Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Non-equilibrium spectral phase transitions in coupled nonlinear optical resonators

Nature Physics volume 19, pages 427–434 (2023)Cite this article

Subjects

Abstract

Coupled systems with multiple interacting degrees of freedom provide a fertile ground for emergent dynamics, which is otherwise inaccessible in their solitary counterparts. Here we show that coupled nonlinear optical resonators can undergo self-organization in their spectrum leading to a first-order phase transition. We experimentally demonstrate such a spectral phase transition in time-multiplexed coupled optical parametric oscillators. We switch the nature of mutual coupling from dispersive to dissipative and access distinct spectral regimes of the parametric oscillator dimer. We observe abrupt spectral discontinuity at the first-order transition point. Furthermore, we show how non-equilibrium phase transitions can lead to enhanced sensing, where the applied perturbation is not resolvable by the underlying linear system. Our approach could be exploited for sensing applications that use nonlinear driven-dissipative systems, leading to performance enhancements without sacrificing sensitivity.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Non-equilibrium phase transitions in single and coupled OPOs.
Fig. 2: First-order spectral phase transition in coupled OPOs.
Fig. 3: Supermodes of the coupled OPOs.
Fig. 4: Dispersive versus dissipative coupling.
Fig. 5: Enhanced sensing using non-equilibrium phase transitions.
Fig. 6: Sensitivity near the spectral phase transition points.

Similar content being viewed by others

Data availability

Source data are available for this paper and can be found at https://doi.org/10.6084/m9.figshare.21252147. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The codes that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’networks. Nature 393, 440–442 (1998).

    MATH  ADS  Google Scholar 

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

    Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  5. Zhang, Y. et al. Squeezed light from a nanophotonic molecule. Nat. Commun. 12, 2233 (2021).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  8. Roy, A. et al. Nondissipative non-Hermitian dynamics and exceptional points in coupled optical parametric oscillators. Optica 8, 415–421 (2021).

    ADS  Google Scholar 

  9. Okawachi, Y. et al. Demonstration of chip-based coupled degenerate optical parametric oscillators for realizing a nanophotonic spin-glass. Nat. Commun. 11, 4119 (2020).

    ADS  Google Scholar 

  10. Guo, X. et al. Distributed quantum sensing in a continuous-variable entangled network. Nat. Phys. 16, 281–284 (2020).

    Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

  12. Marandi, A., Wang, Z., Takata, K., Byer, R. L. & Yamamoto, Y. Network of time-multiplexed optical parametric oscillators as a coherent ising machine. Nat. Photonics 8, 937–942 (2014).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  14. Fruchart, M., Hanai, R., Littlewood, P. B. & Vitelli, V. Non-reciprocal phase transitions. Nature 592, 363–369 (2021).

    ADS  Google Scholar 

  15. Roy, A., Jahani, S., Langrock, C., Fejer, M. & Marandi, A. Spectral phase transitions in optical parametric oscillators. Nat. Commun. 12, 835 (2021).

    ADS  Google Scholar 

  16. Haken, H. Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems. Rev. Mod. Phys. 47, 67 (1975).

    MathSciNet  ADS  Google Scholar 

  17. Vaupel, M., Maitre, A. & Fabre, C. Observation of pattern formation in optical parametric oscillators. Phys. Rev. Lett. 83, 5278 (1999).

    ADS  Google Scholar 

  18. Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851 (1993).

    MATH  ADS  Google Scholar 

  19. Ropp, C., Bachelard, N., Barth, D., Wang, Y. & Zhang, X. Dissipative self-organization in optical space. Nat. Photonics 12, 739–743 (2018).

    ADS  Google Scholar 

  20. Taranenko, V. B., Staliunas, K. & Weiss, C. O. Pattern formation and localized structures in degenerate optical parametric mixing. Phys. Rev. Lett. 81, 2236 (1998).

    ADS  Google Scholar 

  21. Oppo, G.-L., Yao, A. M. & Cuozzo, D. Self-organization, pattern formation, cavity solitons, and rogue waves in singly resonant optical parametric oscillators. Phys. Rev. A 88, 043813 (2013).

    ADS  Google Scholar 

  22. Wu, F. O., Hassan, A. U. & Christodoulides, D. N. Thermodynamic theory of highly multimoded nonlinear optical systems. Nat. Photonics 13, 776–782 (2019).

