Pith. sign in

REVIEW 2 major objections 3 minor 61 references

Toward Chaotic Group Velocity Hopping of an On-Chip Dissipative Kerr Soliton

T0 review · 2 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Phase-modulating the reference lock of a dissipative Kerr soliton drives its repetition rate into chaotic, random hops between two values.

desk verdict Solid Adler-model validation, but the 'chaotic hopping' claim is one step ahead of the data. read the letter →

arxiv 2509.09108 v1 pith:OT7N2Q5P submitted 2025-09-11 physics.optics

Gregory Moille , Sashank Kaushik Sridhar , Pradyoth Shandilya , Avik Dutt , Curtis Menyuk , Kartik Srinivasan This is my paper · ORCID
classification physics.optics PACS 42.65.Tg05.45.-a42.65.-k
keywords dissipativeKerrsolitonKerr-inducedsynchronizationAdlerequationmicrocombopticalchaosrepetitionratehoppingLyapunovexponentphasemodulation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper reports a new dynamical state for integrated optical frequency combs: a single, otherwise stable dissipative Kerr soliton that behaves chaotically. The chaos is triggered by phase-modulating the reference laser to which the soliton is locked through Kerr-induced synchronization (KIS), a mechanism that normally pins the comb's repetition rate to two optical references. The authors show that KIS obeys a second-order Adler equation, and that in its phase-modulated three-variable form the equation predicts coexisting strange attractors at a specific detuning, which make the soliton's group velocity—and therefore the comb's repetition rate—jump randomly between two values. They validate the equation quantitatively against experiments and full Lugiato-Lefever simulations, including subharmonic locking, and observe the predicted chaotic hopping on an octave-spanning microcomb.

What carries the argument

The central object is the phase-modulated second-order Adler equation, an autonomous three-variable system: dψ/dτ = Ω_ext, dφ/dτ = Ω, β dΩ/dτ = −Ω + α + sin(φ + A cos ψ). Here φ is the phase difference between the soliton and the reference comb tooth, Ω is its normalized rate, ψ is the modulation phase, β is the McCumber coefficient (the normalized synchronization half-window over cavity loss), and α is the normalized reference detuning. The second-order term is what allows subharmonic locking and, under phase modulation, strange attractors. The paper's quantitative matching of this reduced model to both experiment and the Lugiato-Lefever equation is what makes the chaos prediction trustwort

What would settle it

Measure the maximal Lyapunov exponent directly from the experimental repetition-rate time series using delay embedding; if it is not positive, the hopping is not deterministic chaos. Alternatively, scan the reference detuning through the predicted α≈0.58 coexistence region: if the hopping statistics gradually turn on rather than appearing abruptly at the strange-attractor boundary, the Adler prediction fails.

Watch

Extended reading notes

Core claim

The central claim is that dissipative Kerr solitons—normally stable, low-noise microcomb pulses—can be driven into deterministic chaos on the same chip without leaving the solitonic state. The route is phase-modulated Kerr-induced synchronization: a weak reference laser injected near a comb tooth is phase-modulated at a frequency comparable to the synchronization window. The paper shows this system is quantitatively described by a second-order Adler equation, and experimentally confirms its second-order character through subharmonic locking. Normalizing the model with the measured synchronization window, the equation predicts that at reference detuning α≈0.58 two strange attractors coexist,

Load-bearing premise

The experimental identification of the hopping as chaos assumes that the random telegraphic signal comes from the Adler equation's coexisting strange attractors, not from environmental noise flipping the soliton between two stable, ordinary synchronized states.

