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arxiv: 2603.25829 · v2 · submitted 2026-03-26 · ❄️ cond-mat.stat-mech · cond-mat.quant-gas

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Quantum Thermalization beyond Non-Integrability and Quantum Scars in a Multispecies Bose-Josephson Junction

Francesco Di Menna , Sergio Ciuchi , Simone Paganelli
Authors on Pith no claims yet

Pith reviewed 2026-05-15 00:04 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.quant-gas
keywords quantum thermalizationBose-Josephson junctionquantum scarsEigenstate Thermalization Hypothesisintegrable systemsquantum chaosultracold atomsergodicity breaking
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The pith

Quantum thermalization occurs in both chaotic and integrable regimes of a three-species Bose-Josephson junction but fails in the separable limit.

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

The paper studies an isolated three-species Bose-Josephson junction with mutual interactions to test when quantum thermalization occurs. It finds that states thermalize according to the Eigenstate Thermalization Hypothesis in both chaotic and integrable regimes, contradicting the usual assumption that non-integrability is required. Thermalization breaks down only when the system enters a separable limit where components decouple completely. The work also locates athermal states inside the chaotic regime that match the pattern of quantum scars.

Core claim

In this multispecies Bose-Josephson junction, quantum thermalization occurs in the chaotic and integrable regimes while it breaks down in the separable limit, showing that non-integrability is not a necessary condition for thermalization. Athermal states identified in the chaotic regime are classifiable as quantum scars and remain consistent with a weak form of the Eigenstate Thermalization Hypothesis.

What carries the argument

The three-species Bose-Josephson junction with mutual interactions, classified into chaotic, integrable, and separable regimes and examined via the Eigenstate Thermalization Hypothesis.

If this is right

  • Thermalization can appear in integrable quantum systems that lack chaos.
  • The separable limit produces ergodicity breaking that prevents thermalization.
  • Quantum scars act as stable athermal states inside otherwise chaotic regimes.
  • Collective semiclassical behavior in the junction produces identifiable scars.

Where Pith is reading between the lines

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

  • Similar junctions realized with ultracold atoms could be used to test thermalization without external baths.
  • The result suggests thermalization criteria may apply more broadly to integrable models than previously thought.
  • Extensions to other multispecies systems could reveal whether scars and thermalization coexist in the same parameter space.

Load-bearing premise

The classification into chaotic, integrable, and separable regimes correctly describes the isolated internal dynamics without hidden external effects.

What would settle it

Direct observation of thermalization in the separable limit or its clear absence in the integrable regime under the same isolation conditions would falsify the central claim.

Figures

Figures reproduced from arXiv: 2603.25829 by Francesco Di Menna, Sergio Ciuchi, Simone Paganelli.

Figure 1
Figure 1. Figure 1: Unfolded level spacing distribution with parameters [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Averaged level spacing ratio in the (U, V)-parameter space with S α = 5. The black triangle and square in the colorbar indicate the theoretical values of ⟨r⟩ for the Poisson and Wigner￾Dyson distributions, respectively. 4.2 Quantum Scars Semiclassical model In the study of quantum thermalization, collective oscillations are studied so much because their relevance in the breaking ergodicity phenomena relate… view at source ↗
Figure 3
Figure 3. Figure 3: Region of stability (sky blue) and the unstability (yellow) for (a) ’ [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Survival probability for both the scar states [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the Husimi distribution at time t = 45.50 for the π00-mode and its corresponding random coherent state and at t = 49.30 for the case of ππ0-mode. These two time are highlited by red points in the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Rescaled EE computed for every eigenstate in function of the corresponding rescaled [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Average level spacing ratio as a function of the magnitude of each spin, and so the [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

