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Emergent nonlinear phenomena in a driven dissipative photonic dimer


Collective effects leading to spatial, temporal or spatiotemporal pattern formation in complex nonlinear systems driven out of equilibrium cannot be described at the single-particle level and are therefore often called emergent phenomena. They are characterized by length scales exceeding the characteristic interaction length and by spontaneous symmetry breaking. Recent advances in integrated photonics have indicated that the study of emergent phenomena is possible in complex coupled nonlinear optical systems. Here we demonstrate that the out-of-equilibrium driving of a strongly coupled pair of photonic integrated Kerr microresonators (‘dimer’)—which, at the ‘single particle’ (that is, individual resonator) level, generate well-understood dissipative Kerr solitons—exhibits emergent nonlinear phenomena. By exploring the dimer phase diagram, we find regimes of soliton hopping, spontaneous symmetry breaking and periodically emerging (in)commensurate dispersive waves. These phenomena are not included in the single-particle description and are related to the parametric frequency conversion between the hybridized supermodes. Moreover, by electrically controlling the supermode hybridization, we achieve wide tunability of spectral interference patterns between the dimer solitons and dispersive waves. Our findings represent a step towards the study of emergent nonlinear phenomena in soliton networks and multimodal lattices.

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Fig. 1: DKS formation in single- and coupled-ring resonators.
Fig. 2: Emergent dynamics revealed by numerical simulations of the photonic dimer.
Fig. 3: Kerr comb reconstruction of the photonic dimer states.
Fig. 4: Electrical control of GSs and DW interference.

Data availability

Source data are provided with this paper. All data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request as well as at

Code availability

Numerical codes used in this study are available from the corresponding authors upon reasonable request.


  1. Bak, P. How Nature Works: The Science of Self-Organized Criticality (Copernicus, 1996).

  2. Anderson, P. W. More is different. Science 177, 393–396 (1972).

    ADS  Google Scholar 

  3. Schulman, L. S. & Seiden, P. E. Percolation and galaxies. Science 233, 425–431 (1986).

    ADS  Google Scholar 

  4. Chialvo, D. R. Emergent complex neural dynamics. Nat. Phys. 6, 744–750 (2010).

    Google Scholar 

  5. Kevrekidis, P. G., Frantzeskakis, D. J. & Carretero-González, R. Emergent Nonlinear Phenomena in Bose–Einstein Condensates: Theory and Experiment Vol. 45 (Springer Science & Business Media, 2007).

  6. Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).

    ADS  Google Scholar 

  7. Nishimori, H. & Ortiz, G. Elements of Phase Transitions and Critical Phenomena (OUP, 2011).

  8. Prigogine, I. & Lefever, R. Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48, 1695–1700 (1968).

    ADS  Google Scholar 

  9. Akhmediev, N. & Ankiewicz, A. Dissipative Solitons: From Optics to Biology and Medicine Vol. 751 (Springer Science & Business Media, 2008).

  10. Haus, H. A. Mode-locking of lasers. IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).

    ADS  Google Scholar 

  11. Grelu, P. & Akhmediev, N. Dissipative solitons for mode-locked lasers. Nat. Photon. 6, 84–92 (2012).

    ADS  Google Scholar 

  12. Cundiff, S. T. & Ye, J. Colloquium: femtosecond optical frequency combs. Rev. Mod. Phys. 75, 325–342 (2003).

    ADS  Google Scholar 

  13. Wright, L. G. et al. Mechanisms of spatiotemporal mode-locking. Nat. Phys. 16, 565–570 (2020).

    Google Scholar 

  14. Kippenberg, T. J., Gaeta, A. L., Lipson, M. & Gorodetsky, M. L. Dissipative Kerr solitons in optical microresonators. Science 361, eaan8083 (2018).

    Google Scholar 

  15. Herr, T. et al. Temporal solitons in optical microresonators. Nat. Photon. 8, 145–152 (2014).

    ADS  Google Scholar 

  16. Yi, X., Yang, Q.-F., Yang, K. Y., Suh, M.-G. & Vahala, K. Soliton frequency comb at microwave rates in a high-Q silica microresonator. Optica 2, 1078–1085 (2015).

    ADS  Google Scholar 

  17. Brasch, V. et al. Photonic chip–based optical frequency comb using soliton Cherenkov radiation. Science 351, 357–360 (2016).

