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Hyperbolic Lattices in Circuit Quantum Electrodynamics
Alicia J. Kollár, Mattias Fitzpatrick, Andrew A. Houck
TL;DR
Realizing regular lattices in negatively curved space is difficult because hyperbolic geometry cannot be embedded isometrically in flat Euclidean space. The paper uses deformable CPW resonators to construct effective hyperbolic tight-binding lattices, finding isolated flat bands in hyperbolic kagome analogs and demonstrating a finite heptagon-kagome device.
Problem
Negative spatial curvature cannot be isometrically embedded in Euclidean space, while existing hyperbolic metamaterial simulations are classical and only weakly interacting.
Method
The paper uses deformable CPW resonators and numerical tight-binding simulations to realize and study hyperbolic kagome-lattice analogs.
Results
Hyperbolic kagome analogs exhibit a flat band, isolated from the remaining spectrum for odd-sided polygons, and a finite non-interacting heptagon-kagome device was constructed.
Takeaways & Limitations
The work demonstrates a route to on-chip quantum simulation of materials science and interacting particles in curved space.
Takeaways & Limitations
The technique is limited by resonator overlap, finite meander density, and the requirement that achievable lattices be medial or line graphs.
Abstract
from arXiv · showhide
After close to two decades of research and development, superconducting circuits have emerged as a rich platform for both quantum computation and quantum simulation. Lattices of superconducting coplanar waveguide (CPW) resonators have been shown to produce artificial materials for microwave photons, where weak interactions can be introduced either via non-linear resonator materials or strong interactions via qubit-resonator coupling. Here, we highlight the previously-overlooked property that these lattice sites are deformable and allow the realization of tight-binding lattices which are unattainable, even in conventional solid-state systems. In particular, we show that networks of CPW resonators can create a new class of materials which constitute regular lattices in an effective hyperbolic space with constant negative curvature. We present numerical simulations of a series of hyperbolic analogs of the kagome lattice which show unusual densities of states with a spectrally-isolated degenerate flat band. We also present a proof-of-principle experimental realization of one of these lattices. This paper represents the first step towards on-chip quantum simulation of materials science and interacting particles in curved space.
INTRODUCTION
The paper motivates hyperbolic lattice simulation by the difficulty of realizing negative curvature and the limitations of existing classical, weakly interacting approaches. It proposes cQED networks as a route to hyperbolic photonic materials and studies kagome generalizations with unusual band structures.
- Motivation: Negative spatial curvature cannot be isometrically embedded in Euclidean space, making hyperbolic particle simulations experimentally difficult.Existing simulations of negatively curved space use hyperbolic metamaterials with spatially varied dielectric constants, but are classical and only weakly interacting.
- Motivation: Prior curved-space analogs have been pursued with acoustic waves, optical pulses, condensates, ion traps, optical waveguides, and optical lattices.These approaches include both experimental and theoretical proposals for wave, condensate, and Dirac-equation simulations.
- Contribution: The paper proposes cQED networks of superconducting microwave resonators as periodic photonic lattices in the two-dimensional hyperbolic plane.Qubits or nonlinear resonator materials can introduce quantum-mechanical or classical photon-photon interactions.
- Broader context: Hyperbolic networks are also studied in computer science because their connectivity can support robust communication networks and lower-overhead logical-qubit encoding.The supplied passages connect these properties to efficient connections, resistance to node removal, and surface codes.
- Contribution: The authors study hyperbolic generalizations of the kagome lattice through numerical non-interacting band-structure calculations and a finite experimental realization.The studied spectra include a spectrally isolated degenerate flat band, and the device uses heptagons instead of hexagons.
- Contribution: CPW resonator sites are deformable, allowing inter-site spacings and hopping rates to be decoupled in tight-binding models.The paper presents this property as enabling curved-space lattices on flat substrates and opening access to novel Euclidean and non-flat lattices.
CIRCUIT QED LATTICES
Circuit QED lattices use coupled superconducting CPW resonators to implement tight-binding models for microwave photons. Because CPW resonators can be bent while preserving their model parameters, their geometry can be deformed without changing the effective hopping graph.
- Platform: Circuit quantum electrodynamics couples lithographically defined superconducting qubits to microwave resonators and can reach the strong-coupling regime.The platform also supports high-fidelity gate operations and relatively long coherence times.