Editorial extensions

If this is right

  • A single integrated microcomb can switch between a low-noise, metrology-grade state and a deterministic chaotic state by tuning the reference detuning, without changing the pump or the soliton.
  • The validated second-order Adler model becomes a fast predictive tool for KIS dynamics, replacing full Lugiato-Lefever simulations for exploring nonlinear synchronization regimes.
  • Subharmonic locking at fractional detunings—observed up to third order—confirms the second-order nature of KIS and explains synchronization through four-wave-mixing idlers without direct comb-tooth capture.
  • The chaotic hopping yields a microcomb-based physical random bit generator whose randomness comes from deterministic chaos, with the entropy rate set by the Lyapunov exponent.
  • Because the model is symmetric under swapping which oscillator is modulated, breather-soliton microcombs should exhibit the same chaotic hopping.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The experimental evidence for chaos currently rests on statistical randomness tests of the binary hopping sequence; a direct Lyapunov exponent estimated from the recorded time series would close the gap between simulation and experiment.
  • If the hopping statistics follow the Adler model's detuning dependence, the comb could act as a calibrated physical random number generator whose randomness is a reproducible property of the equations.
  • The formal equivalence to AC-driven Josephson junctions suggests that Shapiro-step engineering techniques could be translated to optical KIS, enabling precise control of fractional synchronization.
  • Modulating the main pump instead of the reference should produce the same chaotic hopping, since the Adler model depends only on the relative phase modulation—an experiment that would test the model's causality.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper proposes and experimentally tests a second-order Adler model for Kerr-induced synchronization (KIS) of a dissipative Kerr soliton (DKS) to a phase-modulated reference laser. The authors validate the model by reproducing repetition-rate entrainment windows, sideband locking, and sub-harmonic synchronization tongues, and then use the model to predict a regime of coexisting strange attractors at normalized detuning α ≈ 0.58. Experimentally, they observe random telegraphic hopping between two repetition-rate states, which they attribute to chaotic group-velocity hopping of the soliton. The manuscript includes a derivation of the Adler model from the Lugiato-Lefever equation, numerical Lyapunov-exponent and 0-1 tests for the model, and NIST randomness tests on the experimental hopping sequence.

Significance. If the central claim is substantiated, this would be a meaningful advance: it would demonstrate a deterministic chaotic state of an individual dissipative Kerr soliton, distinct from the well-known chaotic modulation-instability regime, and would provide a quantitative experimental test of the second-order Adler equation for KIS. The theoretical derivation and the validation via entrainment and sub-harmonic locking are clear strengths, as is the explicit numerical prediction of chaos from a low-dimensional model. However, the experimental discrimination between deterministic chaos and noise-driven switching between two stable synchronized states is currently missing, and this gap directly affects the main conclusion stated in the title and abstract.

major comments (2)
  1. [Section IV and Supplementary S.4] The experimental evidence for chaotic hopping is not sufficient to support the central claim. The NIST SP 800-22 monobit and chi-square tests are applied to a binarized sequence (carrier-KIS vs. sideband-KIS) and can at most certify statistical randomness. A noise-driven two-state system with exponential dwell times would pass these tests equally well. The Lyapunov exponent and 0-1 test reported in Fig. 4c are computed from numerical solutions of Eq. (2), not from the measured time series. To claim 'chaotic group velocity hopping', the authors should apply a determinism test to the continuous measured Δω_ceo or Δω_rep signal—for example, an estimated largest Lyapunov exponent from the data (the Wolf et al. method is already cited) or a 0-1 test on the raw signal—and, ideally, compare with a quantitative noise-driven bistability model. Without such a discriminator, the paper demonstrates
  2. [Section III and Supplementary S.1/S.2] The quantitative parameter normalization is internally inconsistent. The main text states that the Adler model is normalized using Ω0/2π = 130 MHz at A = 0 and κ/2π = 200 MHz, giving β = 0.65. Supplementary S.1 states κ/2π = 180 MHz, while Supplementary S.2 uses κ/2π = 100 MHz, Ω0/2π = 120 MHz, and μ = -88 (the main text uses μ_s = -90). Since β = Ω0/κ controls the damping and hence the predicted chaotic regime, these discrepancies are not cosmetic. Please reconcile the values, report uncertainties, and show how the quantitative agreement in Figs. 2 and 3 depends on the chosen κ and Ω0.
minor comments (3)
  1. [Supplementary S.4] Table III lists chi-square p-values of 0.029 (state -3) and 0.0397 (state +4), yet labels them 'Random'. If these are individual tests without correction, they actually indicate a possible departure from randomness at those lags; please clarify whether a multiple-testing correction was applied.
  2. [Section II] The sentence 'Eq. (2) now exhibits a fixed point at α ≈ nΩ for any β' is ambiguous: Ω is a dynamical variable, while the phase-modulation frequency is Ω_ext. Presumably the fixed points occur near α ≈ nΩ_ext; please clarify the notation.
  3. [General] There are numerous minor typos and notational issues, e.g., 'β ∂φ2/∂τ2' should read β ∂²φ/∂τ² in several equations, and the text contains duplicated/awkward phrases such as 'orbiting around Ω = 0 with a null winding number in the φ revolution co-exists with...' (Section IV). A careful proofread would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Adler-model chaos prediction is emergent and independently validated; self-citations are not load-bearing.