    ADS  MathSciNet  MATH  Google Scholar 

  18. Guo, H. et al. Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators. Nat. Phys. 13, 94–102 (2017).

    Google Scholar 

  19. Lucas, E., Karpov, M., Guo, H., Gorodetsky, M. L. & Kippenberg, T. J. Breathing dissipative solitons in optical microresonators. Nat. Commun. 8, 736 (2017).

    ADS  Google Scholar 

  20. Cole, D. C. et al. Kerr-microresonator solitons from a chirped background. Optica 5, 1304–1310 (2018).

    ADS  Google Scholar 

  21. Karpov, M. et al. Dynamics of soliton crystals in optical microresonators. Nat. Phys. 15, 1071–1077 (2019).

    Google Scholar 

  22. Karpov, M. et al. Raman self-frequency shift of dissipative Kerr solitons in an optical microresonator. Phys. Rev. Lett. 116, 103902 (2016).

    ADS  Google Scholar 

  23. Herr, T. et al. Mode spectrum and temporal soliton formation in optical microresonators. Phys. Rev. Lett. 113, 1–6 (2014).

    Google Scholar 

  24. Xue, X. et al. Mode-locked dark pulse Kerr combs in normal-dispersion microresonators. Nat. Photon. 9, 594–600 (2015).

    Google Scholar 

  25. Yi, X. et al. Single-mode dispersive waves and soliton microcomb dynamics. Nat. Commun. 8, 14869 (2017).

    ADS  Google Scholar 

  26. Chembo, Y. K. & Menyuk, C. R. Spatiotemporal Lugiato–Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators. Phys. Rev. A 87, 053852 (2013).

    ADS  Google Scholar 

  27. Zhang, M. et al. Electronically programmable photonic molecule. Nat. Photon. 13, 36–40 (2019).

    ADS  Google Scholar 

  28. Casteels, W. & Ciuti, C. Quantum entanglement in the spatial-symmetry-breaking phase transition of a driven-dissipative Bose–Hubbard dimer. Phys. Rev. A 95, 013812 (2017).

    ADS  MathSciNet  Google Scholar 

  29. Chembo, Y. K. Quantum dynamics of Kerr optical frequency combs below and above threshold: spontaneous four-wave mixing, entanglement, and squeezed states of light. Phys. Rev. A 93, 033820 (2016).

    ADS  Google Scholar 

  30. Milián, C., Kartashov, Y. V., Skryabin, D. V. & Torner, L. Cavity solitons in a microring dimer with gain and loss. Opt. Lett. 43, 979–982 (2018).

    ADS  Google Scholar 

  31. Jang, J. K. et al. Synchronization of coupled optical microresonators. Nat. Photon. 12, 688–693 (2018).

    ADS  Google Scholar 

  32. Xue, X., Zheng, X. & Zhou, B. Super-efficient temporal solitons in mutually coupled optical cavities. Nat. Photon. 13, 616–622 (2019).

    ADS  Google Scholar 

  33. Godey, C., Balakireva, I. V., Coillet, A. & Chembo, Y. K. Stability analysis of the spatiotemporal Lugiato–Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes. Phys. Rev. A 89, 063814 (2014).

    ADS  Google Scholar 

  34. Qi, Z. et al. Dissipative cnoidal waves (Turing rolls) and the soliton limit in microring resonators. Optica 6, 1220–1232 (2019).

    ADS  Google Scholar 

  35. Bao, C. et al. Spatial mode-interaction induced single soliton generation in microresonators. Optica 4, 1011–1015 (2017).

    ADS  Google Scholar 

  36. Leisman, K. P., Zhou, D., Banks, J. W., Kovačič, G. & Cai, D. Effective dispersion in the focusing nonlinear Schrödinger equation. Phys. Rev. E 100, 022215 (2019).

    ADS  MathSciNet  Google Scholar 

  37. Grigoriev, V. & Biancalana, F. Resonant self-pulsations in coupled nonlinear microcavities. Phys. Rev. A 83, 043816 (2011).

    ADS  Google Scholar 

  38. Abdollahi, S. & Van, V. Analysis of optical instability in coupled microring resonators. J. Opt. Soc. Am. B 31, 3081–3087 (2014).

    ADS  Google Scholar 

  39. Kelly, S. Characteristic sideband instability of periodically amplified average soliton. Electron. Lett. 28, 806–807 (1992).

