Skip to main content

Thank you for visiting 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.

Hyperbolic lattices in circuit quantum electrodynamics


After two decades of development, cavity quantum electrodynamics with superconducting circuits has emerged as a rich platform for quantum computation and simulation. Lattices of coplanar waveguide resonators constitute artificial materials for microwave photons, in which interactions between photons can be incorporateded either through the use of nonlinear resonator materials or through coupling between qubits and resonators. Here we make use of the previously overlooked property that these lattice sites are deformable and permit tight-binding lattices that are unattainable even in solid-state systems. We show that networks of coplanar waveguide resonators can create a class of materials that constitute lattices in an effective hyperbolic space with constant negative curvature. We present numerical simulations of hyperbolic analogues of the kagome lattice that show unusual densities of states in which a macroscopic number of degenerate eigenstates comprise a spectrally isolated flat band. We present a proof-of-principle experimental realization of one such lattice. This paper represents a step towards on-chip quantum simulation of materials science and interacting particles in curved space.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Circuit QED lattices.
Fig. 2: The graph is everything.
Fig. 3: Schematic diagram of Euclidean and non-Euclidean lattices in circuit QED.
Fig. 4: Layout graph versus effective kagome graph.
Fig. 5: Tight-binding simulations.
Fig. 6: The heptagon-kagome device.

Data availability

The data and codes are available from the corresponding author upon reasonable request.


  1. 1.

    Cannon, J. W., Floyd, W. J., Kenyon, R. & Parry, W. R. in Flavors of Geometry 31, 59–115 (MSRI, 1997).

  2. 2.

    Leonhardt, U. & Philbin, T. G. General relativity in electrical engineering. New J. Phys. 8, 247 (2006).

    ADS  Article  Google Scholar 

  3. 3.

    Batz, S. & Peschel, U. Linear and nonlinear optics in curved space. Phys. Rev. A 78, 043821 (2008).

    ADS  Article  Google Scholar 

  4. 4.

    Smolyaninov, I. I. & Narimanov, E. E. Metric signature transitions in optical metamaterials. Phys. Rev. Lett. 105, 067402 (2010).

    ADS  Article  Google Scholar 

  5. 5.

    Genov, D. A., Zhang, S. & Zhang, X. Mimicking celestial mechanics in metamaterials. Nat. Phys. 5, 687–692 (2009).

    CAS  Article  Google Scholar 

  6. 6.

    Chen, H., Miao, R.-X. & Li, M. Transformation optics that mimics the system outside a Schwarzschild black hole. Opt. Express 18, 15183–15188 (2010).

    ADS  CAS  Article  Google Scholar 

  7. 7.

    Bekenstein, R. et al. Control of light by curved space in nanophotonic structures. Nat. Photon. 11, 664–670 (2017).

    ADS  CAS  Article  Google Scholar 

  8. 8.

    Bekenstein, R., Schley, R., Mutzafi, M., Rotschild, C. & Segev, M. Optical simulations of gravitational effects in the Newton–Schrödinger system. Nat. Phys. 11, 872–878 (2015).

    CAS  Article  Google Scholar 

  9. 9.

    Unruh, W. G. Experimental black-hole evaporation. Phys. Rev. Lett. 46, 1351–1353 (1981).

    ADS  Article  Google Scholar 

  10. 10.

    Weinfurtner, S., Tedford, E. W., Penrice, M. C. J., Unruh, W. G. & Lawrence, G. A. Measurement of stimulated Hawking emission in an analogue system. Phys. Rev. Lett. 106, 021302 (2011).

    ADS  Article  Google Scholar 

  11. 11.

    Philbin, T. G. et al. Fiber-optical analog of the event horizon. Science 319, 1367–1370 (2008).

    ADS  CAS  Article  Google Scholar 

  12. 12.

    Steinhauer, J. Observation of quantum Hawking radiation and its entanglement in an analogue black hole. Nat. Phys. 12, 959–965 (2016).

    Article  Google Scholar 

  13. 13.