This work investigates the relationship between quantum chaos and thermalization in a three-species Bose-Josephson Junction (BJJ) with mutual interactions, without coupling to any external environment. The analysis is grounded in the Eigenstate Thermalization Hypothesis (ETH), the modern framework for quantum thermalization, in which non-integrability and chaos are historically assumed as prerequisites. After a thorough characterization of quantum chaos in this system, we examine the occurrence of thermal behavior expected when ETH holds. We identify three distinct regimes: chaotic, integrable, and separable. Remarkably, quantum thermalization occurs in both the chaotic and integrable regimes, while it breaks down in the separable limit - supporting that non-integrability is not a necessary condition for thermalization. Furthermore, since the system exhibits collective phenomena in the semiclassical limit, we identify athermal states in the chaotic regime classifiable as quantum scars, which show no signs of thermalization, consistently with a weak form of ETH. These findings contribute to the understanding of ergodicity breaking, emerging statistical behavior, and non-equilibrium dynamics in ultracold many-body quantum systems.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript examines quantum thermalization and chaos in a three-species Bose-Josephson junction with mutual interactions and no external coupling. Grounded in the Eigenstate Thermalization Hypothesis (ETH), it identifies chaotic, integrable, and separable regimes, claiming that thermalization occurs in both chaotic and integrable cases (but fails in the separable limit), thereby arguing that non-integrability is not required for thermalization. It further identifies athermal states in the chaotic regime as quantum scars consistent with weak ETH.

Significance. If the central claims hold, the work is significant for challenging the historical link between non-integrability/chaos and ETH-style thermalization, offering a concrete example where thermalization appears in an integrable regime. This could broaden understanding of ergodicity breaking and statistical behavior in ultracold many-body systems with collective semiclassical dynamics, while the quantum-scar identification adds to the literature on weak ETH violations.

major comments (3)
  1. [Abstract and integrable-regime analysis] Abstract and integrable-regime section: The central claim that quantum thermalization occurs in the integrable regime (and thus that non-integrability is unnecessary) is load-bearing. Integrable systems generically relax to a generalized Gibbs ensemble (GGE) fixed by all conserved charges rather than the standard canonical/microcanonical ensemble assumed under ETH. The manuscript must explicitly compare long-time averages and eigenstate expectations of local observables against both the canonical thermal predictions and the full GGE constructed from the integrals of motion; without this test, apparent thermalization may simply reflect GGE physics.
  2. [Regime characterization] Regime-identification section: The separation into chaotic, integrable, and separable regimes underpins all conclusions. The diagnostics (level statistics, Lyapunov exponents, or similar) must be shown to correctly isolate a truly integrable regime with the expected conserved quantities, and the absence of external coupling must be verified to exclude hidden environmental effects that could induce effective non-integrability.
  3. [Athermal states and quantum scars] Quantum-scars subsection: The classification of athermal states in the chaotic regime as quantum scars requires quantitative support beyond qualitative statements. The manuscript should include semiclassical diagnostics such as overlaps with unstable periodic orbits or scarring measures in the collective phase space to distinguish these states from other ergodicity-breaking mechanisms.
minor comments (2)
  1. [Abstract] The abstract states the central result but omits any mention of system sizes, interaction parameters, or numerical convergence; adding one sentence on these would improve accessibility.
  2. [Figures] Figure captions should be self-contained, explicitly labeling the three regimes and the observables used to diagnose thermalization versus scarring.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments raise important points about the strength of our central claims, and we address each one below. We will revise the manuscript to incorporate the requested clarifications and additional analyses.

read point-by-point responses

  1. Referee: [Abstract and integrable-regime analysis] Abstract and integrable-regime section: The central claim that quantum thermalization occurs in the integrable regime (and thus that non-integrability is unnecessary) is load-bearing. Integrable systems generically relax to a generalized Gibbs ensemble (GGE) fixed by all conserved charges rather than the standard canonical/microcanonical ensemble assumed under ETH. The manuscript must explicitly compare long-time averages and eigenstate expectations of local observables against both the canonical thermal predictions and the full GGE constructed from the integrals of motion; without this test, apparent thermalization may simply reflect GGE physics.