- Platform: Networks of coupled resonators form artificial photonic materials using two-dimensional CPW resonators or three-dimensional stub resonators.The lattices are built by connecting many resonators through capacitive or inductive couplings.
- Platform: CPW resonators are planar coaxial structures with an isolated centre pin and two ground planes, commonly compactified with meanders.Their millimetre-scale length and capacitive gaps make them suitable for microfabricated lattice devices.
- Tight-binding model: The coupling capacitance at resonator junctions determines the hopping strength, which is naturally negative in these systems.The resulting band structures are inverted relative to conventional solid-state systems.
- Tight-binding model: In the noninteracting limit, the lattice is described by a tight-binding model with on-site energy ω0 and nearest-neighbor hopping rate t < 0.The geometry is encoded in the nearest-neighbor hopping term.
- Deformability: Unlike atom or ion platforms, CPW resonators can be deformed while maintaining fixed total length and therefore unchanged tight-binding parameters.Keeping the end capacitors unchanged preserves the model when resonators or coupler locations are moved.
- Deformability: Figure 1 illustrates that differently shaped resonators can retain the same resonance frequencies and hopping rates, while junction capacitance sets the effective coupling.The figure links physical resonator shape and coupler geometry to the parameters of the effective model.
- Deformability: Figure 2 contrasts geometric displacement that preserves hopping rates with distance-dependent hopping that changes the tight-binding graph.CPW coupling geometry permits distorted-looking realizations of graphs that cannot be fabricated regularly in flat two-dimensional space.
HYPERBOLIC LATTICES
Hyperbolic geometry permits regular tilings forbidden in Euclidean space, and deformable CPW resonators can realize corresponding effective lattices while preserving tight-binding parameters. The paper focuses on heptagon-kagome structures, whose finite sections are comparatively feasible to fabricate.
- Regular tilings with three pentagons or heptagons meeting at each vertex are forbidden in Euclidean space but possible in curved space.
- Hyperbolic space has negative curvature and a natural length scale R, so tile size cannot be scaled independently of curvature.Each hyperbolic tiling has an intrinsic ratio between curvature and lattice spacing.
- CPW resonators can compensate for stretched or shortened projected edges by modifying their shapes while preserving frequencies and the tight-binding model.
- Fabrication limits on CPW meander density typically rule out complete spherical or radically curved hyperbolic tilings, but finite hyperbolic sections remain readily achievable.
- Because resonators occupy edges, the effective circuit lattice is the medial, or line-graph, lattice of the resonator layout.A hexagonal layout therefore produces a kagome effective lattice.
- The heptagon-kagome lattice is obtained from a hyperbolic heptagon-graphene layout and has the weakest curvature among single-tile hyperbolic tilings.Its layout and effective inter-site spacings are 0.566 R and 0.492 R, respectively.
Tight-Binding Simulations
Because hyperbolic lattices lack a Bravais lattice and Bloch theory, the authors numerically diagonalize finite tight-binding systems. Hyperbolic kagome analogs exhibit flat bands, with odd-sided layouts producing a persistent spectral gap.
- Numerical approach: Hyperbolic translation groups are non-commutative, so no Bravais lattice or general Bloch-theory procedure exists for calculating their band structures.Numerical diagonalization provides eigenvalues and eigenvectors but no momentum-resolved dispersion relations.
- Numerical approach: Finite hyperbolic lattices are constructed cylindrically by adding shells of neighboring polygons, then diagonalizing the truncated hopping matrix.Hopping to resonators outside the simulation is neglected, and the calculation uses a localized delta-function basis.
- Spectra: All simulated kagome and hyperbolic kagome spectra contain a flat band at −2|t|, while the remaining bands fill the range from −2|t| to 4|t|.The Euclidean finite-size spectrum retains the flat-band signature but loses the detailed dispersive-band structure.
- Spectra: Odd-sided heptagon- and nonagon-kagome lattices show a spectral gap separating the flat band from the remaining eigenstates.The gap is independent of system size and decreases as the layout polygon gains more sides; other visible gaps close with increasing size.
- Flat-band structure: Flat bands arise from infinitely many localized eigenstates, with odd-sided polygons requiring alternating closed loops extending over two tiles because of geometric frustration.The origin of the gap isolating these states in odd-sided hyperbolic lattices remains unknown.