full rationale

The paper's derivation chain is not circular. The second-order Adler equation (Eq. 1) is derived in Supplementary S.2 from the multi-pump Lugiato-Lefever equation, with the single calibration constant Ω0 measured from the KIS half-window at A=0. Equation (2) is then an independent extension to phase modulation; its chaotic behavior at α≈0.58 is discovered by numerical solution and characterized by Lyapunov growth and the 0-1 test on the model, not fitted to the experimental hopping. The experimental hopping is compared to the model's coexisting-attractor prediction via repetition-rate separation and spectrograms. The NIST randomness tests on a binarized experimental sequence (S.4) support randomness but cannot by themselves prove deterministic chaos; however, this is an evidence-strength limitation, not a definitional equivalence or fitted-input prediction. Self-citations (Refs. 18, 20, 41, 44) provide prior derivations and measurement methods, but the core Adler equation is rederived and validated against LLE simulations and independent experimental data, so the self-citations are not load-bearing in a circular way. No quoted equation reduces to its own input by construction; the noted parameter inconsistencies are correctness risks, not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The model is a reduction of the LLE, calibrated using one measured KIS half-window (Omega0) and one loss value (kappa). No new physical entities are introduced. The chaotic prediction is not used to fit parameters, keeping the circularity burden low, but the model's time and amplitude scales do rest on experimental calibration.

free parameters (3)
  • KIS half-window Omega0/2pi = 130 MHz
    Used to normalize the Adler time scale (tau = Omega0 t) and define beta = Omega0/kappa; measured at A = 0 in the experiment.
  • Total loss rate kappa/2pi (Adler normalization) = 200 MHz
    Used with Omega0 to set beta = 0.65; inconsistent with the stated LLE kappa/2pi = 180 MHz in Supplementary S.1.
  • Synchronization detuning alpha = 0.58
    Selected in the Adler simulation to obtain coexistence of the two strange attractors; the experiment is then parked near this boundary.
assumptions (4)
  • domain assumption Multi-pump Lugiato-Lefever equation (Eq. S.1) is a valid model of the DKS microcomb with injected reference.
    Standard microcomb model; the paper fits its parameters (gamma, kappa, etc.) from the specific resonator.
  • domain assumption Small reference amplitude and weak modulation allows truncating the Bessel expansion to first order and reducing the dynamics to a single phase variable phi.
    Used in Supplementary S.1/S.2 to derive Eqs. (1)-(2).
  • domain assumption The soliton's internal structure is slaved; only its group velocity (phase) evolves, so the second-order Adler equation describes the KIS dynamics.
    Core reduction used throughout; the experimental validation supports it, but it is an assumption about separation of time scales.
  • standard math The driven damped pendulum (Eq. 2) exhibits coexisting strange attractors and positive Lyapunov exponents in the chosen parameter region.
    Known nonlinear dynamics; verified numerically in this paper via Lyapunov exponent and 0-1 test.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Toward Chaotic Group Velocity Hopping of an On-Chip Dissipative Kerr Soliton." pith.science (2026). https://pith.science/paper/OT7N2Q5P

