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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Leo, K. Interband optical investigation of Bloch oscillations in semiconductor superlattices. Semicond. Sci. Technol. 13, 249–263 (1998).
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).
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).
Bersch, C., Onishchukov, G. & Peschel, U. Experimental observation of spectral Bloch oscillations. Opt. Lett. 34, 2372–2374 (2009).
Chen, H. et al. Real-time observation of frequency Bloch oscillations with fibre loop modulation. Light Sci. Appl. 10, 48 (2021).
Szameit, A. & Nolte, S. Discrete optics in femtosecond-laser-written photonic structures. J. Phys. B 43, 163001 (2010).
Bloch, I., Dalibard, J. & Nascimbene, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012).
Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).
Efremidis, N. K. & Christodoulides, D. N. Bloch oscillations in optical dissipative lattices. Opt. Lett. 29, 2485–2487 (2004).
Trombettoni, A. & Smerzi, A. Discrete solitons and breathers with dilute Bose–Einstein condensates. Phys. Rev. Lett. 86, 2353 (2001).
Konotop, V. & Salerno, M. Modulational instability in Bose–Einstein condensates in optical lattices. Phys. Rev. A 65, 021602 (2002).
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).
Salerno, M., Konotop, V. & Bludov, Y. V. Long-living Bloch oscillations of matter waves in periodic potentials. Phys. Rev. Lett. 101, 030405 (2008).
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).
Longstaff, B. & Graefe, E.-M. Bloch oscillations in a Bose–Hubbard chain with single-particle losses. J. Phys. B 53, 195302 (2020).
Atala, M. et al. Direct measurement of the Zak phase in topological Bloch bands. Nat. Phys. 9, 795–800 (2013).
Wimmer, M., Price, H. M., Carusotto, I. & Peschel, U. Experimental measurement of the Berry curvature from anomalous transport. Nat. Phys. 13, 545–550 (2017).
Wintersperger, K. et al. Realization of an anomalous Floquet topological system with ultracold atoms. Nat. Phys. 16, 1058–1063 (2020).
Li, T. et al. Bloch state tomography using Wilson lines. Science 352, 1094–1097 (2016).
Hoeller, J. & Alexandradinata, A. Topological Bloch oscillations. Phys. Rev. B 98, 024310 (2018).
Di Liberto, M., Goldman, N. & Palumbo, G. Non-abelian Bloch oscillations in higher-order topological insulators. Nat. Commun. 11, 1–9 (2020).
Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).
Jukić, D. & Buljan, H. Four-dimensional photonic lattices and discrete tesseract solitons. Phys. Rev. A 87, 013814 (2013).
Celi, A. et al. Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001 (2014).
Mancini, M. et al. Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. Science 349, 1510–1513 (2015).
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).
Chalopin, T. et al. Probing chiral edge dynamics and bulk topology of a synthetic Hall system. Nat. Phys. 16, 1017–1021 (2020).
Bell, B. A. et al. Spectral photonic lattices with complex long-range coupling. Optica 4, 1433–1436 (2017).
Yuan, L., Lin, Q., Xiao, M. & Fan, S. Synthetic dimension in photonics. Optica 5, 1396–1405 (2018).
Lustig, E. & Segev, M. Topological photonics in synthetic dimensions. Adv. Opt. Photon. 13, 426–461 (2021).
Dutt, A. et al. Experimental band structure spectroscopy along a synthetic dimension. Nat. Commun. 10, 3122 (2019).
Wang, K. et al. Multidimensional synthetic chiral-tube lattices via nonlinear frequency conversion. Light Sci. Appl. 9, 132 (2020).
Leefmans, C. et al. Topological dissipation in a time-multiplexed photonic resonator network. Nat. Phys. 18, 442–449 (2022).
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).
Dutt, A. et al. A single photonic cavity with two independent physical synthetic dimensions. Science 367, 59–64 (2020).
Balčytis, A. et al. Synthetic dimension band structures on a Si CMOS photonic platform. Sci. Adv. 28, 1–9 (2022).
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).
Wang, K. et al. Generating arbitrary topological windings of a non-Hermitian band. Science 371, 1240–1245 (2021).