    Carusotto, I., Fagnocchi, S., Recati, A., Balbinot, R. & Fabbri, A. Numerical observation of Hawking radiation from acoustic black holes in atomic Bose–Einstein condensates. New J. Phys. 10, 103001 (2008).

    ADS  Article  Google Scholar 

  14. 14.

    Gerace, D. & Carusotto, I. Analog Hawking radiation from an acoustic black hole in a flowing polariton superfluid. Phys. Rev. B 86, 144505 (2012).

    ADS  Article  Google Scholar 

  15. 15.

    Sabín, C. Mapping curved spacetimes into Dirac spinors. Sci. Rep. 7, 40346 (2017).

    ADS  Article  Google Scholar 

  16. 16.

    Pedernales, J. S. et al. Dirac equation in (1+1)-dimensional curved spacetime and the multiphoton quantum Rabi model. Phys. Rev. Lett. 120, 160403 (2018).

    ADS  CAS  Article  Google Scholar 

  17. 17.

    Koke, C., Noh, C. & Angelakis, D. G. Dirac equation in 2-dimensional curved spacetime, particle creation, and coupled waveguide arrays. Ann. Phys. 374, 162–178 (2016).

    ADS  MathSciNet  CAS  Article  Google Scholar 

  18. 18.

    Boada, O., Celi, A., Latorre, J. I. & Lewenstein, M. Dirac equation for cold atoms in artificial curved spacetimes. New J. Phys. 13, 035002 (2011).

    ADS  Article  Google Scholar 

  19. 19.

    Houck, A. A., Türeci, H. E. & Koch, J. On-chip quantum simulation with superconducting circuits. Nat. Phys. 8, 292–299 (2012).

    CAS  Article  Google Scholar 

  20. 20.

    Anderson, B. M., Ma, R., Owens, C., Schuster, D. I. & Simon, J. Engineering topological many-body materials in microwave cavity arrays. Phys. Rev. X 6, 041043 (2016).

    Google Scholar 

  21. 21.

    Fitzpatrick, M., Sundaresan, N. M., Li, A. C. Y., Koch, J. & Houck, A. A. Observation of a dissipative phase transition in a one-dimensional circuit QED lattice. Phys. Rev. X 7, 011016 (2017).

    Google Scholar 

  22. 22.

    Schmidt, S. & Koch, J. Circuit QED lattices: towards quantum simulation with superconducting circuits. Ann. Phys . (Berl.) 525, 395–412 (2013).

    Article  Google Scholar 

  23. 23.

    Underwood, D. L., Shanks, W. E., Koch, J. & Houck, A. A. Low-disorder microwave cavity lattices for quantum simulation with photons. Phys. Rev. A 86, 023837 (2012).

    ADS  Article  Google Scholar 

  24. 24.

    Coxeter, H. S. M. Regular honeycombs in hyperbolic space. In Proc. ICM Amsterdam 3, 155–169 (1954).

    Google Scholar 

  25. 25.

    Schmidt, S., Gerace, D., Houck, A. A., Blatter, G. & Türeci, H. E. Nonequilibrium delocalization-localization transition of photons in circuit quantum electrodynamics. Phys. Rev. B 82, 100507 (2010).

    ADS  Article  Google Scholar 

  26. 26.

    Raftery, J., Sadri, D., Schmidt, S., Türeci, H. E. & Houck, A. A. Observation of a dissipation-induced classical to quantum transition. Phys. Rev. X 4, 031043 (2014).

    Google Scholar 

  27. 27.

    Woess, W. Context-free languages and random walks on groups. Discrete Math. 67, 81–87 (1987).

    MathSciNet  Article  Google Scholar 

  28. 28.

    Sunada, T. Group C-algebra sand the spectrum of a periodic Schrödinger operator on a manifold. Can. J. Math. 44, 180–193 (1992).

    Article  Google Scholar 

  29. 29.