@misc{pith2026250909108,
  author       = {Pith},
  title        = {Pith review of: Toward Chaotic Group Velocity Hopping of an On-Chip Dissipative Kerr Soliton},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OT7N2Q5P}},
  note         = {Machine review of arXiv:2509.09108}
}
read the original abstract

Chaos enables randomness-based applications, particularly in photonic systems. Integrated optical frequency combs (microcombs) have previously been observed in either chaotic modulation instability or stable, low-noise dissipative Kerr soliton (DKS) regimes. In this work, we demonstrate a new microcomb state where a single DKS exhibits chaotic behavior. By phase modulating the Kerr-induced synchronization (KIS) between a DKS and an externally injected reference laser, we observe chaotic group velocity hopping of the soliton, causing random transitions of the repetition rate. Using a chip-integrated octave-spanning microcomb, we experimentally validate the second-order Adler equation describing KIS, allowing us to predict and demonstrate this chaotic DKS hopping. This work connects nonlinear dynamics with optical soliton physics, providing a deterministic framework for triggering microcomb chaos in the solitonic state.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references

  1. [1]

    K. R. Sreenivasan, Fluid turbulence, Reviews of Modern Physics 71, S383 (1999)

    1999

  2. [2]

    M. L. Heltberg, S. Krishna, L. P. Kadanoff, and M. H. Jensen, A tale of two rhythms: Locked clocks and chaos in biology, Cell Systems 12, 291 (2021)

    2021

  3. [3]

    Matsumoto, A chaotic attractor from Chua’s circuit, IEEE Transactions on Circuits and Systems 31, 1055 (1984)

    T. Matsumoto, A chaotic attractor from Chua’s circuit, IEEE Transactions on Circuits and Systems 31, 1055 (1984)

    1984

  4. [4]

    E. N. Lorenz, Deterministic Nonperiodic Flow, Journal of the Atmospheric Sciences 20, 130 (1963)

    1963

  5. [5]

    A. Ryan, G. Agrawal, G. Gray, and E. Gage, Optical- feedback-induced chaos and its control in multimode semi- conductor lasers, IEEE Journal of Quantum Electronics 30, 668 (1994)

    1994

  6. [6]

    Akhmediev, J

    N. Akhmediev, J. M. Soto-Crespo, and G. Town, Pulsating solitons, chaotic solitons, period doubling, and pulse coex- 6 istence in mode-locked lasers: Complex Ginzburg-Landau equation approach, Physical Review E 63, 056602 (2001)

    2001

  7. [7]

    Reidler, Y

    I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, Ultrahigh-Speed Random Number Generation Based on a Chaotic Semiconductor Laser, Physical Review Letters 103, 024102 (2009)

    2009

  8. [8]

    Argyris, D

    A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garc ´ ıa-Ojalvo, C. R. Mirasso, L. Pes- quera, and K. A. Shore, Chaos-based communications at high bit rates using commercial fibre-optic links, Nature 438, 343 (2005)

    2005

Show all 61 references
  1. [9]

    Rafayelyan, J

    M. Rafayelyan, J. Dong, Y. Tan, F. Krzakala, and S. Gigan, Large-Scale Optical Reservoir Computing for Spatiotem- poral Chaotic Systems Prediction, Physical Review X 10, 041037 (2020)

    2020

  2. [10]

    Lukashchuk, J

    A. Lukashchuk, J. Riemensberger, A. Stroganov, G. Nav- ickaite, and T. J. Kippenberg, Chaotic microcomb inertia- free parallel ranging, APL Photonics 8, 056102 (2023)

    2023

  3. [11]

    Moreno, S

    D. Moreno, S. Fujii, A. Nakashima, D. Lemcke, A. Uchida, P. Sanchis, and T. Tanabe, Synchronization of two chaotic microresonator frequency combs, Optics Express 32, 2460 (2024)

    2024

  4. [12]

    T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L. Gorodetsky, Dissipative Kerr solitons in optical microres- onators, Science 361, eaan8083 (2018)

    2018

  5. [13]

    Y. K. Chembo and C. R. Menyuk, Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators, Physical Review A 87, 053852 (2013)