Experiment
The authors fabricated a two-shell heptagon-kagome circuit-QED device from coupled CPW resonators and measured its microwave transmission. Disorder-aware theoretical reconstructions show reasonable agreement with experiment.
- Device realization: The device realizes a finite heptagon-kagome lattice containing one central heptagon and two shells of neighboring tiles.The layout uses CPW resonators patterned in niobium on sapphire and includes microwave coupling ports.
- Device realization: The 140-resonator device has 8 GHz fundamental modes, 16 GHz second harmonics, and a second-harmonic hopping rate of −136.2 MHz.The second harmonic realizes the heptagon-kagome lattice used for measurements.
- Device realization: High-frequency λ/4 resonators on the outer ring maintain consistent loading and uniform on-site energies.Four additional CPW lines at the device corners couple microwaves into and out of the lattice.
- Measurement and modeling: The experimental transmission agrees reasonably with theoretical curves from models including small systematic offsets and realistic disorder.The comparison supports the device-level realization of the intended finite lattice.
- Measurement and modeling: Transmission simulations incorporate systematic on-site-energy offsets, random disorder in energies and hopping rates, Lorentzian mode profiles, and packaging leakage.Theoretical curves are generated for 15 disorder realizations and compared with measured transmission near the second harmonic.
CONCLUSION
The paper demonstrates that CPW-resonator circuit QED can realize artificial photonic materials in effective curved space. Simulations reveal isolated flat bands in odd-sided hyperbolic kagome lattices, while an experimental heptagon-kagome device provides a proof of principle.
- CONCLUSION: Circuit QED lattices of two-dimensional CPW resonators can produce artificial photonic materials in effective curved space.The approach exploits the deformability of CPW lattice sites.
- CONCLUSION: Numerical tight-binding simulations show flat bands in hyperbolic kagome analogs, with odd-sided polygons isolating the flat band from the rest of the spectrum.The simulations cover a class of hyperbolic kagome-like lattices.
- CONCLUSION: A proof-of-principle device realizes a finite section of a non-interacting heptagon-kagome lattice.The experimental demonstration establishes a circuit implementation of one curved-space lattice.
- CONCLUSION: The technique may also realize other graphs and lattices, including ripple-distorted two-dimensional sheets and Cayley trees.These possibilities extend beyond the curved-space lattices studied here.
METHODS
The methods fabricate and calibrate CPW resonators, mount the disorder-compensated device in a dilution refrigerator, and reconstruct transmission from numerically calculated modes. Leakage through the PCB housing limits measurements near 15–17 GHz.
- Fabrication: CPW resonators are patterned by photolithography and reactive-ion etching in 200 nm niobium on a 500 µm sapphire substrate.The resonators are 7.5 mm long with target fundamental and second-harmonic frequencies of 8 GHz and 16 GHz.
- Calibration: Each resonator shape is measured separately, then lengths are adjusted to compensate parasitic frequency offsets against a 7.99 GHz reference.The u-shaped outer-ring resonators provide the reference for matching the other geometries.
- Measurement setup: The final device is mounted in a microwave PCB with indium seals and aluminum wire bonds, then cooled on the base plate of a dilution refrigerator.Microwave excitation is supplied by a high-frequency generator.
- Transmission modeling: Theoretical transmission is reconstructed by computing eigenmodes and eigenenergies, assigning port-overlap-dependent quality factors and Lorentzian lineshapes, and summing the resulting contributions.The reconstruction models the expected electric field transmitted through the lattice.
- Measurement limitation: Leakage transmission through the PCB housing is the main measurement limitation from 15 to 17 GHz, producing a coherent background varying over 50–100 MHz.The background cannot be measured independently and is estimated empirically from filtered experimental data.
CPW Resonators and Circuit QED Lattices
CPW resonators map to tight-binding lattices, but their mode symmetry and hopping signs impose important constraints. Full-wave modes avoid sign inconsistencies that frustrate odd-sided hyperbolic kagome lattices, while the flat band remains robust.
- Circuit-to-lattice mapping: Connecting CPW resonators end to end produces a tight-binding lattice whose hopping is set by capacitive coupling.The resonator resonance frequency sets the on-site energy, while capacitive coupling determines the hopping rate.
