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.

Programmable large-scale simulation of bosonic transport in optical synthetic frequency lattices

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

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

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

Fig. 1: Simulations of large-scale bosonic transport with a programmable photonic simulator.
Fig. 2: Optical bandstructure measurements of 2D and 3D lattices.
Fig. 3: Input state preparation.
Fig. 4: Time-reversal symmetry-breaking in a 2D triangular lattice due to an effective gauge field.
Fig. 5: Simulations of bosonic transport in a tree-like geometry with a graph comprising over 100,000 sites.

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

  1. Lahini, Y. et al. Observation of a localization transition in quasiperiodic photonic lattices. Phys. Rev. Lett. 103, 013901 (2009).

    Article  ADS  Google Scholar 

  2. 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).

    Article  ADS  Google Scholar 

  3. Harris, N. C. et al. Quantum transport simulations in a programmable nanophotonic processor. Nat. Photon. 11, 447–452 (2017).

    Article  ADS  Google Scholar 

  4. 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).

    Article  ADS  Google Scholar 

  5. Wang, P. et al. Localization and delocalization of light in photonic moiré lattices. Nature 577, 42–46 (2020).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  7. Hung, J. S. C. et al. Quantum simulation of the bosonic Creutz ladder with a parametric cavity. Phys. Rev. Lett. 127, 100503 (2021).

    Article  ADS  Google Scholar 

  8. Karamlou, A. H. et al. Quantum transport and localization in 1D and 2D tight-binding lattices. npj Quantum Inf. 8, 35 (2022).

    Article  ADS  Google Scholar 

  9. Periwal, A. et al. Programmable interactions and emergent geometry in an array of atom clouds. Nature 600, 630–635 (2021).

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  11. Lustig, E. et al. Photonic topological insulator in synthetic dimensions. Nature 567, 356–360 (2019).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  15. Eichelkraut, T. et al. Mobility transition from ballistic to diffusive transport in non-Hermitian lattices. Nat. Commun. 4, 2533 (2013).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  17. Weidemann, S., Kremer, M., Longhi, S. & Szameit, A. Coexistence of dynamical delocalization and spectral localization through stochastic dissipation. Nat. Photon. 15, 576–581 (2021).

    Article  ADS  Google Scholar 

  18. Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).

    Article  Google Scholar 

  19. Hokmabadi, M. P., Schumer, A., Christodoulides, D. N. & Khajavikhan, M. Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity. Nature 576, 70–74 (2019).

    Article  ADS  Google Scholar 

  20. Schwartz, A. & Fischer, B. Laser mode hyper-combs. Opt. Express 21, 6196–6204 (2013).

    Article  ADS  Google Scholar 

  21. 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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  23. 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).

    Article  ADS  Google Scholar 

  24. Dutt, A. et al. Creating boundaries along a synthetic frequency dimension. Nat. Commun. 13, 3377 (2022).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  26. Lee, N. R. et al. Propagation of microwave photons along a synthetic dimension. Phys. Rev. A 101, 053807 (2020).

    Article  ADS  Google Scholar 

  27. Englebert, N. et al. Bloch oscillations of driven dissipative solitons in a synthetic dimension. Nat. Phys. 1–8 (2023)

  28. Wang, K., Dutt, A., Wojcik, C. C. & Fan, S. Topological complex-energy braiding of non-Hermitian bands. Nature 598, 59–64 (2021).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  30. Yuan, L., Dutt, A. & Fan, S. Synthetic frequency dimensions in dynamically modulated ring resonators. APL Photonics 6, 071102 (2021).

    Article  ADS  Google Scholar 

  31. 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).

    Article  MathSciNet  ADS  Google Scholar 

  32. Mak, K. F., McGill, K. L., Park, J. & McEuen, P. L. The valley Hall effect in MoS2 transistors. Science 344, 1489–1492 (2014).

    Article  ADS  Google Scholar 

  33. 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).

    Article  ADS  Google Scholar 

  34. Gubser, S. S., Jepsen, C., Ji, Z. & Trundy, B. Continuum limits of sparse coupling patterns. Phys. Rev. D 98, 045009 (2018).

    Article  MathSciNet  ADS  Google Scholar 

  35. Bentsen, G. et al. Treelike interactions and fast scrambling with cold atoms. Phys. Rev. Lett. 123, 130601 (2019).

    Article  ADS  Google Scholar 

  36. 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).

    Article  ADS  Google Scholar 

  37. D’Errico, A. et al. Bloch-Landau-Zener dynamics induced by a synthetic field in a photonic quantum walk. APL Photonics 6, 020802 (2021).

    Article  ADS  Google Scholar 

  38. Li, G. et al. Dynamic band structure measurement in the synthetic space. Sci. Adv. 7, eabe4335 (2021).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  40. Yuan, L. & Fan, S. Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator. Optica 3, 1014–1018 (2016).

    Article  ADS  Google Scholar 

  41. Qin, C. et al. Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials. Phys. Rev. Lett. 120, 133901 (2018).

    Article  ADS  Google Scholar 

  42. 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).

    Article  ADS  Google Scholar 

  43. 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).

    Article  ADS  Google Scholar 

  44. 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).

    Article  ADS  Google Scholar 

  45. 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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  47. Battiston, F. et al. The physics of higher-order interactions in complex systems. Nat. Phys. 17, 1093–1098 (2021).

    Article  Google Scholar 

  48. Fan, L. et al. Multidimensional convolution operation with synthetic frequency dimensions in photonics. Phys. Rev. Appl. 10, 34088 (2022).

    Article  Google Scholar 

  49. 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).

    Article  ADS  Google Scholar 

  50. Kippenberg, T. J., Gaeta, A. L., Lipson, M. & Gorodetsky, M. L. Dissipative Kerr solitons in optical microresonators. Science 361, eaan8083 (2018).

    Article  Google Scholar 

Download references

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

Authors

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

Correspondence to Alen Senanian or Peter L. McMahon.

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.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-023-02075-7

This article is cited by

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