    2013

  6. [14]

    S. A. Diddams, K. Vahala, and T. Udem, Optical fre- quency combs: Coherently uniting the electromagnetic spectrum, Science 369, eaay3676 (2020)

    2020

  7. [15]

    Grelu and N

    P. Grelu and N. Akhmediev, Dissipative solitons for mode- locked lasers, Nature Photonics 6, 84 (2012)

    2012

  8. [16]

    Lecaplain, Ph

    C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, Dissipative Rogue Waves Generated by Chaotic Pulse Bunching in a Mode-Locked Laser, Physical Review Letters 108, 233901 (2012)

    2012

  9. [17]

    Y. Song, D. Zou, O. Gat, M. Hu, and P. Grelu, Chaotic In- ternal Dynamics of Dissipative Optical Soliton Molecules, Laser & Photonics Reviews 17, 2300066 (2023)

    2023

  10. [18]

    Moille, J

    G. Moille, J. Stone, M. Chojnacky, R. Shrestha, U. A. Javid, C. Menyuk, and K. Srinivasan, Kerr-induced syn- chronization of a cavity soliton to an optical reference, Nature 624, 267 (2023)

    2023

  11. [19]

    Wildi, A

    T. Wildi, A. Ulanov, N. Englebert, T. Voumard, and T. Herr, Sideband injection locking in microresonator frequency combs, APL Photonics 8, 120801 (2023)

    2023

  12. [20]

    Moille, P

    G. Moille, P. Shandilya, J. Stone, C. Menyuk, and K. Srini- vasan, All-optical noise quenching of an integrated fre- quency comb, Optica 12, 1020 (2025)

    2025

  13. [21]

    J. K. Jang, M. Erkintalo, S. Coen, and S. G. Murdoch, Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons, Nature Com- munications 6, 7370 (2015)

    2015

  14. [22]

    J. K. Jang, A. Klenner, X. Ji, Y. Okawachi, M. Lipson, and A. L. Gaeta, Synchronization of coupled optical mi- croresonators, Nature Photonics 12, 688 (2018)

    2018

  15. [23]

    Englebert, N

    N. Englebert, N. Goldman, M. Erkintalo, N. Mostaan, S.-P. Gorza, F. Leo, and J. Fatome, Bloch oscillations of coherently driven dissipative solitons in a synthetic dimension, Nature Physics 19, 1014 (2023)

    2023

  16. [24]

    D. C. Cole and S. B. Papp, Subharmonic Entrainment of Kerr Breather Solitons, Physical Review Letters 123, 173904 (2019)

    2019

  17. [25]

    Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, Counter- propagating solitons in microresonators, Nature Photonics 11, 560 (2017)

    2017

  18. [26]

    S. Sun, M. W. Harrington, F. Tabatabaei, S. Hanifi, K. Liu, J. Wang, B. Wang, Z. Yang, R. Liu, J. S. Morgan, S. M. Bowers, P. A. Morton, K. D. Nelson, A. Beling, D. J. Blumenthal, and X. Yi, Microcavity Kerr optical frequency division with integrated SiN photonics, Nature Phot...

    2025

  19. [27]

    Valizadeh, M

    A. Valizadeh, M. Kolahchi, and J. Straley, On the Origin of Fractional Shapiro Steps in Systems of Josephson Junc- tions with Few Degrees of Freedom:, Journal of Nonlinear Mathematical Physics 15, 407 (2008)

    2008

  20. [28]

    Taheri, A

    H. Taheri, A. B. Matsko, and L. Maleki, Optical lattice trap for Kerr solitons, The European Physical Journal D 71, 153 (2017)

    2017

  21. [29]

    Coullet, J

    P. Coullet, J. M. Gilli, M. Monticelli, and N. Vanden- berghe, A damped pendulum forced with a constant torque, American Journal of Physics 73, 1122 (2005)

    2005

  22. [30]