    Authors: We agree that an explicit comparison to the GGE is necessary to substantiate the claim that thermalization occurs beyond GGE physics. In our model the integrable regime possesses a small number of conserved quantities (species populations and total energy) whose associated GGE reduces to the microcanonical ensemble for the local observables examined. To make this rigorous we will add direct comparisons of long-time averages and eigenstate expectations against both the canonical ensemble and the full GGE in the revised manuscript. revision: yes

  2. Referee: [Regime characterization] Regime-identification section: The separation into chaotic, integrable, and separable regimes underpins all conclusions. The diagnostics (level statistics, Lyapunov exponents, or similar) must be shown to correctly isolate a truly integrable regime with the expected conserved quantities, and the absence of external coupling must be verified to exclude hidden environmental effects that could induce effective non-integrability.

    Authors: We will expand the regime-identification section with additional figures and discussion that explicitly connect the level statistics and Lyapunov exponents to the conserved quantities present in the integrable regime. The Hamiltonian is defined without external coupling; we will add an explicit verification that no hidden environmental terms are present. revision: yes

  3. Referee: [Athermal states and quantum scars] Quantum-scars subsection: The classification of athermal states in the chaotic regime as quantum scars requires quantitative support beyond qualitative statements. The manuscript should include semiclassical diagnostics such as overlaps with unstable periodic orbits or scarring measures in the collective phase space to distinguish these states from other ergodicity-breaking mechanisms.

    Authors: We will strengthen the quantum-scars subsection by adding quantitative semiclassical diagnostics, including overlaps with unstable periodic orbits and scarring measures evaluated in the collective phase space, to provide rigorous support for the scar identification. revision: yes

Circularity Check

0 steps flagged

No circularity: regimes and thermalization diagnostics are independently defined

full rationale

The paper first characterizes the three regimes (chaotic, integrable, separable) via independent diagnostics such as level-spacing statistics, eigenstate delocalization measures, and the structure of the Hamiltonian in the absence of external coupling. Thermalization is subsequently tested by direct comparison of long-time averages and diagonal ensemble expectations against the microcanonical/canonical predictions for local observables. No equation or claim reduces a 'prediction' to a fitted input by construction, nor does the central result rely on a self-citation chain whose validity is presupposed. The abstract and manuscript structure treat the ETH framework as an external reference rather than deriving it internally, and the observation of thermalization in the integrable regime is presented as an empirical outcome of the numerics rather than a definitional tautology. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Eigenstate Thermalization Hypothesis as the definition of thermalization and on the ability to cleanly partition the parameter space into chaotic, integrable, and separable regimes without external baths.

axioms (1)
  • domain assumption Eigenstate Thermalization Hypothesis provides the correct criterion for quantum thermalization in isolated systems
    Invoked in the abstract as the modern framework for quantum thermalization

pith-pipeline@v0.9.0 · 5502 in / 1156 out tokens · 32312 ms · 2026-05-15T00:04:06.560739+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · cited by 1 Pith paper

  1. [1]

    Functions of Bounded Variation and Free Discontinuity Problems

    L. Pitaevskii and S. Stringari,Bose-Einstein Condensation and Superfluidity, Oxford Uni- versity Press, ISBN 9780198758884, doi:10.1093/acprof:oso/9780198758884.001.0001 (2016)

    work page doi:10.1093/acprof:oso/9780198758884.001.0001 2016

  2. [2]

    Melé-Messeguer, S

    M. Melé-Messeguer, S. Paganelli, B. Juliá-Díaz, A. Sanpera and A. Polls,Spin-driven spatial symmetry breaking of spinor condensates in a double well, Phys. Rev. A86, 053626 (2012), doi:10.1103/PhysRevA.86.053626

    work page doi:10.1103/physreva.86.053626 2012

  3. [3]

    Karle, N

    V . Karle, N. Defenu and T. Enss,Coupled superfluidity of binary bose mixtures in two dimensions, Phys. Rev. A99, 063627 (2019), doi:10.1103/PhysRevA.99.063627

    work page doi:10.1103/physreva.99.063627 2019

  4. [4]

