Mind and space bending physics on a convenient chip

Thanks to Einstein, we know that our three-dimensional space is warped and curved. And in curved space, the general ideas of geometry and straight lines are broken, creating an opportunity to discover unfamiliar landscapes governed by new rules. But studying how physics plays out in a curved space is challenging: just like real estate, location is everything.

“We know from general relativity that the universe itself is curved in different places,” says JQI Fellow Alicia Cole, who is also a professor of physics at the University of Maryland (UMD). “But, any place where there really is a lab is very weakly curved because if you have to go to one of these places where gravity is strong it will just rip the lab apart.”

There are more geometric rules than the spaces we normally recognize, called non-Euclidean. If you can explore the atmosphere other than Euclid, you will find amazing landscapes. Space can be compressed so that straight, parallel lines draw hard distances together instead of maintaining solid distances. Or they can be extended so that they move ahead of each other forever. In such a world, four equal-length roads that are all connected by right turns at right angles may fail to form a square block that will take you back to your initial intersection.

This environment overturns the basic assumptions of general research and may be impossible to visualize accurately. Non-Euclidean geometry is so alien that it has been used in videogames and horror stories as unnatural landscapes that challenge or unsettle audiences.

But this unfamiliar geometry is much more than just distant, other equine abstractions. Physicists are interested in new physics that can reveal curved space, and geometry other than Eucliden can also help improve the design of certain technologies. One type of non-Euclidean geometry that is interesting is hyperbolic space, also called negative-curved space. Even a two-dimensional, physical version of the hypersensitive space is impossible to create in our normal, “flat” atmosphere. But scientists can still mimic hyperbolic environments to explore how certain physics plays into a negatively curved space.

In a recent paper from Physical Review A, the collaboration between the groups of Kolar and JQI Fellow Alexei Gorshkov, a physicist at the National Institute of Standards and Technology and a Fellow of the Joint Center for Quantum Information and Computer Science, is presented. New mathematical tools to better understand the simulation of hyperbolic spaces. Cole is researching previous experiments to mimic orderly grids in hyperbolic space using microwave lights contained on chips. Their new toolbox includes what they call “a dictionary between different and continuous geometry” to help researchers translate experimental results into a more useful form. With the help of these tools, researchers can better discover the Topsy-Turvy world of hypersensitive space.

The situation is not like coming down from a rabbit hole like Alice, but these experiments are an opportunity to discover a new world where amazing discoveries are hiding behind any corner and the very meaning of turning a corner should be reconsidered.

“There are really a lot of applications to these experiments,” says Igor Botcher, a JQI postdoctoral researcher who was the first author of the new paper. “At the moment, it’s unbelievable what all that can be done, but I expect it will have a lot of rich applications and a lot of cooling physics.”

A curved New World

In a flat space, the shortest distance between two points is a straight line, and parallel lines never intersect, no matter how long. In a curved space, these basics of geometry are no longer true. When the mathematical definition of flat and curve is applied to two-dimensions, the meanings of days are the same. You can get a sense of the basics of curved spaces by imagining or actually playing around with a piece of paper or a map.

For example, the surface of a globe (or any ball) is an example of a two-dimensional positive curved space. And if you try to make a flat map in a globe, you get more paper wrinkles as you turn the sphere. To get a smooth sphere you have to lose a lot of space, resulting in parallel lines eventually meeting, lines of longitude that start parallel to the equatorial meeting at two poles. Because of this disadvantage, you may want to consider a positive-curved space with less space than a flat one.

Hyperbolic space is a positive curved space as opposed to the over-spaced space. A hypersensitive space turns away from itself at every point. Unfortunately, there is not exactly a hypersensitivity of the ball in which you can push on a two-dimensional sheet; It will literally not fit into the space in which we live.

The best you can do is shape a saddle (or a pringle) where the surrounding sheet is hypersensitively bent away from the center point. It is impossible to make every point on the sheet equally hyperbolic; Without crowding and distorting the first hypersensitive saddle point, there is no way to add paper and add paper to create a second perfect saddle point.

The extra space of hyperbolic geometry makes it especially interesting because it means there is more scope to form connections. The difference in potential pathways between the points affects how the particles interact and what types of similar grids can be created, such as the heptagon grid as shown above. Taking advantage of the additional connections that are possible in a hyperbolic space can make it difficult to cut parts of the grid completely apart, which affects the design of networks such as the Internet.

Explore Labyrinth Circuits

Since it is physically impossible to create hyperbolic space on Earth, researchers must settle to create lab experiments that reproduce some of the features of curved space. Caller and colleagues have previously demonstrated that they can simulate the same, two-dimensional curved space. Similarities are made using circuits (as shown above) that serve as a very systematic way to travel through microwaves.

One characteristic of circuits is that microwaves are indifferent to the shape of the resonators in which they are inserted and are only affected by the total length. It doesn’t matter which angle the different routes connect to. Keller realized that these facts meant that the physical space of a circuit could be effectively stretched or squeezed to create a non-Euclidean space, as far as microwaves were concerned.

In their previous work, Claire and colleagues were able to create mazes with different zig-zagging path shapes and simulate circuits hyperbolic space. Despite the convenience and orderliness of the circuits they used, physics still represents a fantastic new world that requires new mathematical tools to navigate efficiently.

Hyperbolic spaces present physicists with different mathematical challenges than the spaces in which they normally operate. For instance, researchers cannot use the standard physics trick to visualize what happens to an infinitely small grid, a grid that gets smaller and smaller to figure out what works like a simple, continuous space. This is because in hyperbolic space, the shape of the lattice changes with its size due to the curvature of space. The new paper establishes mathematical tools to intercept these issues and make the results of the simulations meaningful, such as a dictionary between independent and continuous geometry.

With the new tools, researchers can obtain accurate mathematical descriptions and predictions rather than just making qualitative observations. Although the simulation is only of the grid, the dictionary allows them to study hyperbolic spaces continuously. With the dictionary, researchers can take descriptions of microwaves traveling between specific points on the grid and translate them into equations describing simple propagation, or convert mathematical sums into integrals at all sites on the grid, which is more convenient in certain situations. .

“If you let me experiment with a certain number of sites, this dictionary tells you how to translate it into a setting in a constantly hyperbolic space.” “With the dictionary, we can find all the relevant parameters you need to know in a laboratory setup, especially for limited or small systems, which are always experimentally important.”

With the help of new tools to help understand simulation results, researchers are better equipped to answer questions and make discoveries with simulations. Botcher says he is optimistic that simulations will be useful for investigating ADS / CFT correspondence, a physics hypothesis for combining quantum gravity theories and quantum field theories using a non-Euclidean description of the universe. And Keller plans to explore whether these experiments can reveal even more physics by incorporating interactions into the simulation.

“Hardware opened a new door,” says Keller. “And now we want to see what physics this will let us go.”