pith. sign in

arxiv: 2509.20922 · v7 · submitted 2025-09-25 · 🪐 quant-ph · cond-mat.quant-gas· nlin.CD

Classical and quantum chaotic synchronization in coupled dissipative time crystals

Eli\v{s}ka Postavov\'a , Gianluca Passarelli , Procolo Lucignano , Angelo Russomanno This is my paper

Pith reviewed 2026-05-18 14:24 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gasnlin.CD
keywords chaotic synchronizationdissipative time crystalsquantum trajectoriesGaussian Unitary EnsembleLyapunov exponentPearson correlationentanglement entropyz-magnetization crossover
0
0 comments X

The pith

Coupled dissipative time crystals enter a regime of chaotic synchronization in both their classical mean-field limit and finite-size quantum dynamics.

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

The paper studies two coherently coupled dissipative time crystals. In the classical limit of infinite spin length, a regime appears where the subsystems synchronize chaotically, identified by a positive largest Lyapunov exponent together with a Pearson correlation coefficient approaching one. At the edge of this regime the correlation jumps sharply, separating a staggered z-magnetization phase from a uniform one. For finite quantum systems, quantum-trajectory simulations produce histograms of subsystem z-magnetizations whose maxima trace an analogous staggered-to-uniform crossover; the steady-state density matrix obeys Gaussian Unitary Ensemble level statistics, which the authors read as the quantum counterpart of the classical chaos.

Core claim

In the classical mean-field limit the coupled time crystals exhibit chaotic synchronization marked by a positive largest Lyapunov exponent and a Pearson correlation coefficient close to one, with an abrupt variation of the coefficient at the boundary that signals the crossover between staggered and uniform z-magnetization. In the quantum regime, time- and trajectory-resolved histograms of subsystem z-magnetizations display maxima that undergo the same staggered-to-uniform crossover, interpreted as quantum chaotic synchronization; the steady-state density matrix follows Gaussian Unitary Ensemble statistics, while entanglement entropy and the non-commutativity of the infinite-time and infinite

What carries the argument

Chaotic synchronization diagnosed classically by Lyapunov exponents and Pearson correlation, and quantum-mechanically by trajectory histograms of z-magnetizations together with GUE statistics of the steady-state density matrix.

If this is right

  • The location of the classical and quantum crossover points differs because the infinite-time and infinite-spin limits do not commute.
  • Entanglement entropy between the two subsystems quantifies an additional quantum ingredient that shifts the synchronization boundary.
  • Dissipative quantum chaos is diagnosed by the appearance of Gaussian Unitary Ensemble statistics in the steady-state density matrix.
  • Quantum-trajectory histograms of local observables can reveal synchronization crossovers that mirror the classical mean-field behavior.

Where Pith is reading between the lines

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

  • The same trajectory-based diagnostics could be applied to other pairs of coupled dissipative many-body systems to search for quantum chaotic synchronization.
  • Varying the coherent coupling strength or the dissipation rate offers a concrete experimental knob to move between the staggered and uniform synchronization regimes.
  • Because entanglement is explicitly quantified, the setup supplies a natural testbed for asking how entanglement suppresses or enhances classical chaotic features.
  • Larger spin lengths or longer evolution times could be simulated to map how the quantum crossover point approaches the classical one.

Load-bearing premise

The maxima in the quantum trajectory histograms of subsystem z-magnetizations and the GUE statistics of the steady-state density matrix constitute direct evidence of chaotic synchronization analogous to the classical case, even though the infinite-time and infinite-spin limits do not commute and entanglement is present.

What would settle it

A numerical or experimental run in which the largest Lyapunov exponent remains non-positive or the Pearson coefficient stays well below one throughout the purported chaotic regime, or in which the quantum histograms lack the staggered-to-uniform crossover while the density-matrix spectrum deviates from GUE statistics.

Figures

Figures reproduced from arXiv: 2509.20922 by Angelo Russomanno, Eli\v{s}ka Postavov\'a, Gianluca Passarelli, Procolo Lucignano.

