognize that the absence of these benchmarks leaves the 2D claims less substantiated than they could be. In the revised manuscript we will add fidelity and overlap comparisons between NQS and exact Lindblad dynamics for 3×3 and 4×4 2D arrays. We will also include the scaling of the number of variational parameters (hidden units and network width) with system size for both 1D and 2D geometries. revision: yes
Referee: [Methods] § on the variational method and ansatz: The vectorized density-matrix NQS (likely RBM or autoregressive form) is applied to the Lindblad equation, yet no convergence tests with respect to network depth, width, or optimization hyperparameters are reported for the 2D cases. This is load-bearing for the claim that the method remains accurate throughout the dissipative evolution, as insufficient expressivity would directly undermine the reported subradiant dynamics.

Authors: We acknowledge that explicit convergence tests for the 2D cases are necessary to support the claim of sustained accuracy during the dissipative evolution. The original manuscript contains a description of the vectorized density-matrix ansatz and optimization procedure but does not report systematic sweeps of network depth, width, or hyperparameters specifically for 2D arrays. In the revision we will add these convergence tests for representative 2D system sizes, demonstrating that the late-time subradiant behavior is robust with respect to increases in expressivity and is not an artifact of under-parameterization. revision: yes

Circularity Check

0 steps flagged

Numerical application of established NQS ansatz to open-system dynamics; no derivation reduces to inputs

full rationale

The paper presents a computational demonstration applying neural quantum states (an existing variational method for density matrices) to Lindblad evolution in 1D/2D atomic arrays of size ~40. The central claim is feasibility for subradiant regimes beyond tensor-network reach, supported by reported simulations rather than any closed derivation. No equations define a quantity in terms of itself, no fitted parameters are relabeled as predictions, and no load-bearing premise rests on self-citation chains. The method's expressivity is an empirical question addressed by the numerics, not a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard open-quantum-system theory and the variational representability of neural quantum states; no new physical entities or ad-hoc parameters are introduced.

axioms (2)

domain assumption The dynamics are governed by a Lindblad master equation incorporating coherent dipole-dipole interactions and collective decay.
Standard framework for Markovian light-matter systems invoked throughout the abstract.
domain assumption A neural-network parametrization can variationally approximate the time-evolved many-body state for the chosen system sizes.
Core methodological assumption enabling the reported simulations.

pith-pipeline@v0.9.0 · 5456 in / 1217 out tokens · 49685 ms · 2026-05-08T17:53:46.579589+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation (J = ½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Late time dynamics correspond to a power law decay, fitted on largest system sizes ... exponent ... very similar and close to 1/3.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

49 extracted references · 8 canonical work pages · 1 internal anchor

[1]

R. H. Dicke, Coherence in Spontaneous Radiation Pro- cesses, Phys. Rev.93, 99 (1954)

1954
[2]

Gonzalez-Tudela, V

A. Gonzalez-Tudela, V. Paulisch, H. J. Kimble, and J. I. Cirac, Efficient Multiphoton Generation in Waveguide Quantum Electrodynamics, Phys. Rev. Lett.118, 213601 (2017)

2017
[3]

Guerin, M

W. Guerin, M. O. Araujo, and R. Kaiser, Subradiance in a Large Cloud of Cold Atoms, Phys. Rev. Lett.116, 083601 (2016)

2016
[5]

Holzinger and S

R. Holzinger and S. F. Yelin, Beyond Dicke superradi- ance: Universal scaling of the peak emission rate (2025), arXiv:2506.12649 [quant-ph] version: 1

work page arXiv 2025
[6]

Pi˜ neiro Orioli, J

A. Pi˜ neiro Orioli, J. Thompson, and A. Rey, Emergent Dark States from Superradiant Dynamics in Multilevel Atoms in a Cavity, Phys. Rev. X12, 011054 (2022)

2022
[7]

Pi˜ neiro Orioli and A

A. Pi˜ neiro Orioli and A. M. Rey, Subradiance of multi- level fermionic atoms in arrays with fillingn≥2, Phys. Rev. A101, 043816 (2020)

2020
[8]

S. J. Masson and A. Asenjo-Garcia, Universality of Dicke superradiance in arrays of quantum emitters, Nature Commun.13, 2285 (2022)

2022
[9]

J. T. Lee, S. Cardenas-Lopez, S. J. Masson, R. Trivedi, and A. Asenjo-Garcia, Exact Many-body Quantum Dy- 6 namics in One-Dimensional Baths via ”Superspins” (2025), arXiv:2505.00588 [quant-ph]

work page arXiv 2025
[10]

Qu and A

C. Qu and A. M. Rey, Spin squeezing and many-body dipolar dynamics in optical lattice clocks, Phys. Rev. A 100, 041602 (2019)

2019
[11]

Olmos and I

B. Olmos and I. Lesanovsky, Hybrid sub- and superradi- ant states in emitter arrays with quantized motion, Phys. Rev. Lett.134, 243602 (2025)

