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# Hyperbolic lattices in circuit quantum electrodynamics

## Abstract

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.

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## Data availability

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

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## Acknowledgements

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

Authors

### Contributions

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.

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.

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Kollár, A.J., Fitzpatrick, M. & Houck, A.A. Hyperbolic lattices in circuit quantum electrodynamics. Nature 571, 45–50 (2019). https://doi.org/10.1038/s41586-019-1348-3

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• DOI: https://doi.org/10.1038/s41586-019-1348-3

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