    ADS  Google Scholar 

  40. Benton, C. J., Gorbach, A. V. & Skryabin, D. V. Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires. Phys. Rev. A 78, 033818 (2008).

    ADS  Google Scholar 

  41. Gorbach, A. V. et al. Spatiotemporal nonlinear optics in arrays of subwavelength waveguides. Phys. Rev. A 82, 041802 (2010).

    ADS  Google Scholar 

  42. Pfeiffer, M. H. P. et al. Ultra-smooth silicon nitride waveguides based on the Damascene reflow process: fabrication and loss origins. Optica 5, 884–892 (2018).

    ADS  Google Scholar 

  43. Herr, T. et al. Universal formation dynamics and noise of Kerr-frequency combs in microresonators. Nat. Photon. 6, 480–487 (2012).

    ADS  Google Scholar 

  44. Spencer, D. T. et al. An optical-frequency synthesizer using integrated photonics. Nature 557, 81–85 (2018).

    ADS  Google Scholar 

  45. Miller, S. A. et al. Tunable frequency combs based on dual microring resonators. Opt. Express 23, 21527–21540 (2015).

    ADS  Google Scholar 

  46. Guo, H. et al. Intermode breather solitons in optical microresonators. Phys. Rev. X 7, 041055 (2017).

    Google Scholar 

  47. Marin-Palomo, P. et al. Microresonator-based solitons for massively parallel coherent optical communications. Nature 546, 274–279 (2017).

    ADS  Google Scholar 

  48. McMahon, P. L. et al. A fully programmable 100-spin coherent Ising machine with all-to-all connections. Science 354, 614–617 (2016).

    ADS  MathSciNet  Google Scholar 

  49. Shen, Y. et al. Deep learning with coherent nanophotonic circuits. Nat. Photon. 11, 441–446 (2017).

    ADS  Google Scholar 

  50. Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).

    ADS  MathSciNet  Google Scholar 

  51. Yao, B. et al. Gate-tunable frequency combs in graphene–nitride microresonators. Nature 558, 410–414 (2018).

    ADS  Google Scholar 

  52. Hodaei, H., Miri, M.-A., Heinrich, M., Christodoulides, D. N. & Khajavikhan, M. Parity-time–symmetric microring lasers. Science 346, 975–978 (2014).

    ADS  Google Scholar 

  53. Pfeiffer, M. H. P. et al. Photonic Damascene process for integrated high-Q microresonator based nonlinear photonics. Optica 3, 20–25 (2016).

    ADS  Google Scholar 

  54. Liu, J. et al. Ultralow-power chip-based soliton microcombs for photonic integration. Optica 5, 1347–1353 (2018).

    ADS  Google Scholar 

  55. Liu, J. et al. Double inverse nanotapers for efficient light coupling to integrated photonic devices. Opt. Lett. 43, 3200–3203 (2018).

    ADS  Google Scholar 

  56. Del’Haye, P., Arcizet, O., Gorodetsky, M. L., Holzwarth, R. & Kippenberg, T. J. Frequency comb assisted diode laser spectroscopy for measurement of microcavity dispersion. Nat. Photon. 3, 529–533 (2009).

    ADS  Google Scholar 

  57. Liu, J. et al. Frequency-comb-assisted broadband precision spectroscopy with cascaded diode lasers. Opt. Lett. 41, 3134–3137 (2016).

    ADS  Google Scholar 

  58. Baney, D. M., Szafraniec, B. & Motamedi, A. Coherent optical spectrum analyzer. IEEE Photon. Technol. Lett. 14, 355–357 (2002).

    ADS  Google Scholar 

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The authors thank A. Tusnin and M. Karpov for fruitful discussions. This publication was supported by contract 18AC00032 (DRINQS) from the Defense Advanced Research Projects Agency (DARPA), Defense Sciences Office (DSO). This material is based on the work supported by the Air Force Office of Scientific Research under award no. FA9550-19-1-0250. This work was further supported by the European Union’s Horizon 2020 Programme for Research and Innovation under grant no. 846737 (Marie Skłodowska-Curie IF CoSiLiS), 812818 (Marie Skłodowska-Curie ETN MICROCOMB), 722923 (Marie Skłodowska-Curie ETN OMT) and 732894 (FET Proactive HOT), and by the Swiss National Science Foundation under grant agreement 192293. Si3N4 samples were fabricated and grown in the Center of MicroNanotechnology (CMi) at EPFL. Microheater integration was performed at the Binnig and Rohrer Nanotechnology Center at IBM Research, Europe.