    ADS  Google Scholar 

  23. Turing, A. M. The chemical basis of morphogenesis. Bull. Math. Biol. 52, 153–197 (1990).

    Google Scholar 

  24. DeGiorgio, V. & Scully, M. O. Analogy between the laser threshold region and a second-order phase transition. Phys. Rev. A 2, 1170 (1970).

    ADS  Google Scholar 

  25. Wright, L. G., Christodoulides, D. N. & Wise, F. W. Spatiotemporal mode-locking in multimode fiber lasers. Science 358, 94–97 (2017).

    ADS  Google Scholar 

  26. Gordon, A. & Fischer, B. Phase transition theory of many-mode ordering and pulse formation in lasers. Phys. Rev. Lett. 89, 103901 (2002).

    ADS  Google Scholar 

  27. Stanley, H. E. Phase Transitions and Critical Phenomena. (Clarendon Press, 1971).

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

    ADS  Google Scholar 

  29. Wilczek, F. Quantum time crystals. Phys. Rev. Lett. 109, 160401 (2012).

    ADS  Google Scholar 

  30. Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016).

    ADS  Google Scholar 

  31. Dechoum, K., Rosales-Zárate, L. & Drummond, P. D. Critical fluctuations in an optical parametric oscillator: when light behaves like magnetism. J. Opt. Soc. Am. B 33, 871–883 (2016).

    ADS  Google Scholar 

  32. Drummond, P. D., McNeil, K. J. & Walls, D. F. Non-equilibrium transitions in sub/second harmonic generation. Opt. Acta 27, 321–335 (1980).

    ADS  Google Scholar 

  33. Kuznetsov, A. V. Optical bistability driven by a first order phase transition. Opt. Commun. 81, 106–111 (1991).

    ADS  Google Scholar 

  34. Gol’Tsman, G. N. et al. Picosecond superconducting single-photon optical detector. Appl. Phys. Lett. 79, 705–707 (2001).

    ADS  Google Scholar 

  35. Yang, L.-P. & Jacob, Z. Quantum critical detector: amplifying weak signals using discontinuous quantum phase transitions. Opt. Express 27, 10482–10494 (2019).

    ADS  Google Scholar 

  36. Di Candia, R., Minganti, F., Petrovnin, K. V., Paraoanu, G. S. & Felicetti, S. Critical parametric quantum sensing. Preprint at https://doi.org/2107.04503 (2021).

  37. Del Bino, L., Silver, J. M., Stebbings, S. L. & Del’Haye, P. Symmetry breaking of counter-propagating light in a nonlinear resonator. Sci. Rep. 7, 43142 (2017).

    ADS  Google Scholar 

  38. Wang, C. et al. A nonlinear microresonator refractive index sensor. J. Lightwave Technol. 33, 4360–4366 (2015).

    ADS  Google Scholar 

  39. Kaplan, A. E. & Meystre, P. Enhancement of the Sagnac effect due to nonlinearly induced nonreciprocity. Opt. Lett. 6, 590–592 (1981).

    ADS  Google Scholar 

  40. Wang, C. et al. Nonlinearly enhanced refractive index sensing in coupled optical microresonators. Opt. Lett. 39, 26–29 (2014).

    ADS  Google Scholar 

  41. Hamerly, R. et al. Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators. Phys. Rev. A 94, 063809 (2016).

    ADS  Google Scholar 

  42. Hodaei, H. et al. Enhanced sensitivity at higher-order exceptional points. Nature 548, 187–191 (2017).

    ADS  Google Scholar 

  43. Leefmans, C. et al. Topological dissipation in a time-multiplexed photonic resonator network. Nat. Phys. 18, 442–449 (2022).

    Google Scholar 

  44. Ding, J., Belykh, I., Marandi, A. & Miri, M.-A. Dispersive versus dissipative coupling for frequency synchronization in lasers. Phys. Rev. Appl. 12, 054039 (2019).

    ADS  Google Scholar 

  45. Lee, K. F. et al. Carrier envelope offset frequency of a doubly resonant, nondegenerate, mid-infrared gaas optical parametric oscillator. Opt. Lett. 38, 1191–1193 (2013).

    ADS  Google Scholar 

  46. Krioukov, E., Klunder, D. J. W., Driessen, A., Greve, J. & Otto, C. Sensor based on an integrated optical microcavity. Opt. Lett. 27, 512–514 (2002).

