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.
This is a preview of subscription content, access via your institution
Open Access articles citing this article.
Dissipative Kerr single soliton generation with extremely high probability via spectral mode depletion
Frontiers of Optoelectronics Open Access 01 December 2022
Nature Open Access 10 August 2022
Scientific Reports Open Access 20 June 2022
Subscribe to Nature+
Get immediate online access to Nature and 55 other Nature journal
Subscribe to Journal
Get full journal access for 1 year
only $8.25 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Get time limited or full article access on ReadCube.
All prices are NET prices.
Numerical codes used in this study are available from the corresponding authors upon reasonable request.
Bak, P. How Nature Works: The Science of Self-Organized Criticality (Copernicus, 1996).
Anderson, P. W. More is different. Science 177, 393–396 (1972).
Schulman, L. S. & Seiden, P. E. Percolation and galaxies. Science 233, 425–431 (1986).
Chialvo, D. R. Emergent complex neural dynamics. Nat. Phys. 6, 744–750 (2010).
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).
Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).
Nishimori, H. & Ortiz, G. Elements of Phase Transitions and Critical Phenomena (OUP, 2011).
Prigogine, I. & Lefever, R. Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48, 1695–1700 (1968).
Akhmediev, N. & Ankiewicz, A. Dissipative Solitons: From Optics to Biology and Medicine Vol. 751 (Springer Science & Business Media, 2008).
Haus, H. A. Mode-locking of lasers. IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
Grelu, P. & Akhmediev, N. Dissipative solitons for mode-locked lasers. Nat. Photon. 6, 84–92 (2012).
Cundiff, S. T. & Ye, J. Colloquium: femtosecond optical frequency combs. Rev. Mod. Phys. 75, 325–342 (2003).
Wright, L. G. et al. Mechanisms of spatiotemporal mode-locking. Nat. Phys. 16, 565–570 (2020).
Kippenberg, T. J., Gaeta, A. L., Lipson, M. & Gorodetsky, M. L. Dissipative Kerr solitons in optical microresonators. Science 361, eaan8083 (2018).
Herr, T. et al. Temporal solitons in optical microresonators. Nat. Photon. 8, 145–152 (2014).
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).
Brasch, V. et al. Photonic chip–based optical frequency comb using soliton Cherenkov radiation. Science 351, 357–360 (2016).
Guo, H. et al. Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators. Nat. Phys. 13, 94–102 (2017).
Lucas, E., Karpov, M., Guo, H., Gorodetsky, M. L. & Kippenberg, T. J. Breathing dissipative solitons in optical microresonators. Nat. Commun. 8, 736 (2017).
Cole, D. C. et al. Kerr-microresonator solitons from a chirped background. Optica 5, 1304–1310 (2018).
Karpov, M. et al. Dynamics of soliton crystals in optical microresonators. Nat. Phys. 15, 1071–1077 (2019).
Karpov, M. et al. Raman self-frequency shift of dissipative Kerr solitons in an optical microresonator. Phys. Rev. Lett. 116, 103902 (2016).
Herr, T. et al. Mode spectrum and temporal soliton formation in optical microresonators. Phys. Rev. Lett. 113, 1–6 (2014).
Xue, X. et al. Mode-locked dark pulse Kerr combs in normal-dispersion microresonators. Nat. Photon. 9, 594–600 (2015).
Yi, X. et al. Single-mode dispersive waves and soliton microcomb dynamics. Nat. Commun. 8, 14869 (2017).
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).
Zhang, M. et al. Electronically programmable photonic molecule. Nat. Photon. 13, 36–40 (2019).
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).
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).
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).
Jang, J. K. et al. Synchronization of coupled optical microresonators. Nat. Photon. 12, 688–693 (2018).
Xue, X., Zheng, X. & Zhou, B. Super-efficient temporal solitons in mutually coupled optical cavities. Nat. Photon. 13, 616–622 (2019).
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).
Qi, Z. et al. Dissipative cnoidal waves (Turing rolls) and the soliton limit in microring resonators. Optica 6, 1220–1232 (2019).
Bao, C. et al. Spatial mode-interaction induced single soliton generation in microresonators. Optica 4, 1011–1015 (2017).
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).
Grigoriev, V. & Biancalana, F. Resonant self-pulsations in coupled nonlinear microcavities. Phys. Rev. A 83, 043816 (2011).
Abdollahi, S. & Van, V. Analysis of optical instability in coupled microring resonators. J. Opt. Soc. Am. B 31, 3081–3087 (2014).
Kelly, S. Characteristic sideband instability of periodically amplified average soliton. Electron. Lett. 28, 806–807 (1992).
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).
Gorbach, A. V. et al. Spatiotemporal nonlinear optics in arrays of subwavelength waveguides. Phys. Rev. A 82, 041802 (2010).
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).
Herr, T. et al. Universal formation dynamics and noise of Kerr-frequency combs in microresonators. Nat. Photon. 6, 480–487 (2012).
Spencer, D. T. et al. An optical-frequency synthesizer using integrated photonics. Nature 557, 81–85 (2018).
Miller, S. A. et al. Tunable frequency combs based on dual microring resonators. Opt. Express 23, 21527–21540 (2015).
Guo, H. et al. Intermode breather solitons in optical microresonators. Phys. Rev. X 7, 041055 (2017).
Marin-Palomo, P. et al. Microresonator-based solitons for massively parallel coherent optical communications. Nature 546, 274–279 (2017).
McMahon, P. L. et al. A fully programmable 100-spin coherent Ising machine with all-to-all connections. Science 354, 614–617 (2016).
Shen, Y. et al. Deep learning with coherent nanophotonic circuits. Nat. Photon. 11, 441–446 (2017).
Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).
Yao, B. et al. Gate-tunable frequency combs in graphene–nitride microresonators. Nature 558, 410–414 (2018).
Hodaei, H., Miri, M.-A., Heinrich, M., Christodoulides, D. N. & Khajavikhan, M. Parity-time–symmetric microring lasers. Science 346, 975–978 (2014).
Pfeiffer, M. H. P. et al. Photonic Damascene process for integrated high-Q microresonator based nonlinear photonics. Optica 3, 20–25 (2016).
Liu, J. et al. Ultralow-power chip-based soliton microcombs for photonic integration. Optica 5, 1347–1353 (2018).
Liu, J. et al. Double inverse nanotapers for efficient light coupling to integrated photonic devices. Opt. Lett. 43, 3200–3203 (2018).
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).
Liu, J. et al. Frequency-comb-assisted broadband precision spectroscopy with cascaded diode lasers. Opt. Lett. 41, 3134–3137 (2016).
Baney, D. M., Szafraniec, B. & Motamedi, A. Coherent optical spectrum analyzer. IEEE Photon. Technol. Lett. 14, 355–357 (2002).
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.
The authors declare no competing interests.
Peer review information Nature Physics thanks Aurélien Coillet and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
(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.
(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.
About this article
Cite this article
Tikan, A., Riemensberger, J., Komagata, K. et al. Emergent nonlinear phenomena in a driven dissipative photonic dimer. Nat. Phys. 17, 604–610 (2021). https://doi.org/10.1038/s41567-020-01159-y
This article is cited by
Scientific Reports (2022)
Nature Communications (2022)
Dissipative Kerr single soliton generation with extremely high probability via spectral mode depletion
Frontiers of Optoelectronics (2022)
Towards integrated photonic interposers for processing octave-spanning microresonator frequency combs
Light: Science & Applications (2021)