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Bloch oscillations of coherently driven dissipative solitons in a synthetic dimension

Nature Physics volume 19, pages 1014–1021 (2023)Cite this article

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Abstract

The engineering of synthetic dimensions allows for the construction of fictitious lattice structures by coupling the discrete degrees of freedom of a physical system. This method enables the study of static and dynamical Bloch band properties in the absence of a real periodic lattice structure. In that context, the potentially rich physics and opportunities offered by non-linearities and dissipation have remained largely unexplored. Here we investigate the complex interplay between Bloch band transport, non-linearity and dissipation, exploring how a synthetic dimension realized in the frequency space of a coherently driven optical resonator influences the dynamics of the system. We observe and study non-linear dissipative Bloch oscillations along the synthetic frequency dimension, sustained by localized dissipative structures (solitons) that persist in the resonator. The unique properties of the coherently driven dissipative soliton states can extend the effective size of the synthetic dimension far beyond that achieved in the linear regime, as well as enable long-lived Bloch oscillations and high-resolution probing of the underlying band structure.

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Fig. 1: Concept.
Fig. 2: Synthetic band spectroscopy in the non-linear regime.
Fig. 3: BOs of dissipative solitons.

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Source data are available for this paper. 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.

References

  1. Leo, K. Interband optical investigation of Bloch oscillations in semiconductor superlattices. Semicond. Sci. Technol. 13, 249–263 (1998).

    ADS  Google Scholar 

  2. Dahan, M. B., Peik, E., Reichel, J., Castin, Y. & Salomon, C. Bloch oscillations of atoms in an optical potential. Phys. Rev. Lett. 76, 4508 (1996).

    ADS  Google Scholar 

  3. Morandotti, R., Peschel, U., Aitchison, J., Eisenberg, H. & Silberberg, Y. Experimental observation of linear and nonlinear optical Bloch oscillations. Phys. Rev. Lett. 83, 4756 (1999).

    ADS  Google Scholar 

  4. Bersch, C., Onishchukov, G. & Peschel, U. Experimental observation of spectral Bloch oscillations. Opt. Lett. 34, 2372–2374 (2009).

    ADS  Google Scholar 

  5. Chen, H. et al. Real-time observation of frequency Bloch oscillations with fibre loop modulation. Light Sci. Appl. 10, 48 (2021).

    Google Scholar 

  6. Szameit, A. & Nolte, S. Discrete optics in femtosecond-laser-written photonic structures. J. Phys. B 43, 163001 (2010).

    ADS  Google Scholar 

  7. Bloch, I., Dalibard, J. & Nascimbene, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012).

    Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

  9. Efremidis, N. K. & Christodoulides, D. N. Bloch oscillations in optical dissipative lattices. Opt. Lett. 29, 2485–2487 (2004).

    ADS  Google Scholar 

  10. Trombettoni, A. & Smerzi, A. Discrete solitons and breathers with dilute Bose–Einstein condensates. Phys. Rev. Lett. 86, 2353 (2001).

    ADS  Google Scholar 

  11. Konotop, V. & Salerno, M. Modulational instability in Bose–Einstein condensates in optical lattices. Phys. Rev. A 65, 021602 (2002).

    ADS  Google Scholar 

  12. Fallani, L. et al. Observation of dynamical instability for a Bose–Einstein condensate in a moving 1D optical lattice. Phys. Rev. Lett. 93, 140406 (2004).

    ADS  Google Scholar 

  13. Salerno, M., Konotop, V. & Bludov, Y. V. Long-living Bloch oscillations of matter waves in periodic potentials. Phys. Rev. Lett. 101, 030405 (2008).

    ADS  Google Scholar 

  14. Gaul, C., Lima, R., Díaz, E., Müller, C. & Domínguez-Adame, F. Stable Bloch oscillations of cold atoms with time-dependent interaction. Phys. Rev. Lett. 102, 255303 (2009).

    ADS  Google Scholar 

  15. Longstaff, B. & Graefe, E.-M. Bloch oscillations in a Bose–Hubbard chain with single-particle losses. J. Phys. B 53, 195302 (2020).

    ADS  Google Scholar 

  16. Atala, M. et al. Direct measurement of the Zak phase in topological Bloch bands. Nat. Phys. 9, 795–800 (2013).

    Google Scholar 

  17. Wimmer, M., Price, H. M., Carusotto, I. & Peschel, U. Experimental measurement of the Berry curvature from anomalous transport. Nat. Phys. 13, 545–550 (2017).