    ADS  Google Scholar 

  47. Heideman, R. G. & Lambeck, P. V. Remote opto-chemical sensing with extreme sensitivity: design, fabrication and performance of a pigtailed integrated optical phase-modulated mach–zehnder interferometer system. Sens. Actuators B: Chem. 61, 100–127 (1999).

    Google Scholar 

  48. Ren, J. et al. Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope. Opt. Lett. 42, 1556–1559 (2017).

    ADS  Google Scholar 

  49. Puckett, M. W. et al. 422 Million intrinsic quality factor planar integrated all-waveguide resonator with sub-mhz linewidth. Nat. Commun. 12, 934 (2021).

    ADS  Google Scholar 

  50. Jankowski, M. et al. Ultrabroadband nonlinear optics in nanophotonic periodically poled lithium niobate waveguides. Optica 7, 40–46 (2020).

    ADS  Google Scholar 

  51. Lu, J. et al. Ultralow-threshold thin-film lithium niobate optical parametric oscillator. Optica 8, 539–544 (2021).

    ADS  Google Scholar 

  52. Guo, Q. et al. Femtojoule femtosecond all-optical switching in lithium niobate nanophotonics. Nat. Photonics 16, 625–631 (2022).

    ADS  Google Scholar 

  53. Tusnin, A. K., Tikan, A. M., Komagata, K. & Kippenberg, T. J. Coherent dissipative structures in chains of coupled χ(3) resonators. Preprint at https://arxiv.org/abs/2104.11731 (2021).

  54. Longhi, S. & Geraci, A. Swift-hohenberg equation for optical parametric oscillators. Phys. Rev. A 54, 4581 (1996).

    ADS  Google Scholar 

  55. Okawachi, Y. et al. Dual-pumped degenerate kerr oscillator in a silicon nitride microresonator. Opt. Lett. 40, 5267–5270 (2015).

    ADS  Google Scholar 

  56. Wu, L.-A., Kimble, H. J., Hall, J. L. & Wu, H. Generation of squeezed states by parametric down conversion. Phys. Rev. Lett. 57, 2520 (1986).

    ADS  Google Scholar 

  57. Gatti, A. & Lugiato, L. Quantum images and critical fluctuations in the optical parametric oscillator below threshold. Phys. Rev. A 52, 1675 (1995).

    ADS  Google Scholar 

  58. Longhi, S. Nonadiabatic pattern formation in optical parametric oscillators. Phys. Rev. Lett. 84, 5756 (2000).

    ADS  Google Scholar 

  59. Menotti, M. et al. Nonlinear coupling of linearly uncoupled resonators. Phys. Rev. Lett. 122, 013904 (2019).

    ADS  Google Scholar 

  60. Langrock, C. & Fejer, M. M. Fiber-feedback continuous-wave and synchronously-pumped singly-resonant ring optical parametric oscillators using reverse-proton-exchanged periodically-poled lithium niobate waveguides. Opt. Lett. 32, 2263–2265 (2007).

    ADS  Google Scholar 

Download references

Acknowledgements

We thank S. Jahani and M. Gyun Suh for helpful discussions. We acknowledge support from ARO grant no. W911NF-18-1-0285 (A.M.), AFOSR award FA9550-20-1-0040 (A.M.), NSF grant nos. 1846273 (A.M.) and 1918549 (A.M.), and NASA. We wish to thank NTT Research for their financial and technical support.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, California Institute of Technology, Pasadena, CA, USA

    Arkadev Roy, Rajveer Nehra & Alireza Marandi

  2. Edward L. Ginzton Laboratory, Stanford University, Stanford, CA, USA

    Carsten Langrock & Martin Fejer

Authors
  1. Arkadev Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rajveer Nehra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carsten Langrock
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Martin Fejer
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Alireza Marandi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

A.R. and A.M. conceived the idea. A.R. performed the experiments with help from R.N. A.R. developed the theory and performed the numerical simulations. C.L. fabricated the PPLN waveguide used in the experiment with the supervision of M.F. A.R. and A.M. wrote the manuscript with input from all authors. A.M. supervised the project.

Corresponding author

Correspondence to Alireza Marandi.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary information containing Table 1, Figs. 1–24 and references.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Roy, A., Nehra, R., Langrock, C. et al. Non-equilibrium spectral phase transitions in coupled nonlinear optical resonators. Nat. Phys. 19, 427–434 (2023). https://doi.org/10.1038/s41567-022-01874-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-022-01874-8

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing