Abstract
Photonic simulators using synthetic frequency dimensions have enabled flexible experimental analogues of condensed-matter systems. However, so far, such photonic simulators have been limited in scale, yielding results that suffer from finite-size effects. Here we present an analogue simulator capable of simulating large two-dimensional (2D) and 3D lattices, as well as lattices with non-planar connectivity. Our simulator takes advantage of the broad bandwidth achievable in photonics, allowing our experiment to realize programmable lattices with over 100,000 lattice sites. We showcase the scale of our simulator by demonstrating the extension of bandstructure spectroscopy from 1D to 2D and 3D lattices. We then report the direct observation of time-reversal symmetry-breaking in a triangular lattice in both momentum and real space, as well as site-resolved occupation measurements in a tree-like geometry that serves as a toy model in quantum gravity. Moreover, we demonstrate a method to excite arbitrary multisite states, which we use to study the response of a 2D lattice to both conventional and exotic input states. Our work highlights the scalability and flexibility of optical synthetic frequency dimensions. Future experiments building on our approach will be able to explore non-equilibrium phenomena in high-dimensional lattices and to simulate models with nonlocal higher-order interactions.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
Data availability
All data generated used in this work are available at https://doi.org/10.5281/zenodo.6959554.
Code availability
All code used in this work are available at https://doi.org/10.5281/zenodo.6959554.
References
Lahini, Y. et al. Observation of a localization transition in quasiperiodic photonic lattices. Phys. Rev. Lett. 103, 013901 (2009).
Lin, Q., Xiao, M., Yuan, L. & Fan, S. Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension. Nat. Commun. 7, 13731 (2016).
Harris, N. C. et al. Quantum transport simulations in a programmable nanophotonic processor. Nat. Photon. 11, 447–452 (2017).
Muniz, A. L. M., Wimmer, M., Bisianov, A., Morandotti, R. & Peschel, U. Collapse on the line—how synthetic dimensions influence nonlinear effects. Sci. Rep. 9, 9518 (2019).
Wang, P. et al. Localization and delocalization of light in photonic moiré lattices. Nature 577, 42–46 (2020).
Wang, K. et al. Generating arbitrary topological windings of a non-Hermitian band. Science 371, 1240–1245 (2021).
Hung, J. S. C. et al. Quantum simulation of the bosonic Creutz ladder with a parametric cavity. Phys. Rev. Lett. 127, 100503 (2021).
Karamlou, A. H. et al. Quantum transport and localization in 1D and 2D tight-binding lattices. npj Quantum Inf. 8, 35 (2022).
Periwal, A. et al. Programmable interactions and emergent geometry in an array of atom clouds. Nature 600, 630–635 (2021).
Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).
Lustig, E. et al. Photonic topological insulator in synthetic dimensions. Nature 567, 356–360 (2019).
Dutt, A. et al. A single photonic cavity with two independent physical synthetic dimensions. Science 367, 59–64 (2020).
Lustig, E. & Segev, M. Topological photonics in synthetic dimensions. Adv. Opt. Photon. 13, 426–461 (2021).
Leefmans, C. et al. Topological dissipation in a time-multiplexed photonic resonator network. Nat. Phys. 18, 442–449 (2022).
Eichelkraut, T. et al. Mobility transition from ballistic to diffusive transport in non-Hermitian lattices. Nat. Commun. 4, 2533 (2013).
Hodaei, H. et al. Enhanced sensitivity at higher-order exceptional points. Nature 548, 187–191 (2017).
Weidemann, S., Kremer, M., Longhi, S. & Szameit, A. Coexistence of dynamical delocalization and spectral localization through stochastic dissipation. Nat. Photon. 15, 576–581 (2021).
Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).
Hokmabadi, M. P., Schumer, A., Christodoulides, D. N. & Khajavikhan, M. Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity. Nature 576, 70–74 (2019).
Schwartz, A. & Fischer, B. Laser mode hyper-combs. Opt. Express 21, 6196–6204 (2013).
Ozawa, T., Price, H. M., Goldman, N., Zilberberg, O. & Carusotto, I. Synthetic dimensions in integrated photonics: from optical isolation to four-dimensional quantum Hall physics. Phys. Rev. A 93, 043827 (2016).
Dutt, A. et al. Experimental band structure spectroscopy along a synthetic dimension. Nat. Commun. 10, 3122 (2019).
Hu, Y., Reimer, C., Shams-Ansari, A., Zhang, M. & Lončar, M. Realization of high-dimensional frequency crystals in electro-optic microcombs. Optica 7, 1189–1194 (2020).
Dutt, A. et al. Creating boundaries along a synthetic frequency dimension. Nat. Commun. 13, 3377 (2022).
Bersch, C., Onishchukov, G. & Peschel, U. Experimental observation of spectral Bloch oscillations. Opt. Lett. 34, 2372–2374 (2009).
Lee, N. R. et al. Propagation of microwave photons along a synthetic dimension. Phys. Rev. A 101, 053807 (2020).
Englebert, N. et al. Bloch oscillations of driven dissipative solitons in a synthetic dimension. Nat. Phys. 1–8 (2023)
Wang, K., Dutt, A., Wojcik, C. C. & Fan, S. Topological complex-energy braiding of non-Hermitian bands. Nature 598, 59–64 (2021).
