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The strange deformation geometry helps to push the scientific limits



The strange deformation geometry helps to push the scientific limits

The Princeton researchers have constructed an electronic matrix in a microchip that simulates the interactions of particles in a hyperbolic plane, a geometric surface in which space moves away from itself at each point. Credit: Kollár et al.

The atomic interactions in everyday solids and liquids are so complex that some of the properties of these materials continue to elude physicists' understanding. Solving problems mathematically is beyond the capabilities of modern computers, so scientists at Princeton University have turned to an unusual branch of geometry.

The researchers led by Andrew Houck, professor of electrical engineering, have built an electronic matrix in a microchip that simulates the interactions of particles in a hyperbolic plane, a geometric surface in which space moves away from itself at each point. A hyperbolic plane is hard to imagine: the artist M.C. Escher used hyperbolic geometry in many of his hallucinatory pieces, but is perfect for answering questions about particle interactions and other challenging 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 respond to a wide range of difficult questions by looking at the interactions of photons in the simulated hyperbolic space.

"You can gather particles, activate a very controlled amount of interaction between them and see the complexity emerge," said Houck, who was the lead author of the article published on July 4 in the journal. Nature.

Alicia Kollár, a postdoctoral researcher at the Center for Complex Materials at Princeton and lead author of the study, said the goal is to allow researchers to address complex questions about 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, then computer modeling is very difficult, we are trying to implement a model at the hardware level so that nature makes the most difficult part of the calculation for you." said Kollár.

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

The strange deformation geometry helps to push the scientific limits

A schematic of the resonators in the microchip, which are arranged in a network pattern of heptagons, or seven-sided polygons. The structure exists in a plane, but simulates the unusual geometry of a hyperbolic plane. Credit: Kollár et al.

"In normal 3-D space, a hyperbolic surface does not exist," Houck said. "This material allows us to start thinking about mixing quantum mechanics and curved space in a laboratory environment."

In trying to force a three-dimensional sphere in a two-dimensional plane, it is revealed that space in a spherical plane is smaller than in a plane. This is why the shapes of the countries appear stretched when they are drawn on a flat map of the spherical Earth. In contrast, a hyperbolic plane should be compressed to fit a 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 compressing the hyperbolic space on a flat surface, the researchers used a special type of resonator called a coplanar waveguide resonator. When microwave photons pass through this resonator, they behave in the same way if their path is straight or meandering. The serpentine structure of the resonators offers flexibility to "crush and compress" the sides of the heptagons to create a flat mosaic pattern, Kollár said.

Looking at the central heptagon of the chip is similar to looking through the lens of a fish-eye camera, in which the objects at the edge of the field of view appear smaller than at the center: the heptagons look smaller the more far from the center. This arrangement allows the microwave photons that move through the resonator circuit to behave like particles in a hyperbolic space.

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

But first, Kollár and his colleagues will need to further develop the photonic material, both by continuing to examine its mathematical basis and by introducing elements that allow the interaction of the photons in the circuit.

"By themselves, microwave photons do not interact with each other, they pass through them," Kollár said. Most applications of the material would require "doing something to do it so they can know there is another photon there."


Natural material discovered showing hyperbolicity in the plane.


More information:
Alicia J. Kollár et al, Hyperbolic gratings in quantum electrodynamics of circuits, Nature (2019). DOI: 10.1038 / s41586-019-1348-3

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The geometry of strange deformation helps to push the scientific limits (2019, July 12)
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