    Google Scholar 

  18. Wintersperger, K. et al. Realization of an anomalous Floquet topological system with ultracold atoms. Nat. Phys. 16, 1058–1063 (2020).

    Google Scholar 

  19. Li, T. et al. Bloch state tomography using Wilson lines. Science 352, 1094–1097 (2016).

    ADS  Google Scholar 

  20. Hoeller, J. & Alexandradinata, A. Topological Bloch oscillations. Phys. Rev. B 98, 024310 (2018).

    ADS  Google Scholar 

  21. Di Liberto, M., Goldman, N. & Palumbo, G. Non-abelian Bloch oscillations in higher-order topological insulators. Nat. Commun. 11, 1–9 (2020).

    Google Scholar 

  22. Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).

    Google Scholar 

  23. Jukić, D. & Buljan, H. Four-dimensional photonic lattices and discrete tesseract solitons. Phys. Rev. A 87, 013814 (2013).

    ADS  Google Scholar 

  24. Celi, A. et al. Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001 (2014).

    ADS  Google Scholar 

  25. Mancini, M. et al. Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. Science 349, 1510–1513 (2015).

    MathSciNet  MATH  ADS  Google Scholar 

  26. Stuhl, B., Lu, H.-I., Aycock, L., Genkina, D. & Spielman, I. Visualizing edge states with an atomic Bose gas in the quantum Hall regime. Science 349, 1514–1518 (2015).

    MathSciNet  MATH  ADS  Google Scholar 

  27. Chalopin, T. et al. Probing chiral edge dynamics and bulk topology of a synthetic Hall system. Nat. Phys. 16, 1017–1021 (2020).

    Google Scholar 

  28. Bell, B. A. et al. Spectral photonic lattices with complex long-range coupling. Optica 4, 1433–1436 (2017).

    ADS  Google Scholar 

  29. Yuan, L., Lin, Q., Xiao, M. & Fan, S. Synthetic dimension in photonics. Optica 5, 1396–1405 (2018).

    ADS  Google Scholar 

  30. Lustig, E. & Segev, M. Topological photonics in synthetic dimensions. Adv. Opt. Photon. 13, 426–461 (2021).

    Google Scholar 

  31. Dutt, A. et al. Experimental band structure spectroscopy along a synthetic dimension. Nat. Commun. 10, 3122 (2019).

    ADS  Google Scholar 

  32. Wang, K. et al. Multidimensional synthetic chiral-tube lattices via nonlinear frequency conversion. Light Sci. Appl. 9, 132 (2020).

    Google Scholar 

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

    Google Scholar 

  34. Moille, G., Menyuk, C., Chembo, Y. K., Dutt, A. & Srinivasan, K. Synthetic frequency lattices from an integrated dispersive multi-color soliton. Preprint at http://arxiv.org/abs/2210.09036 (2022).

  35. Dutt, A. et al. A single photonic cavity with two independent physical synthetic dimensions. Science 367, 59–64 (2020).

    ADS  Google Scholar 

  36. Balčytis, A. et al. Synthetic dimension band structures on a Si CMOS photonic platform. Sci. Adv. 28, 1–9 (2022).

    Google Scholar 

  37. Senanian, A., Wright, L. G., Wade, P. F., Doyle, H. K. & McMahon, P. L. Programmable large-scale simulation of bosonic transport in optical synthetic frequency lattices. Preprint at http://arxiv.org/abs/2208.05088 (2022).

  38. Wang, K. et al. Generating arbitrary topological windings of a non-Hermitian band. Science 371, 1240–1245 (2021).

    ADS  Google Scholar 

  39. Dutt, A. et al. A single photonic cavity with two independent physical synthetic dimensions. Science 367, 59–64 (2020).

    ADS  Google Scholar 

  40. Yuan, L. et al. Creating locally interacting Hamiltonians in the synthetic frequency dimension for photons. Photonics Res. 8, B8–B14 (2020).

    Google Scholar 

  41. Smirnova, D., Leykam, D., Chong, Y. & Kivshar, Y. Nonlinear topological photonics. Appl. Phys. Rev. 7, 021306 (2020).

    ADS  Google Scholar 

  42. Xia, S. et al. Nonlinear tuning of PT symmetry and non-Hermitian topological states. Science 372, 72–76 (2021).

    ADS  Google Scholar 

  43. Jürgensen, M., Mukherjee, S. & Rechtsman, M. C. Quantized nonlinear Thouless pumping. Nature 596, 63–67 (2021).

    ADS  Google Scholar 

  44. Pernet, N. et al. Gap solitons in a one-dimensional driven-dissipative topological lattice. Nat. Phys. 18, 678–684 (2022).