    Floyd, W. J. & Plotnik, S. P. Growth functions on Fuchsian groups and the Euler characteristic. Invent. Math. 88, 1–29 (1987).

    ADS  MathSciNet  Article  Google Scholar 

  30. 30.

    Bartholdi, L. & Ceccherini-Silberstein, T. G. Growth series and random walks on some hyperbolic graphs. Monatsh. Math. 136, 181–202 (2002).

    MathSciNet  Article  Google Scholar 

  31. 31.

    Strichartz, R. S. Harmonic analysis as spectral theory of Laplacians. J. Funct. Anal. 87, 51–148 (1989).

    MathSciNet  Article  Google Scholar 

  32. 32.

    McLaughlin, J. C. Random Walks and Convolution Operators on Free Products. PhD thesis, New York Univ. (1986).

  33. 33.

    Agmon, S. Spectral theory of Schrödinger operators on Euclidean and non-Euclidean spaces. Commun. Pure Appl. Math. 39, S3–S16 (1986).

    MathSciNet  Article  Google Scholar 

  34. 34.

    Krioukov, D., Papadopoulos, F., Kitsak, M. & Vahdat, A. Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106 (2010).

    ADS  MathSciNet  Article  Google Scholar 

  35. 35.

    Boguñá, M., Papadopoulos, F. & Krioukov, D. Sustaining the Internet with hyperbolic mapping. Nat. Commun. 1, 62 (2010).

    ADS  Article  Google Scholar 

  36. 36.

    Lipton, R. J. & Tarjan, R. E. A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 177–189 (1979).

    MathSciNet  Article  Google Scholar 

  37. 37.

    Breuckmann, N. P. & Terhal, B. M. Constructions and noise threshold of hyperbolic surface codes. IEEE Trans. Inf. Theory 62, 3731–3744 (2016).

    MathSciNet  Article  Google Scholar 

  38. 38.

    Breuckmann, N. P., Vuillot, C., Campbell, E., Krishna, A. & Terhal, B. M. Hyperbolic and semi-hyperbolic surface codes for quantum storage. Quantum Sci. Technol. 2, 035007 (2017).

    ADS  Article  Google Scholar 

  39. 39.

    Blais, A., Huang, R.-S., Wallraff, A., Girvin, S. M. & Schoelkopf, R. J. Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 (2004).

    ADS  Article  Google Scholar 

  40. 40.

    Koch, J. et al. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007).

    ADS  Article  Google Scholar 

  41. 41.

    Reagor, M. et al. Quantum memory with millisecond coherence in circuit QED. Phys. Rev. B 94, 014506 (2016).

    ADS  Article  Google Scholar 

  42. 42.

    Ashcroft, N. W. & Mermin, N. D. Solid State Physics (Thomson Learning, 1976).

  43. 43.

    Kroto, H. W., Heath, J. R., O’Brien, S. C., Curl, R. F. & Smalley, R. E. C60: buckminsterfullerene. Nature 318, 162–163 (1985).

    ADS  CAS  Article  Google Scholar 

  44. 44.

    Bergman, D. L., Wu, C. & Balents, L. Band touching from real-space topology in frustrated hopping models. Phys. Rev. B 78, 125104 (2008).

    ADS  Article  Google Scholar 

  45. 45.

    Leykam, D., Andreanov, A. & Flach, S. Artificial flat band systems: from lattice models to experiments. Adv. Phys. X 3, 1473052 (2018).

    Google Scholar 

  46. 46.

    Kollár, A. J., Fitzpatrick, M., Sarnak, P. & Houck, A. A. Line-graph lattices: Euclidean and non-Euclidean flat bands, and implementations in circuit quantum electrodynamics. Preprint at (2019).

  47. 47.

    Annunziata, A. J. et al. Tunable superconducting nanoinductors. Nanotechnology 21, 445202 (2010).

    Article  Google Scholar 

  48. 48.

    Rotzinger, H. et al. Aluminium-oxide wires for superconducting high kinetic inductance circuits. Supercond. Sci. Technol. 30, 025002 (2016).

