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Warping geometry pushes scientific boundaries

Warping geometry pushes scientific boundaries

Atomic interactions in common solids and liquids are so intricate that physicists are still baffled by some of these materials’ characteristics. Because solving the issues theoretically is beyond the capability of contemporary computers, Princeton University scientists have turned to an uncommon area of geometry instead.

Researchers lead by electrical engineering professor Andrew Houck created an electronic array on a microchip that replicates particle interactions on a hyperbolic plane, a geometric surface in which space bends away from itself at every point. A hyperbolic plane is difficult to visualize (the artist M.C. Escher employed hyperbolic geometry in many of his mind-bending creations), but it is ideal for addressing concerns concerning particle interactions and other complex mathematical problems.

The researchers created a lattice that works as a hyperbolic space using superconducting circuits. When the researchers inject photons into the lattice, they can see the photons’ interactions in simulated hyperbolic space to answer a variety of challenging issues.

“You can throw particles together, turn on a very controlled amount of interaction between them, and see the complexity emerge,” said Houck.

The study’s primary author, Alicia Kollár, a postdoctoral research associate at the Princeton Center for Complicated Materials, said the objective is to enable researchers to answer complex problems regarding quantum interactions, which regulate the behavior of atomic and subatomic particles.

“The problem is that if you want to study a very complicated quantum mechanical material, then that computer modeling is very difficult. We’re trying to implement a model at the hardware level so that nature does the hard part of the computation for you.”

Strange warping geometry helps to push scientific boundaries
A schematic of the resonators on the microchip, which are arranged in a lattice pattern of heptagons, or seven-sided polygons. The structure exists on a flat plane, but simulates the unusual geometry of a hyperbolic plane. Credit: Kollár et al.

A circuit of superconducting resonators provides routes for microwave photons to move and interact on the centimeter-sized device. The chip’s resonators are organized in a lattice pattern of heptagons, or seven-sided polygons. The structure occurs on a flat plane, yet its geometry is that of a hyperbolic plane.

“In normal 3-D space, a hyperbolic surface doesn’t exist,” said Houck. “This material allows us to start to think about mixing quantum mechanics and curved space in a lab setting.”

The graph is everything. a, Colour plot of the lattice potential of a regular 2D square lattice. b, The corresponding regular tight-binding graph of the potential in a. c, Alternate drawing of the tight-binding graph in a. Despite the visible displacement of the nodes, the hopping Hamiltonian is unmodified because the hopping rates, indicated by the color of the graph edges, are identical to b. d, New tight-binding graph with the same nodes as c but with hopping rates dependent on the distance between the nodes. e, Highly disordered lattice potential which gives rise to the tight-binding graph in d. In systems where the effective hopping rate is determined by the distance between sites, any displacement of the lattice sites results in a disordered model with modified properties. In CPW lattices, however, the hopping rates are determined by the geometry of the coupling capacitors at each end of the resonators. In some cases, such as the curved-space lattices shown in Fig. 3d-i, the regular tight-binding graph is impossible to produce in 2D flat space, whereas a mathematically identical but distorted-looking graph like c can be fabricated using CPW resonators. This would not be possible in systems where hopping rates are solely determined by the distance between sites.
Schematic diagram of Euclidean and non-Euclidean lattices in circuit QED. a, One vertex of a successful attempt to tile the Euclidean plane with regular hexagons. b, Resulting hexagonal lattice. c, Euclidean circuit QED lattice with the resonators laid out in regular hexagons. Because this is a valid Euclidean tiling the resonator network is highly regular, and all resonators look the same. (Photograph modified from [19].) d, One vertex of a failed attempt to tile the Euclidean plane with regular pentagons. A gap is left between the tiles, so this tiling is only valid in spherical space (positive curvature). e, Projection of a spherical soccer-ball lattice into the Euclidean plane. Some tiles must be stretched to cover the missing space. f, Schematic of a circuit QED lattice which realizes the soccer ball tiling. Resonator shapes are modified at different points in the lattice to bridge the stretched distances while preserving the hopping rates and on-site energies. g, One vertex of a failed attempt to tile the Euclidean plane with regular heptagons. The tiles overlap, so this tiling is only valid in hyperbolic space (negative curvature). h, Conformal projection of a hyperbolic heptagon lattice into the Euclidean plane. i, Schematic of a circuit QED lattice which realizes a section of the hyperbolic lattice in h. Resonator shapes are modified at different points in the lattice to permit tighter packing while preserving the hopping rates and on-site energies.

“It’s a space that you can mathematically write down, but it’s very difficult to visualize because it’s too big to fit in our space,” explained Kollár.

The researchers utilized a particular form of resonator called a coplanar waveguide resonator to replicate the impact of compressing hyperbolic space onto a flat surface. Microwave photons act the same way whether their route is straight or meandering via this resonator. The resonators’ meandering construction allows Kollár to “squish and scrunch” the sides of the heptagons to generate a flat tiling pattern.

Looking at the chip’s core heptagon is like looking through a fisheye camera lens, where things near the outside of the field of vision appear smaller than those in the center—the heptagons appear smaller as they move away from the center. Microwave photons moving through the resonator circuit can now act as particles in a hyperbolic environment thanks to this design.

The heptagon-kagome device. a, Resonator layout (dark blue) and effective lattice (light blue) for a circuit that realizes two shells of the heptagon-kagome lattice. Orange circles indicate three-way capacitive couplers. b, Photograph of a physical device that realizes the layout and effective graphs in a. The device was fabricated in a 200 nm niobium film on a sapphire substrate and consists of 140 CPW resonators with fundamental resonance frequencies of 8 GHz, second harmonic frequencies of 16 GHz, and a hopping rate of −136.2 MHz at the second harmonic. Four additional CPW lines have been included at each corner of the device to couple microwaves into and out of the device for transmission measurements. Short stubs protruding inward from the outermost three-way couplers in the device are high-frequency λ/4 resonators that maintain a consistent loading of the sites in the outer ring–ensuring uniform on-site energies. c, Experimental transmission (S21) for the device in b is shown in dark blue. The red curves show theoretical transmission for an ensemble of theoretical models including small systematic offsets in the on-site energies and realistic disorder levels, demonstrating reasonable agreement between theory
and experiment.

But first, Kollár and her colleagues must advance the photonic material by continuing to investigate its mathematical foundation and incorporating components that allow photons in the circuit to interact.

“By themselves, microwave photons don’t interact with each other—they pass right through,” said Kollár. Most applications of the material would require “doing something to make it so that they can tell there’s another photon there.”

Hyperbolic lattices in circuit quantum electrodynamics, Alicia J. Kollár, Mattias Fitzpatrick & Andrew A. Houck

Published: July 2019

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