    Albiez, R

    M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani and M. K. Oberthaler,Direct observation of tunneling and nonlinear self-trapping in a single bosonic josephson junction, Phys. Rev. Lett.95, 010402 (2005), doi:10.1103/PhysRevLett.95.010402

    work page doi:10.1103/physrevlett.95.010402 2005

  5. [5]

    S. Levy, E. Lahoud, I. Shomroni and J. Steinhauer,The ac and dc josephson effects in a bose–einstein condensate, Nature449(7162), 579 (2007), doi:10.1038/nature06186

    work page doi:10.1038/nature06186 2007

  6. [6]

    F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi and M. Inguscio,Josephson junction arrays with bose-einstein condensates, Science293(5531), 843 (2001), doi:10.1126/science.1062612,https://www.science.org/doi/pdf/10. 1126/science.1062612

    work page doi:10.1126/science.1062612 2001

  7. [7]

    Kinoshita, T

    T. Kinoshita, T. Wenger and D. S. Weiss,A quantum newton’s cradle, Nature440(7086), 900 (2006), doi:10.1038/nature04693

    work page doi:10.1038/nature04693 2006

  8. [8]

    Hofferberth, I

    S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm and J. Schmiedmayer,Non- equilibrium coherence dynamics in one-dimensional bose gases, Nature449(7160), 324 (2007), doi:10.1038/nature06149. 16 SciPost Physics Submission

    work page doi:10.1038/nature06149 2007

  9. [9]

    Cheneau, P

    M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T. Fukuhara, C. Gross, I. Bloch, C. Kollath and S. Kuhr,Light-cone-like spreading of correlations in a quantum many-body system, Nature481(7382), 484 (2012), doi:10.1038/nature10748

    work page doi:10.1038/nature10748 2012

  10. [10]

    A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss and M. Greiner, Quantum thermalization through entanglement in an isolated many-body system, Science 353(6301), 794 (2016), doi:10.1126/science.aaf6725,https://www.science.org/doi/ pdf/10.1126/science.aaf6725

    work page doi:10.1126/science.aaf6725 2016

  11. [11]

    Sinha and S

    S. Sinha and S. Sinha,Chaos and quantum scars in bose-josephson junction coupled to a bosonic mode, Phys. Rev. Lett.125, 134101 (2020), doi:10.1103/PhysRevLett.125.134101

    work page doi:10.1103/physrevlett.125.134101 2020

  12. [12]

    Russomanno, M

    A. Russomanno, M. Fava and R. Fazio,Weak ergodicity breaking in josephson-junction arrays, Phys. Rev. B106, 035123 (2022), doi:10.1103/PhysRevB.106.035123

    work page doi:10.1103/physrevb.106.035123 2022

  13. [13]

    Sinha and S

    S. Sinha and S. Sinha,Dissipative bose-josephson junction coupled to bosonic baths, Phys. Rev. E100, 032115 (2019), doi:10.1103/PhysRevE.100.032115

    work page doi:10.1103/physreve.100.032115 2019

  14. [14]

    Mondal, S

    D. Mondal, S. Sinha, S. Ray, J. Kroha and S. Sinha,Classical route to ergodicity and scarring phenomena in a two-component bose-josephson junction, Phys. Rev. A106, 043321 (2022), doi:10.1103/PhysRevA.106.043321

    work page doi:10.1103/physreva.106.043321 2022

  15. [15]

    Taglieber, A.-C

    M. Taglieber, A.-C. V oigt, T. Aoki, T. W. Hänsch and K. Dieckmann,Quantum degenerate two-species fermi-fermi mixture coexisting with a bose-einstein condensate, Phys. Rev. Lett. 100, 010401 (2008), doi:10.1103/PhysRevLett.100.010401

    work page doi:10.1103/physrevlett.100.010401 2008

  16. [16]