Figure 1
Figure 1. Figure 1: Largest Lyapunov exponent ΛL versus Γ and Ω for initial conditions mz j = 1 and m x,y j = 0 for all j (panel a), averaged over random initial conditions as given in Sec. 3.1.2 with a = 0.1 (panel b), and with a = 1 (panel c). The averages are over Nr = 50 initial conditions. See the main text and [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Largest Lyapunov exponent ⟨ΛL⟩ (top row) and Pearson correlation coefficient CP (bottom row) versus Γ along the line Γ + Ω = 1.5 (left column), 2Ω + Γ = 2 (center column), and 2Ω + Γ = 1.5 (right column). These values are calculated as averages over Nr = 100 initial conditions generated on a spherical cap with parameter a = 0.1 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Heat maps of various quantities versus Γ and Ω. (a,b) Pearson correlation coefficient. (c,d) Difference between the magnetizations averaged over time and initial conditions, ⟨δm⟩ = ⟨mz B ⟩ − ⟨mz A ⟩. (e,f) Relative sign s = sign(⟨mA z ⟩ ⟨mB z ⟩) of the two magnetizations. The calculations are performed by averaging over Nr = 100 random initial conditions generated on a spherical cap with parameter a = 0.1 … view at source ↗
Figure 4
Figure 4. Figure 4: Time and initial-condition average values of mz j (t), ⟨mz j ⟩ versus Γ, along the parameter-space line Ω + Γ = 1.5 (a, left column), 2Ω + Γ = 2 (b, center column), and 2Ω + Γ = 1.5 (c, right column). We average over a set of Nr = 100 initial conditions generated either for a = 0.1 (top row) or a = 1 (bottom row). The legend is shared. The background shadowing marks the Lyapunov exponent with the same colo… view at source ↗
Figure 5
Figure 5. Figure 5: (a,c) S z A and S z B versus Ω for different N. (b,d) Time-averaged connected correlator Czz AB. Average over Nr ≥ 48 trajectories. On the left column we consider the curves for the line 2Ω + Γ = 2 and on the right one 2Ω + Γ = 1.5. The vertical line marks the classical staggered-uniform crossover as found in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) Histogram of (S z A) r (t) as it changes along time t and across trajectories r. (b) Histogram of (S z B) r (t) as it changes along time t and across trajectories r. (c) Histogram of the entanglement entropy E r (t) as it changes along time t and across trajectories r. We fix the constraint 2Ω + Γ = 1.5, δt = 10−4 and S = 20. synchronization, we interpret these quantum observations as evidence of quant… view at source ↗
Figure 7
Figure 7. Figure 7: (a,c) Maximum points of the histograms of S z A and S z B along and across trajectories versus Γ. (b,d) Maximum point of the histogram of the entanglement entropy versus Γ. We fix the constraint 2Ω + Γ = 2 in the left column and 2Ω + Γ = 1.5 in the right column. We take also δt = 10−4 . The vertical line marks the classical staggered-uniform crossover as found in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Average level spacing ratio of the NESS density matrix versus Γ for different values of S, compared with the average level spacing ratio for the Gaussian Unitary Ensemble as evaluated in Ref. [53]. The flatness of the curve and the agreement with the GUE value increase for increasing subsystem spin magnitude S. We fix the constraint 2Ω + Γ = 2 and evaluate ˆρNESS as ˆρt with t = 100 in units of 1/κ. In Eq.… view at source ↗
read the original abstract

We investigate the dynamics of two coherently coupled dissipative time crystals. In the classical mean-field limit of infinite spin length, we identify a regime of chaotic synchronization, marked by a positive largest Lyapunov exponent and a Pearson correlation coefficient close to one. At the boundary of this regime, the Pearson coefficient varies abruptly, marking a crossover between staggered and uniform $z$-magnetization. To address finite-size quantum dynamics, we employ a quantum-trajectory approach and study the trajectory-resolved expectations of subsystem $z$-magnetizations. Their histograms over time and trajectory realizations exhibit maxima that undergo a staggered-to-uniform crossover analogous to the classical one. In analogy with the classical case, we interpret this behavior as quantum chaotic synchronization, with dissipative quantum chaos highlighted by the steady-state density matrix exhibiting Gaussian Unitary Ensemble statistics. The classical and quantum crossover points are different due to the noncommutativity of the infinite-time and infinite-spin-magnitude limits and the role played by entanglement in the quantum case, quantified via the two-subsystem entanglement entropy.