- Mode choice: Full-wave modes are used because their symmetry avoids the hopping-sign problem that can occur for half-wave modes on hyperbolic lattices.For many hyperbolic lattices, local gauge transformations cannot make all half-wave hopping rates have the same sign.
- Geometric frustration: Odd-sided hyperbolic kagome lattices frustrate half-wave sign assignments, whereas even-sided polygons permit gauging away the asymmetry.The dispersive bands are affected by this frustration, but the flat band is not.
- Geometric frustration: The flat band remains immune to frustration because its states are formed from loops containing an even number of edges.These loop states preserve the required phase pattern despite the odd-sided plaquettes.
- System sizes: Theoretical spectra use three shells of neighbors, while the experimental device contains two shells.The system-size caption reports rapid density-of-states convergence despite small systems and effective hard-wall confinement.
Tight-Binding Models: System Size Effects
Finite heptagon-kagome tight-binding models are built by adding shells of neighboring polygons. The flat-band gap is visible at small sizes, while higher-energy gaps progressively close as the system grows.
- System construction: Finite-size simulations begin with a central heptagon and one neighboring shell, then iteratively add shells and recompute the effective lattice.Links to sites outside each chosen system size are neglected.
- Spectral convergence: The flat band is visible even in the one-shell simulation, although its gap is initially difficult to distinguish from finite-size gaps.The flat-band gap becomes readily visible for systems with two or more neighbor layers.
- Spectral convergence: Higher-lying gaps close progressively with increasing system size, and it remains unclear whether any stay open at infinite size.The simulations therefore establish finite-size behavior without resolving the thermodynamic limit of those gaps.
- Experimental scale: The experimental device was constructed with the depth at which the flat-band gap is readily visible.The cited passage identifies this depth as two or more layers of neighboring polygons.
Numerical Eigenstates
Three-shell heptagon-kagome eigenstates combine Euclidean-like mode patterns with distinctive hyperbolic behavior. They include edge-localized states, azimuthal modulation, and compact-support flat-band states.
- Eigenstate structure: The highest-energy state has uniform phase but radially varying amplitude caused by effective hard-wall confinement at the boundary.Because hopping is negative, the uniform-phase configuration appears at the highest rather than lowest energy.
- Eigenstate structure: The next-highest states resemble Laguerre-type or particle-in-a-cylindrical-box modes from flat Euclidean space.The qualitative similarity persists for multiple modes across the spectrum.
- Hyperbolic features: A macroscopic fraction of sites lies in the outermost ring, producing many states concentrated near the system edge.This edge-state abundance reflects the hyperbolic lattice geometry.
- Hyperbolic features: Some intermediate states show azimuthal amplitude and phase modulation with independent periods.The cited example is the state shown in Fig. 8f.
- Flat band: The flat-band state is a localized compact-support loop whose neighboring phases alternate by π and whose destructive interference prevents hopping.Translated copies are orthogonal and form a degenerate manifold with multiplicity proportional to system size.
Lattice Curvatures
Hyperbolic lattices have a curvature-dependent length scale, so valid tilings require specific polygon sizes. The paper uses Poincaré-disc projections to construct graphene-like layouts and their kagome-like effective lattices.
- Curved-space geometry: Unlike Euclidean space, hyperbolic space has a natural length scale R determined by Gaussian curvature K = ±1/R2.Polygon shape depends on its size relative to R.
- Curved-space geometry: Hyperbolic polygons have smaller internal angles than their Euclidean counterparts and become progressively pointier as their size increases.This size dependence distinguishes hyperbolic tilings from scale-invariant Euclidean tilings.
- Tiling constraints: For fixed curvature, polygons must have a precise size to tile without gaps or overlaps, so each non-Euclidean lattice has a specific tile size.The corresponding lattice geometry depends on the relation between site spacing and R.
- Construction method: The Poincaré disc model maps the hyperbolic plane of curvature −1 onto the Euclidean unit disc and is used to determine the lattice curvatures.The construction focuses on graphene generalizations that generate kagome-like effective models.
- Construction method: The effective-lattice spacing is obtained from distances between midpoints of layout-polygon edges.For the heptagon case, the graphene-like spacing is 0.566R and the heptagon-kagome spacing is 0.492R.