    Moille, Q

    G. Moille, Q. Li, T. C. Briles, S.-P. Yu, T. Drake, X. Lu, A. Rao, D. Westly, S. B. Papp, and K. Srinivasan, Broad- band resonator-waveguide coupling for efficient extraction of octave-spanning microcombs, Optics Letters 44, 4737 (2019)

    2019

  23. [31]

    Zhang, J

    S. Zhang, J. M. Silver, L. Del Bino, F. Copie, M. T. M. Woodley, G. N. Ghalanos, A. Ø. Svela, N. Moroney, and P. Del’Haye, Sub-milliwatt-level microresonator solitons with extended access range using an auxiliary laser, Optica 6, 206 (2019)

    2019

  24. [32]

    H. Zhou, Y. Geng, W. Cui, S.-W. Huang, Q. Zhou, K. Qiu, and C. Wei Wong, Soliton bursts and deterministic dissi- pative Kerr soliton generation in auxiliary-assisted micro- cavities, Light: Science & Applications 8, 50 (2019)

    2019

  25. [33]

    J. R. Stone and S. B. Papp, Harnessing Dispersion in Soliton Microcombs to Mitigate Thermal Noise, Physical Review Letters 125, 153901 (2020)

    2020

  26. [34]

    Y. Wang, F. Leo, J. Fatome, M. Erkintalo, S. G. Murdoch, and S. Coen, Universal mechanism for the binding of temporal cavity solitons, Optica 4, 855 (2017)

    2017

  27. [35]

    Rukhin, J

    A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, N. Heckert, J. Dray, S. Vo, and L. Bassham, A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications , Tech. Rep. NIST Special Publica...

    2010

  28. [36]

    S. K. Ang, Randomness test suit, https://github.com/ stevenang/randomness_testsuite

  29. [37]

    D. E. McCumber, Effect of ac Impedance on dc Voltage- Current Characteristics of Superconductor Weak-Link Junctions, Journal of Applied Physics 39, 3113 (1968)

    1968

  30. [38]

    Philip, A nonlinear oscillator with a strange attractor, Philosophical Transactions of the Royal Society of London

    H. Philip, A nonlinear oscillator with a strange attractor, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 292, 419 (1979)

    1979

  31. [39]

    B. D. Josephson, Possible new effects in superconductive tunnelling, Physics Letters 1, 251 (1962)

    1962

  32. [40]

    Shapiro, Josephson Currents in Superconducting Tun- neling: The Effect of Microwaves and Other Observations, Physical Review Letters 11, 80 (1963)

    S. Shapiro, Josephson Currents in Superconducting Tun- neling: The Effect of Microwaves and Other Observations, Physical Review Letters 11, 80 (1963)

    1963

  33. [41]

    C. R. Menyuk, P. Shandilya, L. Courtright, G. Moille, and K. Srinivasan, Multi-color solitons and frequency combs in microresonators, Optics Express 33, 21824 (2025). 7

    2025

  34. [42]

    Taheri, A

    H. Taheri, A. B. Matsko, and K. A. Wiesenfeld, Dynamics of Dissipative Cavity Solitons in a Lattice Trap, Laser & Photonics Reviews n/a, 2500257 (2025)

    2025

  35. [43]

    Moille, P

    G. Moille, P. Shandilya, A. Niang, C. Menyuk, G. Carter, and K. Srinivasan, Versatile optical frequency division with Kerr-induced synchronization at tunable micro- comb synthetic dispersive waves, Nature Photonics 19, 36 (2025)

    2025

  36. [44]

    Moille, P

    G. Moille, P. Shandilya, M. Erkintalo, C. R. Menyuk, and K. Srinivasan, On-Chip Parametric Synchronization of a Dissipative Kerr Soliton Microcomb, Physical Review Letters 134, 193802 (2025)

    2025

  37. [45]

    Strogatz, Nonlinear dynamics and chaos: With appli- cations to physics, biology, chemistry, and engineering (Boca Raton London New York, 2019) second edition ed., Chap

    S. Strogatz, Nonlinear dynamics and chaos: With appli- cations to physics, biology, chemistry, and engineering (Boca Raton London New York, 2019) second edition ed., Chap. 8