    Google Scholar 

  45. Haelterman, M., Trillo, S. & Wabnitz, S. Dissipative modulation instability in a nonlinear dispersive ring cavity. Opt. Commun. 91, 401–407 (1992).

    ADS  Google Scholar 

  46. Coen, S. & Haelterman, M. Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber. Phys. Rev. Lett. 79, 4139–4142 (1997).

    ADS  Google Scholar 

  47. Coen, S. & Haelterman, M. Continuous-wave ultrahigh-repetition-rate pulse-train generation through modulational instability in a passive fiber cavity. Opt. Lett. 26, 39–41 (2001).

    ADS  Google Scholar 

  48. Leo, F. et al. Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer. Nat. Photonics 4, 471–476 (2010).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  50. Jang, J. K., Erkintalo, M., Coen, S. & Murdoch, S. G. Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons. Nat. Commun. 6, 7370 (2015).

    ADS  Google Scholar 

  51. Pasquazi, A. et al. Micro-combs: a novel generation of optical sources. Phys. Rep. 729, 1–81 (2018).

    MathSciNet  MATH  ADS  Google Scholar 

  52. Nielsen, A. U. et al. Nonlinear localization of dissipative modulation instability. Phys. Rev. Lett. 127, 123901 (2021).

    ADS  Google Scholar 

  53. Xu, G. et al. Spontaneous symmetry breaking of dissipative optical solitons in a two-component Kerr resonator. Nat. Commun. 12, 4023 (2021).

    ADS  Google Scholar 

  54. Englebert, N. et al. Parametrically driven Kerr cavity solitons. Nat. Photonics 15, 857–861 (2021).

    ADS  Google Scholar 

  55. Englebert, N., Mas Arabí, C., Parra-Rivas, P., Gorza, S.-P. & Leo, F. Temporal solitons in a coherently driven active resonator. Nat. Photonics 15, 536–541 (2021).

    ADS  Google Scholar 

  56. Erkintalo, M., Murdoch, S. G. & Coen, S. Phase and intensity control of dissipative Kerr cavity solitons. J. R. Soc. N. Z. 52, 149–167 (2022).

    Google Scholar 

  57. Del’Haye, P. et al. Optical frequency comb generation from a monolithic microresonator. Nature 450, 1214–1217 (2007).

  58. Coen, S. & Erkintalo, M. Universal scaling laws of Kerr frequency combs. Opt. Lett. 38, 1790–1792 (2013).

    ADS  Google Scholar 

  59. Englebert, N. et al. Observation of temporal cavity solitons in a synthetic photonic lattice. In 2021 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference (Optical Society of America, 2021).

  60. Hu, Y. et al. Realization of high-dimensional frequency crystals in electro-optic microcombs. Optica 7, 1189–1194 (2020).

    ADS  Google Scholar 

  61. Tusnin, A. K., Tikan, A. M. & Kippenberg, T. J. Nonlinear states and dynamics in a synthetic frequency dimension. Phys. Rev. A 102, 023518 (2020).

    ADS  Google Scholar 

  62. Lugiato, L. A. & Lefever, R. Spatial dissipative structures in passive optical systems. Phys. Rev. Lett. 58, 2209–2211 (1987).

    MathSciNet  ADS  Google Scholar 

  63. Zhang, M. et al. Broadband electro-optic frequency comb generation in a lithium niobate microring resonator. Nature 568, 373–377 (2019).

    ADS  Google Scholar 

  64. Goda, K. & Jalali, B. Dispersive Fourier transformation for fast continuous single-shot measurements. Nat. Photonics 7, 102–112 (2013).

    ADS  Google Scholar 

  65. Mahjoubfar, A. et al. Time stretch and its applications. Nat. Photonics 11, 341–351 (2017).

    ADS  Google Scholar 

  66. He, Y. et al. High-speed tunable microwave-rate soliton microcomb. Preprint at http://arxiv.org/abs/2208.08046 (2022).

  67. Matsko, A. B. & Maleki, L. On timing jitter of mode locked Kerr frequency combs. Opt. Express 21, 28862–28876 (2013).

    ADS  Google Scholar 

  68. Yi, X., Yang, Q.-F., Yang, K. Y. & Vahala, K. Theory and measurement of the soliton self-frequency shift and efficiency in optical microcavities. Opt. Lett. 41, 3419–3422 (2016).

