Thanks to Einstein, we know that our three-dimensional space is warped and curved. And in the curved space, the normal ideas of geometry and straight lines break, creating the possibility of exploring an unknown landscape governed by new rules. But studying how physics works in a curved space is challenging – just like in real estate, location is everything.
“We know from general relativity that the universe itself is curved in various places,” says Alicia Kollár, a colleague at JQI, who is also a professor of physics at the University of Maryland (UMD). “But any place where there is actually a laboratory is very weakly curved because if you were to go to one of these places where gravity is strong, the laboratory would fall apart.”
Spaces that have geometric rules other than those we usually take for granted are called non-Euclidean. If you could explore non-Euclidean environments, you will find landscapes that are perplexing. The space could contract so that straight, parallel lines join together instead of rigidly maintaining fixed spacing. Or it could expand so that they recede forever. In such a world, four streets of equal length that are all connected by right-angled right turns may fail to form a square block that takes you back to your initial intersection.
These environments overturn the fundamental assumptions of normal navigation and may be impossible to visualize accurately. Non-Euclidean geometries are so alien that they have been used in video games and horror stories as unnatural landscapes that challenge or upset audiences.
But these unknown geometries are much more than distant and otherworldly abstractions. Physicists are interested in the new physics that curved space can reveal, and non-Euclidean geometries could even help improve the designs of some technologies. One type of non-Euclidean geometry of interest is hyperbolic space, also called space with negative curvature. Even a two-dimensional physical version of a hyperbolic space is impossible to achieve in our normal “flat” environment. But scientists can still mimic hyperbolic environments to explore how certain physics plays out in space with negative curvature.
In a recent article in Physical Review A, a collaboration was presented between Kollár’s groups and JQI Fellow Alexey Gorshkov, who is also a physicist at the National Institute of Standards and Technology and a Fellow of the Joint Center for Quantum Information and Computer Science , presented new mathematical tools to better understand the simulations of hyperbolic spaces. The research builds on Kollár’s previous experiments to simulate ordered grids in hyperbolic space using the microwave light contained on the chips. Their new toolbox includes what they call a “dictionary between discrete and continuous geometry” to help researchers translate experimental results into a more useful form. With these tools, researchers can better explore the upside-down world of hyperbolic space.
The situation isn’t quite like Alice falling down the rabbit hole, but these experiments are an opportunity to explore a new world where surprising discoveries may be hiding around every corner and the very meaning of turning a corner needs to be reconsidered.
“There are really a lot of applications of these experiments,” says JQI postdoctoral researcher Igor Boettcher, who is the first author of the new paper. “At this point, it is unpredictable what can be done, but I expect it will have many rich applications and a lot of interesting physics.”
A new curved world
In flat space, the shortest distance between two points is a straight line, and parallel lines will never intersect, no matter how long they are. In a curved space, these foundations of geometry are no longer valid. The mathematical definitions of flat and curved are similar to everyday meaning when applied to two dimensions. You can get an idea of the basics of curved spaces by imagining – or actually playing with – pieces of paper or maps.
For example, the surface of a globe (or any ball) is an example of a positively curved two-dimensional space. And if you try to turn a flat map into a globe, you end up with the excess paper wrinkling as you fold it into a sphere. To have a smooth sphere it is necessary to lose excess space, resulting in parallel lines that eventually meet, such as longitude lines that start parallel to the equator that meet at the two poles. Because of this loss, you can think of a positively curved space as less spacious than the flat space.
Hyperbolic space is the opposite of a positively curved space, a larger space. A hyperbolic space moves away from itself at every point. Unfortunately, there is no hyperbolic equivalent of a ball into which you can force a two-dimensional sheet; it literally won’t fit the type of space we live in.
The best you can do is create a saddle (or Pringle) shape where the surrounding sheet curves hyperbolically from the center point. Making every point on a sheet similarly hyperbolic is impossible; there is no way to keep curving and adding paper to create a perfect second saddle point without it clumping and distorting the hyperbolic first saddle point.
The extra space of a hyperbolic geometry makes it particularly interesting since it means there is more room to form connections. Differences in possible paths between points affect how particles interact and the type of uniform grid, such as the heptagon grid shown above, can be made. Taking advantage of the extra connections possible in a hyperbolic space can make it more difficult to completely cut sections of a grid from each other, which could affect the design of networks such as the Internet.
Navigate in labyrinthine circuits
Since it is impossible to physically create a hyperbolic space on Earth, researchers have to be content with creating laboratory experiments that reproduce some of the characteristics of curved space. Kollár and colleagues have previously shown that they can simulate a uniform two-dimensional curved space. The simulations are performed using circuits (such as the one shown above) which act as a very organized maze for the passage of microwaves.
A feature of circuits is that microwaves are indifferent to the shapes of the resonators that contain them and are only affected by the total length. Furthermore, it does not matter at which angle the different paths connect. Kollár realized that these facts mean that the physical space of the circuit can be effectively stretched or squashed to create a non-Euclidean space, at least as far as microwaves are concerned.
In their previous work, Kollár and colleagues were able to create mazes with various zigzag path shapes and to demonstrate that the circuits simulated hyperbolic space. Despite the convenience and order of the circuits they used, the physics played in them still represents a strange new world that requires new mathematical tools to navigate efficiently.
Hyperbolic spaces offer physicists different mathematical challenges than the Euclidean spaces in which they normally work. For example, researchers cannot use the standard physicist’s trick of imagining a lattice getting smaller and smaller to figure out what happens to an infinitely small grid, which should act as a continuous, uniform space. This is because in a hyperbolic space the shape of the lattice changes with its size due to the curvature of the space. The new paper establishes mathematical tools, such as a dictionary between discrete and continuous geometry, to circumvent these problems and make sense of the simulation results.
With the new tools, researchers can obtain exact mathematical descriptions and predictions instead of just making qualitative observations. The dictionary allows them to study continuous hyperbolic spaces even if the simulation is only of a grid. With the dictionary, researchers can take a description of the microwaves traveling between distinct points on the grid and translate them into an equation describing regular diffusion, or convert mathematical sums across all sites on the grid to integrals, which is convenient in certain situations.
“If you give me an experiment with a number of sites, this dictionary tells you how to translate it into an environment in a continuous hyperbolic space,” says Boettcher. “With the dictionary, we can deduce all relevant parameters you need to know in the laboratory setup, especially for finite or small systems, which is always experimentally important.”
With new tools to help understand simulation results, researchers are better equipped to answer questions and make discoveries with simulations. Boettcher says he is optimistic that simulations are useful for investigating the AdS / CFT correspondence, a physical conjecture for combining quantum gravity theories and quantum field theories using a non-Euclidean description of the universe. And Kollár intends to explore whether these experiments can reveal even more physics by incorporating interactions into simulations.
“The hardware has opened a new door,” says Kollár. “And now we want to see what physics will allow us to go to.”
Weird warping geometry helps push scientific boundaries
“Quantum simulation of hyperbolic space with quantum circuit electrodynamics: from graphics to geometry”, Igor Boettcher, Przemyslaw Bienias, Ron Belyansky, Alicia J. Kollar, Alexey V. Gorshkov, Phys. Rev.A, 102, 032208 (2020). dx.doi.org/10.1103/PhysRevA.102.032208
Mind and Space Bending Physics on a Convenient Chip (2020, October 8)
recovered on October 8, 2020
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