    ADS  Article  Google Scholar 

  49. 49.

    Kesten, H. Symmetric random walks on groups. Trans. Am. Math. Soc. 92, 336–354 (1959).

    MathSciNet  Article  Google Scholar 

  50. 50.

    Chen, M. S., Onsager, L., Bonner, J. & Nagle, J. Hopping of ions in ice. J. Chem. Phys. 60, 405–419 (1974).

    ADS  CAS  Article  Google Scholar 

  51. 51.

    Carroll, S. M. Lecture notes on general relativity. Preprint at (1997).

  52. 52.

    Dunham, D., Lindgren, J. & Witte, D. Creating repeating hyperbolic patterns. Comput. Graph. 15, 215–223 (1981).

    Article  Google Scholar 

  53. 53.

    Adcock, B. M., Jones, K. C., Reiter, C. A. & Vislocky, L. M. Iterated function systems with symmetry in the hyperbolic plane. Comput. Graph. 24, 791–796 (2000).

    Article  Google Scholar 

Download references


We thank P. Sarnak, J. Kollár, R. Bekenstein, C. Fefferman and S. Parameswaran for discussions. This work was supported by the US National Science Foundation, the Princeton Center for Complex Materials (DMR-1420541) and the Multidisciplinary University Research Initiative (MURI) (W911NF-15-1-0397).

Author information




A.J.K. conceived of the experiment, performed numerical simulations, measured the device and prepared the manuscript. M.F. designed, fabricated and measured the device, and prepared the manuscript. A.A.H. supervised the work.

Corresponding author

Correspondence to Alicia J. Kollár.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Reviewer information Nature thanks Göran Johansson, Prof. Enrique Solano and Martin Weides for their contribution to the peer review of this work.

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

Extended data figures and tables

Extended Data Fig. 1 System-size effects.

a, Numerical eigenenergy spectra for the heptagon-kagome lattice versus system size. The smallest layout tiling consists of a central heptagon and a shell of seven immediate neighbours (one shell). Each successive tiling includes another shell of immediate neighbours of the previous one. The spectra are plotted for one (red), two (blue), three (cyan) and four (gold) shells of neighbours. be, The results for each individual system size. The density of states converges rapidly with number of shells despite relatively small system sizes and effective hard-wall confinement. The theoretical plots elsewhere in this paper are from three-shell simulations. The experimental device consists of two shells.

Extended Data Fig. 2 Colour plots of selected numerical eigenstates of three shells of the heptagon-kagome lattice.

States are ordered from highest to lowest energy, and are plotted by placing a circle on each lattice site of the effective lattice. The size of the circle indicates the amplitude of the state on that site, and the colour its phase. a, The maximally excited state. This state is uniform in phase, but its amplitude varies radially owing to the effective confinement from the missing links at the boundary of the simulation. b, c, Examples of the two next-highest states. They bear a striking resemblance to Laguerre–Gaussian or particle-in-a-cylindrical-box modes found in flat Euclidean space. df, Selected intermediate excited states. Notice that the state in f shows both amplitude and phase modulation in the azimuthal direction, with independent periods. g, The localized eigenstate of compact support that forms the flat band.

Extended Data Fig. 3 Poincaré-disk-model1,52,53 conformal projections of hyperbolic lattices.

ad, Graphene-like lattices formed from heptagons, octagons, nonagons and dodecagons, respectively. eh, The corresponding kagome-like effective lattices that arise when ad are used as the layout lattice. i, A table of inter-site spacings for graphene-like hyperbolic layout lattices and their medial lattices, the kagome-like effective lattices. All distances are given in terms of the curvature length \(R=1/\sqrt{| K| }\). As the number of sides of the layout polygon increases, the intrinsic curvature of the tiling also grows, and the polygons become markedly larger.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kollár, A.J., Fitzpatrick, M. & Houck, A.A. Hyperbolic lattices in circuit quantum electrodynamics. Nature 571, 45–50 (2019).

Download citation

Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


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