    ADS  Google Scholar 

  69. Del’Haye, P., Beha, K., Papp, S. B. & Diddams, S. A. Self-injection locking and phase-locked states in microresonator-based optical frequency combs. Phys. Rev. Lett. 112, 043905 (2014).

    ADS  Google Scholar 

  70. Wimmer, M. et al. Observation of optical solitons in PT-symmetric lattices. Nat. Commun. 6, 7782 (2015).

    ADS  Google Scholar 

  71. Mittal, S., Moille, G., Srinivasan, K., Chembo, Y. K. & Hafezi, M. Topological frequency combs and nested temporal solitons. Nat. Phys. 17, 1169–1176 (2021).

    Google Scholar 

  72. Hu, Y. et al. High-efficiency and broadband on-chip electro-optic frequency comb generators. Nat. Photonics 16, 679–685 (2022).

    ADS  Google Scholar 

  73. Li, Z., Xu, Y., Coen, S., Murdoch, S. G. & Erkintalo, M. Experimental observations of bright dissipative cavity solitons and their collapsed snaking in a Kerr resonator with normal dispersion driving. Optica 7, 1195–1203 (2020).

    ADS  Google Scholar 

  74. Smirnov, S. et al. Layout of NALM fiber laser with adjustable peak power of generated pulses. Opt. Lett. 42, 1732–1735 (2017).

    ADS  Google Scholar 

  75. Oliver, C. et al. Bloch oscillations along a synthetic dimension of atomic trap states. Preprint at http://arxiv.org/abs/2112.10648 (2021).

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Acknowledgements

We thank W. Liao for checking the Lagrangian derivation in this paper. This work was supported by funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme, grant agreement no. 757800 (QuadraComb), no. 716908 (TopoCold) and no. 101044957 (LATIS). The projects (40007560 and 40007526) have received funding from the FWO and F.R.S.-FNRS under the Excellence of Science programme. N.E. acknowledges the support of the Fonds pour la formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-FNRS, Belgium). F.L. and N.G. acknowledge the support of the Fonds de la Recherche Scientifique (FNRS, Belgium). J.F. acknowledges the financial support from the CNRS, IRP Wall-IN project and PO FEDER FSE Bourgogne. M.E acknowledges the Marsden Fund and the Rutherford Discovery Fellowships of The Royal Society of New Zealand Te Apārangi. N.M. acknowledges funding by the Deutsche Forschungsgemeinschaft (German Research Foundation) under Germany’s Excellence Strategy – EXC-2111 – 390814868 and via Research Unit FOR 2414 under project no. 277974659.

Author information

Authors and Affiliations

  1. Service OPERA-Photonique, Université libre de Bruxelles, Brussels, Belgium

    Nicolas Englebert, Simon-Pierre Gorza & François Leo

  2. CENOLI, Université libre de Bruxelles, Brussels, Belgium

    Nathan Goldman & Nader Mostaan

  3. Department of Physics, University of Auckland, Auckland, New Zealand

    Miro Erkintalo & Julien Fatome

  4. The Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin, New Zealand

    Miro Erkintalo & Julien Fatome

  5. Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC), Ludwig-Maximilians-Universität München, Munich, Germany

    Nader Mostaan

  6. Munich Center for Quantum Science and Technology (MCQST), Munich, Germany

    Nader Mostaan

  7. Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS, Université de Bourgogne, Dijon, France

    Julien Fatome

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  1. Nicolas Englebert
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Contributions

All authors contributed to the conception of the research and the analysis and interpretation of the results. N.E. performed the experiments and derived the reduced model with supervision from S.-P.G. and F.L. N.E., J.F. and M.E. performed simulations of the LLE and the reduced model. N.E., N.G., M.E. and J.F. prepared the manuscript, with input from all authors.

Corresponding author

Correspondence to Nicolas Englebert.

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Competing interests

N.E., S.-P.G. and F.L. have filed patent applications on the active resonator design and its use for frequency conversion (European patent office, application no. EP20188731.2). The remaining authors declare no competing interests.

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Nature Physics thanks Avik Dutt, Domenico Bongiovanni and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Sects. A–G.

Source data

Source Data Fig. 2

Soliton state optical spectrum experimental data and linear state optical spectrum experimental data.

Source Data Fig. 3

Soliton BO amplitude experimental data and soliton BO period experimental data.

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Englebert, N., Goldman, N., Erkintalo, M. et al. Bloch oscillations of coherently driven dissipative solitons in a synthetic dimension. Nat. Phys. 19, 1014–1021 (2023). https://doi.org/10.1038/s41567-023-02005-7

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