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

1 major / 2 minor

Summary. The manuscript studies two coherently coupled dissipative time crystals. In the classical mean-field limit of infinite spin length, it reports a regime of chaotic synchronization identified by a positive largest Lyapunov exponent together with a Pearson correlation coefficient near unity; the Pearson coefficient changes abruptly at the regime boundary, marking a staggered-to-uniform crossover in z-magnetization. For finite-size quantum dynamics the authors employ quantum-trajectory unravelings, observe that time-and-trajectory histograms of subsystem z-magnetizations display an analogous staggered-to-uniform crossover, and interpret the behavior as quantum chaotic synchronization, further supported by the claim that the steady-state density matrix obeys Gaussian Unitary Ensemble level statistics. The paper explicitly notes that the classical and quantum crossover locations differ because the infinite-time and infinite-spin limits do not commute and because finite entanglement entropy is present in the quantum case.

Significance. If the central interpretation is sustained, the work supplies a concrete, numerically accessible example of chaotic synchronization that bridges the classical mean-field limit and open quantum many-body dynamics. The combination of Lyapunov diagnostics, Pearson correlations, quantum-trajectory histograms, and GUE statistics, together with the explicit discussion of non-commuting limits and entanglement entropy, offers a useful test-bed for dissipative quantum chaos and synchronization phenomena.

major comments (1)
  1. [quantum trajectories and steady-state density matrix] Quantum section (trajectory histograms and steady-state analysis): the claim that the observed staggered-to-uniform crossover in the maxima of subsystem-z histograms, together with GUE statistics of the steady-state density matrix, constitutes direct evidence of quantum chaotic synchronization analogous to the classical case (positive Lyapunov exponent plus Pearson correlation ≈1) is not supported by a matching quantitative diagnostic. No quantum Lyapunov exponent, inter-trajectory correlation function, or other chaos indicator that survives the non-commuting infinite-time/infinite-N limits is presented; the analogy therefore rests on post-hoc regime identification whose robustness remains to be demonstrated.
minor comments (2)
  1. [figures] Numerical figures should include error bars or convergence checks with respect to trajectory number and integration time; the absence of these makes it difficult to assess the statistical significance of the histogram maxima.
  2. [steady-state analysis] The precise definition of the GUE ensemble (e.g., unfolding procedure, spectral range, and system-size scaling) should be stated explicitly so that the level-statistics claim can be reproduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [quantum trajectories and steady-state density matrix] Quantum section (trajectory histograms and steady-state analysis): the claim that the observed staggered-to-uniform crossover in the maxima of subsystem-z histograms, together with GUE statistics of the steady-state density matrix, constitutes direct evidence of quantum chaotic synchronization analogous to the classical case (positive Lyapunov exponent plus Pearson correlation ≈1) is not supported by a matching quantitative diagnostic. No quantum Lyapunov exponent, inter-trajectory correlation function, or other chaos indicator that survives the non-commuting infinite-time/infinite-N limits is presented; the analogy therefore rests on post-hoc regime identification whose robustness remains to be demonstrated.