    Baldovin, G

    M. Baldovin, G. Gradenigo, A. Vulpiani and N. Zanghì,On the foundations of statistical me- chanics, Physics Reports1132, 1 (2025), doi:https://doi.org/10.1016/j.physrep.2025.05.003, On the foundations of statistical mechanics

    work page doi:10.1016/j.physrep.2025.05.003 2025

  17. [17]

    Srednicki,Chaos and quantum thermalization, Phys

    M. Srednicki,Chaos and quantum thermalization, Phys. Rev. E50, 888 (1994), doi:10.1103/PhysRevE.50.888

    work page doi:10.1103/physreve.50.888 1994

  18. [18]

    J. M. Deutsch,Eigenstate thermalization hypothesis, Reports on Progress in Physics81(8), 082001 (2018), doi:10.1088/1361-6633/aac9f1

    work page doi:10.1088/1361-6633/aac9f1 2018

  19. [19]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore,Colloquium: Nonequilib- rium dynamics of closed interacting quantum systems, Rev. Mod. Phys.83, 863 (2011), doi:10.1103/RevModPhys.83.863

    work page doi:10.1103/revmodphys.83.863 2011

  20. [20]

    J. R. Garrison and T. Grover,Does a single eigenstate encode the full hamiltonian?, Phys. Rev. X8, 021026 (2018), doi:10.1103/PhysRevX.8.021026

    work page doi:10.1103/physrevx.8.021026 2018

  21. [21]

    Sinha, S

    S. Sinha, S. Ray and S. Sinha,Classical route to ergodicity and scarring in collec- tive quantum systems, Journal of Physics: Condensed Matter36(16), 163001 (2024), doi:10.1088/1361-648X/ad1bf5

    work page doi:10.1088/1361-648x/ad1bf5 2024

  22. [22]

    D. A. Abanin, E. Altman, I. Bloch and M. Serbyn,Colloquium: Many-body lo- calization, thermalization, and entanglement, Rev. Mod. Phys.91, 021001 (2019), doi:10.1103/RevModPhys.91.021001

    work page doi:10.1103/revmodphys.91.021001 2019

  23. [23]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. V osk, E. Altman, U. Schneider and I. Bloch,Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science349(6250), 842 (2015), doi:10.1126/science.aaa7432, https://www.science.org/doi/pdf/10.1126/science.aaa7432. 17 SciPost Physics Submission

    work page doi:10.1126/science.aaa7432 2015

  24. [24]

    Desaules, F

    J.-Y . Desaules, F. Pietracaprina, Z. Papi´c, J. Goold and S. Pappalardi,Extensive multipartite entanglement from su(2) quantum many-body scars, Phys. Rev. Lett.129, 020601 (2022), doi:10.1103/PhysRevLett.129.020601

    work page doi:10.1103/physrevlett.129.020601 2022

  25. [25]

    E. J. Heller,Bound-state eigenfunctions of classically chaotic hamiltonian systems: Scars of periodic orbits, Phys. Rev. Lett.53, 1515 (1984), doi:10.1103/PhysRevLett.53.1515

    work page doi:10.1103/physrevlett.53.1515 1984

  26. [26]

    Serbyn, D

    M. Serbyn, D. A. Abanin and Z. Papi ´c,Quantum many-body scars and weak breaking of ergodicity, Nature Physics17(6), 675 (2021), doi:10.1038/s41567-021-01230-2

    work page doi:10.1038/s41567-021-01230-2 2021

  27. [27]

    Cattaneo, M

    M. Cattaneo, M. Baldovin, D. Lucente, P. Muratore-Ginanneschi and A. Vulpiani,Ther- malization is typical in large classical and quantum harmonic systems, Phys. Rev. Res.7, L032002 (2025), doi:10.1103/h9fj-ylgm

    work page doi:10.1103/h9fj-ylgm 2025

  28. [28]

    Alba,Eigenstate thermalization hypothesis and integrability in quantum spin chains, Phys

    V . Alba,Eigenstate thermalization hypothesis and integrability in quantum spin chains, Phys. Rev. B91, 155123 (2015), doi:10.1103/PhysRevB.91.155123

    work page doi:10.1103/physrevb.91.155123 2015

  29. [29]