Tusnin, A. K., Tikan, A. M. & Kippenberg, T. J. Nonlinear states and dynamics in a synthetic frequency dimension. Phys. Rev. A 102, 023518 (2020).
Yuan, L., Dutt, A. & Fan, S. Synthetic frequency dimensions in dynamically modulated ring resonators. APL Photonics 6, 071102 (2021).
Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘Parity Anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).
Mak, K. F., McGill, K. L., Park, J. & McEuen, P. L. The valley Hall effect in MoS2 transistors. Science 344, 1489–1492 (2014).
Jiménez-Galán, Á., Silva, R. E. F., Smirnova, O. & Ivanov, M. Lightwave control of topological properties in 2D materials for sub-cycle and non-resonant valley manipulation. Nat. Photon. 14, 728–732 (2020).
Gubser, S. S., Jepsen, C., Ji, Z. & Trundy, B. Continuum limits of sparse coupling patterns. Phys. Rev. D 98, 045009 (2018).
Bentsen, G. et al. Treelike interactions and fast scrambling with cold atoms. Phys. Rev. Lett. 123, 130601 (2019).
Yuan, L., Xiao, M., Lin, Q. & Fan, S. Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation. Phys. Rev. B 97, 104105 (2018).
D’Errico, A. et al. Bloch-Landau-Zener dynamics induced by a synthetic field in a photonic quantum walk. APL Photonics 6, 020802 (2021).
Li, G. et al. Dynamic band structure measurement in the synthetic space. Sci. Adv. 7, eabe4335 (2021).
Chen, H. et al. Real-time observation of frequency Bloch oscillations with fibre loop modulation. Light Sci. Appl. 10, 48 (2021).
Yuan, L. & Fan, S. Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator. Optica 3, 1014–1018 (2016).
Qin, C. et al. Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials. Phys. Rev. Lett. 120, 133901 (2018).
Qin, C., Yuan, L., Wang, B., Fan, S. & Lu, P. Effective electric-field force for a photon in a synthetic frequency lattice created in a waveguide modulator. Phys. Rev. A 97, 063838 (2018).
Miyake, H., Siviloglou, G. A., Kennedy, C. J., Burton, W. C. & Ketterle, W. Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices. Phys. Rev. Lett. 111, 185302 (2013).
Yuan, L. & Fan, S. Three-dimensional dynamic localization of light from a time-dependent effective gauge field for photons. Phys. Rev. Lett. 114, 243901 (2015).
Peterson, C. W., Benalcazar, W. A., Lin, M., Hughes, T. L. & Bahl, G. Strong nonreciprocity in modulated resonator chains through synthetic electric and magnetic fields. Phys. Rev. Lett. 123, 063901 (2019).
Wright, L. G., Christodoulides, D. N. & Wise, F. W. Spatiotemporal mode-locking in multimode fiber lasers. Science 358, 94–97 (2017).
Battiston, F. et al. The physics of higher-order interactions in complex systems. Nat. Phys. 17, 1093–1098 (2021).
Fan, L. et al. Multidimensional convolution operation with synthetic frequency dimensions in photonics. Phys. Rev. Appl. 10, 34088 (2022).
Yu, D., Peng, B., Chen, X., Liu, X.-J. & Yuan, L. Topological holographic quench dynamics in a synthetic frequency dimension. Light Sci. Appl. 10, 209 (2021).
Kippenberg, T. J., Gaeta, A. L., Lipson, M. & Gorodetsky, M. L. Dissipative Kerr solitons in optical microresonators. Science 361, eaan8083 (2018).
Acknowledgements
P.L.M. acknowledges financial support from a David and Lucile Packard Foundation Fellowship, and also membership of the CIFAR Quantum Information Science Program as an Azrieli Global Scholar. We thank NTT Research for their financial and technical support. Portions of this work were supported by the National Science Foundation (award CCF-1918549). We acknowledge helpful discussions with D. Hathcock, E. Mueller, S. Prabhu, E. Rosenberg and members of the NTT PHI Lab/NSF Expeditions research collaboration. We also thank A. Dutt for helpful discussions and for feedback on a draft of the manuscript. We thank M. Buttolph for assistance with fibre splicing and V. Tjong for contributing to the instrumentation-control code.
Author information
Authors and Affiliations
Contributions
A.S., L.G.W. and P.L.M. developed the concept. A.S. and L.G.W. built the experimental set-up, with early contributions from H.K.D. A.S. performed the experiments, the data analysis and the numerical simulations (theory). P.F.W. performed experimental data collection. A.S., L.G.W. and P.L.M. wrote the manuscript. L.G.W. and P.L.M. supervised the project.
Corresponding authors
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 Figs. 1–20 and discussion.
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.
About this article
Cite this article
Senanian, A., Wright, L.G., Wade, P.F. et al. Programmable large-scale simulation of bosonic transport in optical synthetic frequency lattices. Nat. Phys. 19, 1333–1339 (2023). https://doi.org/10.1038/s41567-023-02075-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41567-023-02075-7
This article is cited by
-
Quantized topological pumping in Floquet synthetic dimensions with a driven dissipative photonic molecule
Nature Physics (2024)
-
Multi-dimensional band structure spectroscopy in the synthetic frequency dimension
Light: Science & Applications (2023)
-
Bloch oscillations of coherently driven dissipative solitons in a synthetic dimension
Nature Physics (2023)