Dutt, A. et al. A single photonic cavity with two independent physical synthetic dimensions. Science 367, 59–64 (2020).
Yuan, L. et al. Creating locally interacting Hamiltonians in the synthetic frequency dimension for photons. Photonics Res. 8, B8–B14 (2020).
Smirnova, D., Leykam, D., Chong, Y. & Kivshar, Y. Nonlinear topological photonics. Appl. Phys. Rev. 7, 021306 (2020).
Xia, S. et al. Nonlinear tuning of PT symmetry and non-Hermitian topological states. Science 372, 72–76 (2021).
Jürgensen, M., Mukherjee, S. & Rechtsman, M. C. Quantized nonlinear Thouless pumping. Nature 596, 63–67 (2021).
Pernet, N. et al. Gap solitons in a one-dimensional driven-dissipative topological lattice. Nat. Phys. 18, 678–684 (2022).
Haelterman, M., Trillo, S. & Wabnitz, S. Dissipative modulation instability in a nonlinear dispersive ring cavity. Opt. Commun. 91, 401–407 (1992).
Coen, S. & Haelterman, M. Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber. Phys. Rev. Lett. 79, 4139–4142 (1997).
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).
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).
Herr, T. et al. Temporal solitons in optical microresonators. Nat. Photonics 8, 145–152 (2014).
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).
Pasquazi, A. et al. Micro-combs: a novel generation of optical sources. Phys. Rep. 729, 1–81 (2018).
Nielsen, A. U. et al. Nonlinear localization of dissipative modulation instability. Phys. Rev. Lett. 127, 123901 (2021).
Xu, G. et al. Spontaneous symmetry breaking of dissipative optical solitons in a two-component Kerr resonator. Nat. Commun. 12, 4023 (2021).
Englebert, N. et al. Parametrically driven Kerr cavity solitons. Nat. Photonics 15, 857–861 (2021).
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).
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).
Del’Haye, P. et al. Optical frequency comb generation from a monolithic microresonator. Nature 450, 1214–1217 (2007).
Coen, S. & Erkintalo, M. Universal scaling laws of Kerr frequency combs. Opt. Lett. 38, 1790–1792 (2013).
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).
Hu, Y. et al. Realization of high-dimensional frequency crystals in electro-optic microcombs. Optica 7, 1189–1194 (2020).
Tusnin, A. K., Tikan, A. M. & Kippenberg, T. J. Nonlinear states and dynamics in a synthetic frequency dimension. Phys. Rev. A 102, 023518 (2020).
Lugiato, L. A. & Lefever, R. Spatial dissipative structures in passive optical systems. Phys. Rev. Lett. 58, 2209–2211 (1987).
Zhang, M. et al. Broadband electro-optic frequency comb generation in a lithium niobate microring resonator. Nature 568, 373–377 (2019).
Goda, K. & Jalali, B. Dispersive Fourier transformation for fast continuous single-shot measurements. Nat. Photonics 7, 102–112 (2013).
Mahjoubfar, A. et al. Time stretch and its applications. Nat. Photonics 11, 341–351 (2017).
He, Y. et al. High-speed tunable microwave-rate soliton microcomb. Preprint at http://arxiv.org/abs/2208.08046 (2022).
Matsko, A. B. & Maleki, L. On timing jitter of mode locked Kerr frequency combs. Opt. Express 21, 28862–28876 (2013).
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).
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).
Wimmer, M. et al. Observation of optical solitons in PT-symmetric lattices. Nat. Commun. 6, 7782 (2015).
Mittal, S., Moille, G., Srinivasan, K., Chembo, Y. K. & Hafezi, M. Topological frequency combs and nested temporal solitons. Nat. Phys. 17, 1169–1176 (2021).
Hu, Y. et al. High-efficiency and broadband on-chip electro-optic frequency comb generators. Nat. Photonics 16, 679–685 (2022).
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).
Smirnov, S. et al. Layout of NALM fiber laser with adjustable peak power of generated pulses. Opt. Lett. 42, 1732–1735 (2017).
Oliver, C. et al. Bloch oscillations along a synthetic dimension of atomic trap states. Preprint at http://arxiv.org/abs/2112.10648 (2021).
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.
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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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