    T. N. Ikeda, Y . Watanabe and M. Ueda,Finite-size scaling analysis of the eigenstate ther- malization hypothesis in a one-dimensional interacting bose gas, Phys. Rev. E87, 012125 (2013), doi:10.1103/PhysRevE.87.012125

    work page doi:10.1103/physreve.87.012125 2013

  30. [30]

    Trotzky, Y .-A

    S. Trotzky, Y .-A. Chen, A. Flesch, I. P. McCulloch, U. Schollwöck, J. Eisert and I. Bloch,Probing the relaxation towards equilibrium in an isolated strongly correlated one- dimensional bose gas, Nature physics8(4), 325 (2012), doi:10.1038/nphys2232

    work page doi:10.1038/nphys2232 2012

  31. [31]

    Lai and K

    H.-H. Lai and K. Yang,Entanglement entropy scaling laws and eigenstate typicality in free fermion systems, Phys. Rev. B91, 081110 (2015), doi:10.1103/PhysRevB.91.081110

    work page doi:10.1103/physrevb.91.081110 2015

  32. [32]

    Storms and R

    M. Storms and R. R. P. Singh,Entanglement in ground and excited states of gapped free- fermion systems and their relationship with fermi surface and thermodynamic equilibrium properties, Phys. Rev. E89, 012125 (2014), doi:10.1103/PhysRevE.89.012125

    work page doi:10.1103/physreve.89.012125 2014

  33. [33]

    Łyd˙ zba, R.´Swi˛ etek, M

    P. Łyd˙ zba, R.´Swi˛ etek, M. Mierzejewski, M. Rigol and L. Vidmar,Normal weak eigenstate thermalization, Phys. Rev. B110, 104202 (2024), doi:10.1103/PhysRevB.110.104202

    work page doi:10.1103/physrevb.110.104202 2024

  34. [34]

    J. M. Magán,Random free fermions: An analytical example of eigenstate thermalization, Phys. Rev. Lett.116, 030401 (2016), doi:10.1103/PhysRevLett.116.030401

    work page doi:10.1103/physrevlett.116.030401 2016

  35. [35]

    J., & Wistrich, A

    R. Nandkishore and D. A. Huse,Many-body localization and thermalization in quantum sta- tistical mechanics, Annu. Rev. Condens. Matter Phys.6(1), 15 (2015), doi:10.1146/annurev- conmatphys-031214-014726

    work page doi:10.1146/annurev- 2015

  36. [36]

    J. M. Magán and S. Paganelli,Codification volume of an operator algebra and its irreversible growth through thermal processes, Phys. Rev. A90, 032103 (2014), doi:10.1103/PhysRevA.90.032103

    work page doi:10.1103/physreva.90.032103 2014

  37. [37]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greineret al.,Probing many-body dynamics on a 51-atom quantum simulator, Nature551(7682), 579 (2017), doi:10.1038/nature24622

    work page doi:10.1038/nature24622 2017

  38. [38]

    Aleksandr and A

    I. Aleksandr and A. Khinchin,Mathematical foundations of statistical mechanics, Courier Corporation (1949)

    work page 1949

  39. [39]

    M. V . Berry,Regular and irregular semiclassical wavefunctions, Journal of Physics A: Mathematical and General10(12), 2083 (1977), doi:10.1088/0305-4470/10/12/016. 18 SciPost Physics Submission

    work page doi:10.1088/0305-4470/10/12/016 2083

  40. [40]

    Biroli, C

    G. Biroli, C. Kollath and A. M. Läuchli,Effect of rare fluctuations on the ther- malization of isolated quantum systems, Phys. Rev. Lett.105, 250401 (2010), doi:10.1103/PhysRevLett.105.250401

    work page doi:10.1103/physrevlett.105.250401 2010

  41. [41]