    Authors: We agree that a direct quantum Lyapunov exponent or inter-trajectory correlation function is not computed, as such quantities are not straightforward to define in a manner that survives the non-commuting infinite-time and infinite-N limits while accounting for finite entanglement. Our interpretation instead rests on the quantitative observation that the maxima of the time-and-trajectory histograms of subsystem z-magnetizations exhibit an abrupt staggered-to-uniform crossover, directly analogous to the abrupt change in the classical Pearson coefficient at the regime boundary. The GUE level statistics of the steady-state density matrix provide additional support for dissipative quantum chaos. The manuscript already discusses the role of entanglement entropy and the non-commutativity of limits. We will revise the text to more explicitly frame the histogram-maxima locations as the matching quantitative diagnostic and to include further checks on the robustness of the identified crossover. revision: partial

Circularity Check

0 steps flagged

No significant circularity: claims rest on direct simulation of model dynamics

full rationale

The paper computes classical chaotic synchronization via explicit integration of the mean-field equations, extracting the largest Lyapunov exponent and Pearson correlation directly from the resulting trajectories. Quantum results follow from independent quantum-trajectory Monte Carlo sampling, producing histograms of subsystem magnetizations and eigenvalue statistics of the steady-state density matrix. Neither diagnostic is obtained by fitting a parameter to a target defined from the same run, nor by re-labeling an input quantity. The non-commutativity of limits and the presence of entanglement are explicitly noted rather than assumed away. No self-citation is invoked as a load-bearing uniqueness theorem or ansatz. The central interpretation is therefore an analogy drawn from separate numerical outputs, not a reduction by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions of open quantum systems and time-crystal models rather than new postulates; a few model parameters are tuned to access the synchronization regime.

free parameters (1)
  • coherent coupling strength
    The inter-crystal coupling is a tunable parameter chosen to place the system inside the chaotic synchronization window.
axioms (2)
  • domain assumption Mean-field theory accurately captures the infinite-spin classical limit
    Invoked to define the classical regime and Lyapunov analysis.
  • domain assumption Quantum trajectories faithfully sample the open-system evolution
    Used to generate the finite-size quantum histograms and steady-state statistics.

pith-pipeline@v0.9.0 · 5726 in / 1426 out tokens · 57543 ms · 2026-05-18T14:24:25.047242+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

56 extracted references · 56 canonical work pages

  1. [1]

    Rosenblum M G, Pikovsky A and Kurths J 2001Synchronization – A universal concept in nonlinear sciences(Cambridge: Cambridge University Press)

    work page

  2. [2]

    Notes Phys.39420–422

    Kuramoto Y 1975Lect. Notes Phys.39420–422

    work page

  3. [3]

    Kuramoto Y 2003Chemical oscillations, waves, and turbulenceChemistry Series (Dover Publications) ISBN 978-0-486-42881-9 originally published: Springer Berlin, New York, Heidelberg, 1984

    work page 1984

  4. [4]

    Mari A, Farace A, Didier N, Giovannetti V and Fazio R 2013Phys. Rev. Lett. 111(10) 103605 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.111. 103605

    work page doi:10.1103/physrevlett.111

  5. [5]

    Giorgi G L, Plastina F, Francica G and Zambrini R 2013Physical Review A88 ISSN 1094-1622 URLhttp://dx.doi.org/10.1103/PhysRevA.88.042115

    work page doi:10.1103/physreva.88.042115

  6. [6]

    Bellomo B, Giorgi G L, Palma G M and Zambrini R 2017Phys. Rev. A95(4) 043807 URLhttps://link.aps.org/doi/10.1103/PhysRevA.95.043807

    work page doi:10.1103/physreva.95.043807

  7. [7]

    Buˇ ca B, Booker C and Jaksch D 2022SciPost Phys.12097 URLhttps: //scipost.org/10.21468/SciPostPhys.12.3.097 REFERENCES16

    work page doi:10.21468/scipostphys.12.3.097

  8. [8]

    Vaidya G M, Mamgain A, Hawaldar S, Hahn W, Kaubruegger R, Suri B and Shankar A 2024Phys. Rev. A109(3) 033718 URLhttps://link.aps.org/doi/ 10.1103/PhysRevA.109.033718

    work page doi:10.1103/physreva.109.033718

  9. [9]

    Murtadho T, Vinjanampathy S and Thingna J 2023Physical Review Letters131 ISSN 1079-7114 URLhttp://dx.doi.org/10.1103/PhysRevLett.131.030401

    work page doi:10.1103/physrevlett.131.030401

  10. [10]

