Strange geometry of deformation helps push the scientific boundaries

The atomic interactions in solids and everyday liquids are so complex that some properties of these materials are beyond the comprehension of physicists. The mathematical resolution of problems exceeds the capabilities of modern computers. That's why scientists at Princeton University have turned to an unusual branch of geometry.

Researchers led by Andrew Houck, professor of electrical engineering, have built an electronic chip that simulates particle interactions in a hyperbolic plane, a geometric surface in which space moves away from itself at each point. It's hard to think of a hyperbolic plan: the artist MC Escher has used hyperbolic geometry in many of his works, but is perfect for answering questions about particle interactions and other difficult mathematical questions. .

The research team used superconducting circuits to create a network that functions as a hyperbolic space. When researchers introduce photons into the network, they can answer a wide range of difficult questions by observing the interactions of photons in a simulated hyperbolic space.

"You can throw particles together, activate a very controlled amount of interaction between them and see the complexity emerge," said Houck, lead author of the paper published July 4 in the newspaper. Nature.

Alicia Kollár, a postdoctoral research associate at the Center for Complex Materials at Princeton and lead author of the study, said the goal was to enable researchers to address complex issues regarding quantum interactions, which govern the behavior of atomic and subatomic particles.

"The problem is that if you want to study a very complicated quantum mechanical material, computer modeling is very difficult, we try to implement a model at the hardware level so that nature does the hard part of the calculation for you. , "said Kollár.

The centimeter size chip is etched with a circuit of superconducting resonators that provide paths for microwave photons to move and interact. The resonators on the chip are arranged in a network of heptagons or seven-sided polygons. The structure exists on a flat plane, but simulates the unusual geometry of a hyperbolic plane.

"In the normal 3D space, there is no hyperbolic surface," Houck said. "This material allows us to start thinking about mixing quantum mechanics and curved space in a lab environment."

Trying to force a three-dimensional sphere on a two-dimensional plane reveals that the space on a spherical plane is smaller than on a flat plane. This is why country shapes appear stretched when they are drawn on a spherical Earth flat map. In contrast, a hyperbolic plane should be compressed to fit on a flat plane.

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

To simulate the effect of hyperbolic space compression on a flat surface, the researchers used a special type of resonator called coplanar waveguide resonator. When the microwave photons pass through this resonator, they behave in the same way, whether their path is straight or sinuous. The sinuous structure of the resonators provides the flexibility to "crush and pinch" the sides of the heptagons to create a flat mosaic pattern, Kollár said.

Watching the central heptagon of the flea returns to looking through a fisheye lens, in which the objects at the edge of the field of view appear smaller than at the center: the heptagons are all the smaller that they are far from the center. This arrangement allows the microwave photons moving in the resonator circuit to behave like particles in a hyperbolic space.

The ability of the chip to simulate a curved space could allow new studies in quantum mechanics, including the properties of energy and matter in space-time deformed around black holes. The material could also be useful for understanding complex networks of relationships in the theory of mathematical graphs and communication networks. Kollár noted that this research could eventually help in the design of new materials.

But first, Kollár and his colleagues will have to further develop the photonic material, both by continuing to examine the mathematical underpinnings and by introducing elements allowing the photons of the circuit to interact.

"By themselves, microwave photons do not interact with each other, they pass directly through," Kollár said. Most applications of the material would require "to do something to make it so that it can detect the presence of another photon."