    A. J. Leggett,Bose-einstein condensation in the alkali gases: Some fundamental concepts, Rev. Mod. Phys.73, 307 (2001), doi:10.1103/RevModPhys.73.307

    work page doi:10.1103/revmodphys.73.307 2001

  42. [42]

    G. J. Milburn, J. Corney, E. M. Wright and D. F. Walls,Quantum dynamics of an atomic bose-einstein condensate in a double-well potential, Phys. Rev. A55, 4318 (1997), doi:10.1103/PhysRevA.55.4318

    work page doi:10.1103/physreva.55.4318 1997

  43. [43]

    Haake,Quantum signatures of chaos, In B

    F. Haake,Quantum signatures of chaos, In B. Kramer, ed.,Quantum Coherence in Mesoscopic Systems, pp. 583–595. Springer US, Boston, MA, ISBN 978-1-4899-3698-1, doi:10.1007/978-1-4899-3698-1_38 (1991)

    work page doi:10.1007/978-1-4899-3698-1_38 1991

  44. [44]

    Bohigas, M

    O. Bohigas, M. J. Giannoni and C. Schmit,Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett.52, 1 (1984), doi:10.1103/PhysRevLett.52.1

    work page doi:10.1103/physrevlett.52.1 1984

  45. [45]

    M. V . Berry and M. Tabor,Level clustering in the regular spectrum, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences356(1686), 375 (1977), doi:10.1098/rspa.1977.0140,https://royalsocietypublishing.org/rspa/ article-pdf/356/1686/375/62525/rspa.1977.0140.pdf

    work page doi:10.1098/rspa.1977.0140 1977

  46. [46]

    Oganesyan and D

    V . Oganesyan and D. A. Huse,Localization of interacting fermions at high temperature, Phys. Rev. B75, 155111 (2007), doi:10.1103/PhysRevB.75.155111

    work page doi:10.1103/physrevb.75.155111 2007

  47. [47]

    Y . Y . Atas, E. Bogomolny, O. Giraud and G. Roux,Distribution of the ratio of consec- utive level spacings in random matrix ensembles, Phys. Rev. Lett.110, 084101 (2013), doi:10.1103/PhysRevLett.110.084101

    work page doi:10.1103/physrevlett.110.084101 2013

  48. [48]

    J. M. Radcliffe,Some properties of coherent spin states, Journal of Physics A: General Physics4(3), 313 (1971), doi:10.1088/0305-4470/4/3/009

    work page doi:10.1088/0305-4470/4/3/009 1971

  49. [49]

    D’Alessio, Y

    L. D’Alessio, Y . Kafri, A. Polkovnikov and M. Rigol,From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Advances in Physics65(3), 239 (2016), doi:10.1080/00018732.2016.1198134

    work page doi:10.1080/00018732.2016.1198134 2016

  50. [50]

    M. V . Berry,The bakerian lecture, 1987. quantum chaology, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences413(1844), 183 (1987), doi:10.1098/rspa.1987.0109,https://royalsocietypublishing.org/rspa/ article-pdf/413/1844/183/66695/rspa.1987.0109.pdf

    work page doi:10.1098/rspa.1987.0109 1987

  51. [51]

    L. F. Santos and M. Rigol,Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization, Phys. Rev. E81, 036206 (2010), doi:10.1103/PhysRevE.81.036206

    work page doi:10.1103/physreve.81.036206 2010

  52. [52]

    E. J. Torres-Herrera, J. A. Méndez-Bermúdez and L. F. Santos,Level repulsion and dy- namics in the finite one-dimensional anderson model, Phys. Rev. E100, 022142 (2019), doi:10.1103/PhysRevE.100.022142

    work page doi:10.1103/physreve.100.022142 2019

  53. [53]

    A. A. Elkamshishy and C. H. Greene,Observation of wigner-dyson level statis- tics in a classically integrable system, Phys. Rev. E103, 062211 (2021), doi:10.1103/PhysRevE.103.062211. 19

    work page doi:10.1103/physreve.103.062211 2021