2025
[12]

C. D. Mink and M. Fleischhauer, Collective radiative in- teractions in the discrete truncated Wigner approxima- tion, SciPost Phys.15, 233 (2023)

2023
[15]

Hosseinabadi, O

H. Hosseinabadi, O. Chelpanova, and J. Marino, User- friendly truncated wigner approximation for dissipative spin dynamics, PRX Quantum6, 030344 (2025)

2025
[16]

Carleo and M

G. Carleo and M. Troyer, Solving the quantum many- body problem with artificial neural networks, Science 355, 602 (2017)

2017
[17]

Schmitt and M

M. Schmitt and M. Heyl, Quantum many-body dynamics in two dimensions with artificial neural networks, Phys. Rev. Lett.125, 100503 (2020)

2020
[18]

Schmitt, M

M. Schmitt, M. M. Rams, J. Dziarmaga, M. Heyl, and W. H. Zurek, Quantum phase transition dynamics in the two-dimensional transverse-field Ising model, Science Ad- vances8, eabl6850 (2022)

2022
[19]

J. Nys, G. Pescia, A. Sinibaldi, and G. Carleo, Ab-initio variational wave functions for the time-dependent many- electron Schr¨ odinger equation, Nature Communications 15, 9404 (2024)

2024
[20]

Chen and M

A. Chen and M. Heyl, Empowering deep neural quantum states through efficient optimization, Nature Physics , 1 (2024)

2024
[21]

Duri´ c, J

T. Duri´ c, J. H. Chung, B. Yang, and P. Sengupta, Spin- 1/2 kagome heisenberg antiferromagnet: Machine learn- ing discovery of the spinon pair-density-wave ground state, Phys. Rev. X15, 011047 (2025)

2025
[22]

C. Roth, A. Chen, A. Sengupta, and A. Georges, Superconductivity in the two-dimensional hubbard model revealed by neural quantum states (2025), arXiv:2511.07566 [cond-mat.supr-con]

work page arXiv 2025
[23]

Y. Gu, W. Li, H. Lin, B. Zhan, R. Li, Y. Huang, D. He, Y. Wu, T. Xiang, M. Qin, L. Wang, and D. Lv, Solving the hubbard model with neural quantum states (2025), arXiv:2507.02644 [cond-mat.str-el]

work page arXiv 2025
[24]

M. J. Hartmann and G. Carleo, Neural-Network Ap- proach to Dissipative Quantum Many-Body Dynamics, Physical Review Letters122, 250502 (2019)

2019
[25]

Vicentini, A

F. Vicentini, A. Biella, N. Regnault, and C. Ciuti, Vari- ational neural-network ansatz for steady states in open quantum systems, Phys. Rev. Lett.122, 250503 (2019)

2019
[26]

M. Reh, M. Schmitt, and M. G¨ arttner, Time-Dependent Variational Principle for Open Quantum Systems with Artificial Neural Networks, Physical Review Letters127, 230501 (2021)

2021
[27]

D. Luo, Z. Chen, J. Carrasquilla, and B. K. Clark, Au- toregressive neural network for simulating open quantum systems via a probabilistic formulation, Phys. Rev. Lett. 128, 090501 (2022)

2022
[28]

T. Vovk, A. V. de Walle, H. Pichler, and A. Bohrdt, Neural quantum states for emitter dynamics in waveg- uide qed, arXiv:2508.08964 (2025)

work page arXiv 2025
[29]

selective radiance

A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. J. Kimble, and D. E. Chang, Exponential improve- ment in photon storage fidelities using subradiance and “selective radiance” in atomic arrays, Phys. Rev. X7, 031024 (2017)

2017
[30]

Gross and S

M. Gross and S. Haroche, Superradiance: An essay on the theory of collective spontaneous emission, Physics Reports93, 301 (1982)

1982
[31]

G. S. Agarwal, Master-equation approach to spontaneous emission, Phys. Rev. A2, 2038 (1970)

2038
[32]

S. J. Masson and A. Asenjo-Garcia, Universality of Dicke superradiance in arrays of quantum emitters, Nat. Comm.13, 2285 (2022)

2022
[33]

Henriet, J

L. Henriet, J. S. Douglas, D. E. Chang, and A. Albrecht, Critical open-system dynamics in a one-dimensional optical-lattice clock, Phys. Rev. A99, 023802 (2019)

2019
[34]

Carrasquilla, G

J. Carrasquilla, G. Torlai, R. G. Melko, and L. Aolita, Reconstructing quantum states with generative models, Nature Machine Intelligence1, 155 (2019)

2019
[35]

Cybenko, Approximation by superpositions of a sig- moidal function, Math

G. Cybenko, Approximation by superpositions of a sig- moidal function, Math. Control. Signals, Syst.2, 303 (1989)

1989
[36]

Hornik, Approximation capabilities of multilayer feed- forward networks, Neural Netw4, 251 (1991)