    Solanki P, Mehdi F M, Hajduˇ sek M and Vinjanampathy S 2023Physical Review A 108ISSN 2469-9934 URLhttp://dx.doi.org/10.1103/PhysRevA.108.022216

    work page doi:10.1103/physreva.108.022216

  11. [11]

    Cabot A, Luca Giorgi G and Zambrini R 2021Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences47720200850 ISSN 1471-2946 URLhttp://dx.doi.org/10.1098/rspa.2020.0850

    work page doi:10.1098/rspa.2020.0850 2020

  12. [13]

    Lee T E, Chan C K and Wang S 2014Physical Review E89ISSN 1550-2376 URLhttp://dx.doi.org/10.1103/PhysRevE.89.022913

    work page doi:10.1103/physreve.89.022913

  13. [14]

    Walter S, Nunnenkamp A and Bruder C 2014Physical Review Letters112ISSN 1079-7114 URLhttp://dx.doi.org/10.1103/PhysRevLett.112.094102

    work page doi:10.1103/physrevlett.112.094102

  14. [15]

    Li X, Li Y and Yan Y 2025 Two-body dissipator engineering: Environment- induced quantum synchronization transitions (Preprint2506.07580) URL https://arxiv.org/abs/2506.07580

    work page arXiv 2025

  15. [16]

    Lee T E and Sadeghpour H R 2013Phys. Rev. Lett.111(23) 234101 URL https://link.aps.org/doi/10.1103/PhysRevLett.111.234101

    work page doi:10.1103/physrevlett.111.234101

  16. [17]

    Nadolny T and Bruder C 2023Phys. Rev. Lett.131(19) 190402 URLhttps: //link.aps.org/doi/10.1103/PhysRevLett.131.190402

    work page doi:10.1103/physrevlett.131.190402

  17. [18]

    Cabot A, Giorgi G L, Galve F and Zambrini R 2019Physical Review Letters123 ISSN 1079-7114 URLhttp://dx.doi.org/10.1103/PhysRevLett.123.023604

    work page doi:10.1103/physrevlett.123.023604

  18. [19]

    W¨ achtler C W and Platero G 2023Physical Review Research5ISSN 2643-1564 URLhttp://dx.doi.org/10.1103/PhysRevResearch.5.023021

    work page doi:10.1103/physrevresearch.5.023021

  19. [20]

    W¨ achtler C W and Moore J E 2024Physical Review Letters132ISSN 1079-7114 URLhttp://dx.doi.org/10.1103/PhysRevLett.132.196601

    work page doi:10.1103/physrevlett.132.196601

  20. [21]

    Nadolny T, Bruder C and Brunelli M 2025Phys. Rev. X15(1) 011010 URL https://link.aps.org/doi/10.1103/PhysRevX.15.011010

    work page doi:10.1103/physrevx.15.011010

  21. [22]

    Delmonte A, Romito A, Santoro G E and Fazio R 2023Phys. Rev. A108(3) 032219 URLhttps://link.aps.org/doi/10.1103/PhysRevA.108.032219

    work page doi:10.1103/physreva.108.032219

  22. [23]

    Laskar A W, Adhikary P, Mondal S, Katiyar P, Vinjanampathy S and Ghosh S 2020Physical Review Letters125ISSN 1079-7114 URLhttp://dx.doi.org/ 10.1103/PhysRevLett.125.013601

    work page doi:10.1103/physrevlett.125.013601

  23. [24]

    Zhang L, Wang Z, Wang Y, Zhang J, Wu Z, Jie J and Lu Y 2023Phys. Rev. Res. 5(3) 033209 URLhttps://link.aps.org/doi/10.1103/PhysRevResearch.5. 033209

    work page doi:10.1103/physrevresearch.5

  24. [25]

    Li Y, Xie Z, Yang X, Li Y, Zhao X, Cheng X, Peng X, Li J, Lutz E, Lin Y and Du J 2025 Experimental realization and synchronization of a quantum van der pol oscillator (Preprint2504.00751) URLhttps://arxiv.org/abs/2504.00751 REFERENCES17

    work page arXiv 2025

  25. [26]