Author information

Authors and Affiliations



T.J.K. initiated the study. A.T. and K.K. developed the idea and performed theoretical and numerical analysis with the assistance of H.G. J.R., K.K. and M.C. performed the experiments and data analysis with the assistance of C.S. and A.T. Metal heaters were fabricated by S.H. R.N.W. and J.L. fabricated the Si3N4 samples. P.S. and T.J.K. supervised the project. A.T. and J.R. wrote the manuscript with contributions from all the authors.

Corresponding authors

Correspondence to A. Tikan or T. J. Kippenberg.

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The authors declare no competing interests.

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Peer review information Nature Physics thanks Aurélien Coillet and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Numerical reconstruction of the dimer phase space.

(left) Superposition of interactivity power traces. For every value of the pump power there are 10 interactivity power traces superimposed. (right) Three examples for three different pump powers: 0.9, 1.2 and 1.5 W.

Extended Data Fig. 2 Linear spectroscopy of the photonic dimers.

(a,b) Normalized frequency dependent transmission of top and bottom waveguides of a photonic dimer. (c) Superimposed echellogram of photonic dimer normalized waveguide transmissions. Successive transmission lines are recessed by the cavity free spectral range of 181.8 GHz, starting at 186 THz in the bottom and ending at 203 THz in the top. Avoided mode crossings (AMX), that is scattering into transverse higher-order dimer modes, are much stronger on the symmetric dimer mode (S mode, left parabola) than on the anti-symmetric dimer modes (AS, right parabola). (d,e) Zoomed in transmission traces of top and bottom waveguide transmissions with fitted coupling, detuning and loss parameters.

Extended Data Fig. 3 Experimental setup for Kerr combs reconstruction of the photonic dimer solitons.

(a) External cavity diode laser (ECDL); phase modulator (PM); erbium doped fiber amplifier (EDFA); Fiber polarization controller (FPC); Optical circulator (CIRC); Vector network analyzer (VNA); Fiber Bragg grating (FBG); Optical spectrum analyzer (OSA); Electrical spectrum analyzer (ESA); low-pass filter (LP); Logarithmic amplifier (LA); Sampling oscilloscope (OSC) (b) Optical spectrum measured with the grating-based optical spectrum analyser. (c) Calibrated Kerr comb reconstruction trace of long wavelength (red) and short wavelength (green) ECDL scans. (d) Zoom into the overlap region of the two laser scans. Grey area indicates spectral region highlighted in the panel below. (e) Kerr comb reconstruction spectrogram. The offset frequency marks the frequency difference of the incommensurable dispersive wave from the soliton frequency comb. (f) Zoom into offset regions around incommensurable dispersive waves (top and bottom) and soliton frequency comb. The spectral precision of Kerr comb reconstruction is limited by slow drifts of the pump laser and photonic dimer frequency.

Extended Data Fig. 4 Numerical investigation of the influence of the inter-resonator coupling dependence on the mode number.

Emergence of incommensurate dispersive waves is observed when the inter-resonator detuning is equal to zero which corresponds to the enhanced efficiency of the even inter-mode processes. (a) Power spectral density of the intraresonator field. (b) Supermode decomposition. (c) Reconstructed nonlinear dispersion relation.

Extended Data Fig. 5 Frequency dependent cavity dissipation and coupling rates of the photonic dimers.

Frequency-dependent results of photonic dimer fitting. (a,c) Loaded cavity loss rates κAS/2π and external κS/2π of symmetric and antisymmetric mode families for sample 1 and 2. (b,d) Linear coupling rate J and relative detuning δ of fundamental resonator modes for samples 1 and 2.

Supplementary information

Supplementary Information

Supplementary Sections 1–3 and Figs. 1–3.

Source data

Source Data Fig. 1

Preprocessed data for the reconstruction of Fig. 1b.

Source Data Fig. 2

Preprocessed data for the reconstruction of Fig. 2b,e,f.

Source Data Fig. 3

Preprocessed data for the reconstruction of Fig. 3b,c,e.

Source Data Fig. 4

Preprocessed data for the reconstruction of Fig. 4b.

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Tikan, A., Riemensberger, J., Komagata, K. et al. Emergent nonlinear phenomena in a driven dissipative photonic dimer. Nat. Phys. 17, 604–610 (2021).

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