K. Hornik, Approximation capabilities of multilayer feed- forward networks, Neural Netw4, 251 (1991)

1991
[37]

Hofmann, G

D. Hofmann, G. Fabiani, J. H. Mentink, G. Carleo, and M. A. Sentef, Role of stochastic noise and generalization error in the time propagation of neural-network quantum states, SciPost Physics12, 165 (2022)

2022
[38]

Schmitt and M

M. Schmitt and M. Reh, jVMC: Versatile and performant variational Monte Carlo leveraging automated differenti- ation and GPU acceleration, SciPost Phys. Codebases , 2 (2022)

2022
[39]

Medvidovi´ c and D

M. Medvidovi´ c and D. Sels, Variational Quantum Dy- namics of Two-Dimensional Rotor Models, PRX Quan- tum4, 040302 (2023)

2023
[40]

WaveNet: A Generative Model for Raw Audio

A. van den Oord, S. Dieleman, H. Zen, K. Simonyan, O. Vinyals, A. Graves, N. Kalchbrenner, A. Senior, and K. Kavukcuoglu, Wavenet: A generative model for raw audio (2016), arXiv:1609.03499 [cs.SD]

work page internal anchor Pith review arXiv 2016
[41]

A. J. Daley, Quantum trajectories and open many-body quantum systems, Advances in Physics63, 77 (2014)

2014
[42]

Kubo, Generalized cumulant expansion method, Jour- nal of the Physical Society of Japan17, 1100 (1962)

R. Kubo, Generalized cumulant expansion method, Jour- nal of the Physical Society of Japan17, 1100 (1962)

1962
[43]

Kr¨ amer and H

S. Kr¨ amer and H. Ritsch, Generalized mean-field ap- proach to simulate the dynamics of large open spin ensembles with long range interactions, The European Physical Journal D69, 282 (2015)

2015
[44]

Robicheaux and D

F. Robicheaux and D. A. Suresh, Beyond lowest order mean-field theory for light interacting with atom arrays, Phys. Rev. A104, 023702 (2021)

2021
[45]

Plankensteiner, C

D. Plankensteiner, C. Hotter, and H. Ritsch, Quantum- Cumulants.jl: A Julia framework for generalized mean- field equations in open quantum systems, Quantum6, 617 (2022)

2022
[46]

Rubies-Bigorda, S

O. Rubies-Bigorda, S. Ostermann, and S. F. Yelin, Char- acterizing superradiant dynamics in atomic arrays via a cumulant expansion approach, Phys. Rev. Res.5, 013091 (2023)

2023
[47]

Kumlin, A

J. Kumlin, A. Srivastava, and T. Pohl, Superradiance of strongly interacting dipolar excitons in moir´ e quantum materials, Phys. Rev. Lett.134, 126901 (2025). 7

2025
[48]

Wang, N.-Y

C.-H. Wang, N.-Y. Tsai, Y.-C. Wang, and H. H. Jen, Light-scattering properties beyond weak-field excitation in atomic ensembles, Phys. Rev. A110, 013708 (2024)

2024
[49]

D. G. Su´ arez-Forero, R. Ni, S. Sarkar, M. Jalali Mehrabad, E. Mechtel, V. Simonyan, A. Grankin, K. Watanabe, T. Taniguchi, S. Park, et al., Chiral flat-band optical cavity with atomically thin mirrors, Science Advances10, eadr5904 (2024)

2024
[50]

Huang, Y.-X

T.-S. Huang, Y.-X. Wang, Y.-Q. Wang, D. Chang, M. Hafezi, and A. Grankin, Collective optical properties of moir´ e excitons, Phys. Rev. Lett.134, 176901 (2025)

2025
[51]

Lagoin, C

C. Lagoin, C. Morin, K. Baldwin, L. Pfeiffer, and F. Du- bin, Evidence for a lattice supersolid of subradiant dipo- lar excitons, arXiv:2410.17162 (2024)

work page arXiv 2024
[52]

iJ ni − Γni 2 ⟨σ+ n σ− i ⟩+ −iJ in − Γin 2 ⟨σ+ i σ− n ⟩ # ,(S4) d dt ⟨σ+ i σ− j ⟩=−Γ 0 ⟨σ+ i σ− j ⟩+ Γji 2 4⟨σ z i σz j ⟩ − ⟨σ z i ⟩ − ⟨σ z j ⟩ + +iJji(σz j −σ z i )+ + X n̸=i,j

C. Morin, C. Lagoin, T. Gupta, N. Reinic, K. Baldwin, L. Pfeiffer, G. Pupillo, and F. Dubin, Supersolid crystals of dipolar excitons in a lattice, arXiv:2512.16509 (2025). END MATTER Different Architectures: Comparison of ResNet and Dilated CNN In order to capture the correlations generated by the super- and sub-radiant dynamics, we consider two com- plem...

work page arXiv 2025