    Whitty, A

    Krithika V R, Solanki P, Vinjanampathy S and Mahesh T S 2022Physical Review A105ISSN 2469-9934 URLhttp://dx.doi.org/10.1103/PhysRevA. 105.062206

    work page doi:10.1103/physreva

  26. [27]

    Koppenh¨ ofer M, Bruder C and Roulet A 2020Physical Review Research2ISSN 2643-1564 URLhttp://dx.doi.org/10.1103/PhysRevResearch.2.023026

    work page doi:10.1103/physrevresearch.2.023026

  27. [28]

    Strogatz S H 2000Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering(Westview Press)

    work page

  28. [29]

    Pecora L M and Carroll T L 1990Phys. Rev. Lett.64(8) 821–824 URLhttps: //link.aps.org/doi/10.1103/PhysRevLett.64.821

    work page doi:10.1103/physrevlett.64.821

  29. [30]

    Pecora L and Carroll T 1990Physical Review Letters64821

    work page

  30. [31]

    Cuomo K M and Oppenheim A V 1993Phys. Rev. Lett.71(1) 65–68 URL https://link.aps.org/doi/10.1103/PhysRevLett.71.65

    work page doi:10.1103/physrevlett.71.65

  31. [32]

    Sacha K and Zakrzewski J 2017Reports on Progress in Physics81016401 ISSN 1361-6633 URLhttp://dx.doi.org/10.1088/1361-6633/aa8b38

    work page doi:10.1088/1361-6633/aa8b38

  32. [33]

    Zaletel M P, Lukin M, Monroe C, Nayak C, Wilczek F and Yao N Y 2023 Rev. Mod. Phys.95(3) 031001 URLhttps://link.aps.org/doi/10.1103/ RevModPhys.95.031001

    work page 2023

  33. [34]

    Else D V, Bauer B and Nayak C 2016Phys. Rev. Lett.117(9) 090402 URL https://link.aps.org/doi/10.1103/PhysRevLett.117.090402

    work page doi:10.1103/physrevlett.117.090402

  34. [35]

    Khemani V, Lazarides A, Moessner R and Sondhi S L 2016Phys. Rev. Lett. 116(25) 250401 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.116. 250401

    work page doi:10.1103/physrevlett.116

  35. [36]

    Iemini F, Russomanno A, Keeling J, Schir` o M, Dalmonte M and Fazio R 2018 Phys. Rev. Lett.121(3) 035301 URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.121.035301

    work page 2018

  36. [37]

    Hajduˇ sek M, Solanki P, Fazio R and Vinjanampathy S 2022Phys. Rev. Lett. 128(8) 080603 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.128. 080603

    work page doi:10.1103/physrevlett.128

  37. [38]

    Solanki P, Krishna M, Hajduˇ sek M, Bruder C and Vinjanampathy S 2024 Phys. Rev. Lett.133(26) 260403 URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.133.260403

    work page 2024

  38. [39]

    Solanki P and Minganti F 2024 Chaos in time: A dissipative continuous quasi time crystals (Preprint2411.07297) URLhttps://arxiv.org/abs/2411.07297

    work page arXiv 2024

  39. [40]

    Prazeres L F d, Souza L d S and Iemini F 2021Phys. Rev. B103(18) 184308 URLhttps://link.aps.org/doi/10.1103/PhysRevB.103.184308

    work page doi:10.1103/physrevb.103.184308

  40. [41]

    Paulino P J, Cabot A, Chiara G D, Antezza M, Lesanovsky I and Carollo F 2025 Thermodynamics of coupled time crystals with an application to energy storage (Preprint2411.04836) URLhttps://arxiv.org/abs/2411.04836

    work page arXiv 2025

  41. [42]

    Li Y, Wang C, Tang Y and Liu Y C 2024Phys. Rev. Lett.132(18) 183803 URL https://link.aps.org/doi/10.1103/PhysRevLett.132.183803

    work page doi:10.1103/physrevlett.132.183803

  42. [43]

