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.”

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.”

“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.

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

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