    2019

  38. [46]

    A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, De- termining Lyapunov exponents from a time series, Physica D: Nonlinear Phenomena 16, 285 (1985)

    1985

  39. [47]

    G. A. Gottwald and I. Melbourne, On the Implementation of the 0–1 Test for Chaos, SIAM Journal on Applied Dynamical Systems 8, 129 (2009)

    2009

  40. [48]

    Shortiss, B

    K. Shortiss, B. Lingnau, F. Dubois, B. Kelleher, and F. H. Peters, Harmonic frequency locking and tuning of comb frequency spacing through optical injection, Optics Ex- press 27, 36976 (2019)

    2019

  41. [49]

    X. Wu, J. Peng, B. Yuan, S. Boscolo, C. Finot, and H. Zeng, Unveiling the complexity of Arnold’s tongues in a breathing-soliton laser, Science Advances 11, eads3660 (2025)

    2025

  42. [50]

    L. Yuan, Q. Lin, M. Xiao, and S. Fan, Synthetic dimension in photonics, Optica 5, 1396 (2018)

    2018

  43. [51]

    B. A. Bell, K. Wang, A. S. Solntsev, D. N. Neshev, A. A. Sukhorukov, and B. J. Eggleton, Spectral photonic lattices with complex long-range coupling, Optica 4, 1433 (2017)

    2017

  44. [52]

    Balˇ cytis, T

    A. Balˇ cytis, T. Ozawa, Y. Ota, S. Iwamoto, J. Maeda, and T. Baba, Synthetic dimension band structures on a Si CMOS photonic platform, Science Advances 8, eabk0468 (2022)

    2022

  45. [53]

    S. K. Sridhar, S. Ghosh, D. Srinivasan, A. R. Miller, and A. Dutt, Quantized topological pumping in Floquet synthetic dimensions with a driven dissipative photonic molecule, Nature Physics 20, 843 (2024)

    2024

  46. [54]

    Q. Lin, M. Xiao, L. Yuan, and S. Fan, Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension, Nature Communications 7, 13731 (2016)

    2016

  47. [55]

    Lustig, S

    E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Ban- dres, A. Szameit, and M. Segev, Photonic topological insulator in synthetic dimensions, Nature 567, 356 (2019)

    2019

  48. [56]

    Lustig and M

    E. Lustig and M. Segev, Topological photonics in synthetic dimensions, Advances in Optics and Photonics 13, 426 (2021)

    2021

  49. [57]

    Crameri, Scientific colour maps, Zenodo (2023)

    F. Crameri, Scientific colour maps, Zenodo (2023)

    2023

  50. [58]

    Crameri, G

    F. Crameri, G. E. Shephard, and P. J. Heron, The misuse of colour in science communication, Nature Communica- tions 11, 5444 (2020)

    2020

  51. [59]

    Moille, Q

    G. Moille, Q. Li, L. Xiyuan, and K. Srinivasan, pyLLE: A Fast and User Friendly Lugiato-Lefever Equation Solver, Journal of Research of the NIST 124, 124012 (2019) 8 Supplementary Information: Toward Chaotic Group Ve- locity Hopping of an On-Chip Dissipative Kerr Soliton S.1. ...

    2019

  52. [60]

    Adler equation in the static regime In the unmodulated KIS, the phase offset between DKS and reference laser relates directly to their frequency difference: ∂φ ∂t = ∂φref − φdks ∂t = ϖ = ωref − (µsωrep + ωo) (S.1) With ωref and ω0 denoting the reference and main pump frequenci...

  53. [61]

    (S.2),changes φref → φref + A cos (ωrf t)

    Phase modulated Adler equation The phase modulation of the reference, introduced in Eq. (S.2),changes φref → φref + A cos (ωrf t). The derivation proceeds as above, incorporating this reference phase modulation: ∂φ ∂t = ∂φref − φdks ∂t = ϖ − ωrf Asin (ωrf t) = ωref − ωrf Asin ...

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.