    1103/PhysRevResearch.7.023082

    Delmonte A, Li Z, Passarelli G, Song E Y, Barberena D, Rey A M and Fazio R 2025Physical Review Research7ISSN 2643-1564 URLhttp://dx.doi.org/10. 1103/PhysRevResearch.7.023082

    work page

  43. [44]

    Plenio M B and Knight P L 1998Rev. Mod. Phys.70(1) 101–144 URLhttps: //link.aps.org/doi/10.1103/RevModPhys.70.101 REFERENCES18

    work page doi:10.1103/revmodphys.70.101

  44. [45]

    Phys.6377 ISSN 1460-6976 URLhttp://dx.doi.org/10

    Daley A J 2014Adv. Phys.6377 ISSN 1460-6976 URLhttp://dx.doi.org/10. 1080/00018732.2014.933502

    work page arXiv 2014

  45. [46]

    Fazio R, Keeling J, Mazza L and Schir` o M 2025 Many-body open quantum systems (Preprint2409.10300) URLhttps://arxiv.org/abs/2409.10300

    work page arXiv 2025

  46. [47]

    Pikovsky A and Politi A 2016Lyapunov Exponents: A Tool to Explore Complex Dynamics(Cambridge University Press)

    work page

  47. [48]

    Benettin G, Galgani L and Strelcyn J M 1976Phys. Rev. A14(6) 2338–2345 URLhttps://link.aps.org/doi/10.1103/PhysRevA.14.2338

    work page doi:10.1103/physreva.14.2338

  48. [49]

    Galve F, Luca Giorgi G and Zambrini R 2017Quantum Correlations and Synchronization Measures(Springer International Publishing) p 393–420 ISBN 9783319534121 URLhttp://dx.doi.org/10.1007/978-3-319-53412-1_18

    work page doi:10.1007/978-3-319-53412-1_18

  49. [50]

    Piccitto G, Rossini D and Russomanno A 2024Eur. Phys. J. B9790 URL https://doi.org/10.1140/epjb/s10051-024-00725-0

    work page doi:10.1140/epjb/s10051-024-00725-0

  50. [51]

    Passarelli G, Lucignano P, Rossini D and Russomanno A 2025Quantum91653 ISSN 2521-327X URLhttp://dx.doi.org/10.22331/q-2025-03-05-1653

    work page doi:10.22331/q-2025-03-05-1653 2025

  51. [52]

    Rufo G, Rufo S, Ribeiro P and Chesi S 2025 Quantum and semi-classical signatures of dissipative chaos in the steady state (Preprint2506.14961) URL https://arxiv.org/abs/2506.14961

    work page arXiv 2025

  52. [53]

    Atas Y Y, Bogomolny E, Giraud O and Roux G 2013Physical Review Letters110 ISSN 1079-7114 URLhttp://dx.doi.org/10.1103/PhysRevLett.110.084101

    work page doi:10.1103/physrevlett.110.084101

  53. [54]

    Passarelli G, Turkeshi X, Russomanno A, Lucignano P, Schir` o M and Fazio R 2024 Phys. Rev. Lett.132(16) 163401 URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.132.163401

    work page 2024

  54. [55]

    Passarelli G, Lucignano P, Fazio R and Russomanno A 2022Phys. Rev. B106(22) 224308 URLhttps://link.aps.org/doi/10.1103/PhysRevB.106.224308

    work page doi:10.1103/physrevb.106.224308

  55. [56]

    Russomanno A 2023Phys. Rev. B108(9) 094305 URLhttps://link.aps.org/ doi/10.1103/PhysRevB.108.094305

    work page doi:10.1103/physrevb.108.094305

  56. [57]

    Passarelli G, Russomanno A and Lucignano P 2025Phys. Rev. A111(6) 062417 URLhttps://link.aps.org/doi/10.1103/d7tm-9hkp

    work page doi:10.1103/d7tm-9hkp