How do you recognize additional dimensions

A look into the fourth dimension

The “fourth dimension” is usually understood to mean the time since Albert Einstein developed the special theory of relativity in Zurich in 1905. But how can you get a fourth spatial Imagine dimension - in addition to top-bottom, right-left, front-back?

In art, for example, Salvador Dalí attempted this: his crucifixion scene, painted in 1954, shows a crucifix that consists of the three-dimensional unfolding of a hypercube in four dimensions (similar to the unfolding of a cube into squares).

Two teams of scientists from Switzerland, the USA, Germany, Italy and Israel have now gained a completely different, but no less fascinating insight into the fourth dimension of space. ETH researcher Oded Zilberberg, professor at the Institute for Theoretical Physics, played a central role in both papers, which were recently published in the journal Nature. He created the theoretical basis for the experiments in which a four-dimensional physical phenomenon could be observed in two dimensions.

The quantum Hall effect

Both experiments dealt with the so-called quantum Hall effect. This effect usually occurs in interfaces between two materials in which electrons can only move in two dimensions. A magnetic field perpendicular to the material then leads to the classic Hall effect. If a current flows through the material, an electrical voltage is created perpendicular to the direction of the current - the larger the magnetic field, the higher the voltage. This is because the magnetic field creates a force that acts at right angles to the direction of motion (the Lorentz force) and deflects the electrons.

At very low temperatures and very strong magnetic fields, however, quantum mechanics comes into play, so that the voltage no longer increases continuously, but in discrete steps. So far, three Nobel Prizes in physics have been awarded for experimental and theoretical work on the quantum Hall effect.

A question of topology

The quantum Hall effect can also be understood as a topological phenomenon. The topology describes, for example, how many "holes" an object has and what other shapes it can be converted into without cutting. Similar laws lead to the quantum Hall effect that the electrons can only move on topologically precisely defined paths. For example, for certain strengths of the magnetic field, the electric current can only flow at the edges of the material, but not inside it.

About twenty years ago it was shown mathematically that there should also be corresponding topological effects in four spatial dimensions. “Back then, it was more like science fiction,” says Oded Zilberberg, “because actually observing something like this in experiments seemed impossible - physical space just has three dimensions. »

Virtual dimensions through topological pumping

But Zilberberg had a clever idea: With the help of so-called topological pumps, it should be possible to add a virtual dimension to each of the two real dimensions of the quantum Hall effect. A topological pump works by modulating a certain control variable of the physical system, whereby its quantum state changes in a characteristic way over time. The end result looks like the system has moved in an additional spatial dimension. In this way one can theoretically transform a two-dimensional system into a four-dimensional one.

An optical image of the fourth dimension

Scientists have now shown in two independent experiments that this also works in practice. Physicists headed by Mikael Rechtsman at Penn State University in the USA realized Oded Zilberberg's idea together with Kevin Chen's research group at the University of Pittsburgh by using laser beams to burn a two-dimensional grid of waveguides into a six-inch-long glass block. These waveguides were not straight, however, but instead wound in a serpentine fashion through the glass, so that the distances between them changed along the glass block. Light waves moving through the waveguide could, depending on the distance, jump over to an adjacent waveguide more or less easily.

These constantly changing couplings between the waveguides acted as topological pumps, doubling the number of dimensions in the experiment from two to four. The researchers were now able to literally "see" the expected four-dimensional quantum Hall effect by feeding light into the waveguides on one side of the glass block and recording with a video camera what came out on the other side. For example, the edge states characteristic of the four-dimensional quantum Hall effect, in which light should only exit from the waveguides at the edge of the grating, became directly visible.

Four-dimensional quantized transport of cold atoms

With the help of extremely cold atoms trapped in optical lattices made of crossed laser beams, Immanuel Bloch and his colleagues at the Max Planck Institute for Quantum Optics in Munich also realized topological pumps. In their experiment, pumping was achieved by periodically changing the properties of the split lattice pots in which the atoms were trapped.

By measuring the two-dimensional movement of the atoms in the lattice, they were able to confirm that they behaved in four dimensions according to the quantum Hall effect. In particular, they were able to directly observe the quantized transport phenomena that had been predicted for this case (and which correspond to the voltage occurring perpendicular to the direction of the current in the ordinary, two-dimensional quantum Hall effect).

Progress in basic research

The practical use of it all? "At the moment, these experiments are still miles away from useful applications," admits Zilberberg. But they represent an important step forward for basic research. Physicists can now investigate not only on paper but also experimentally what effects phenomena from four (or even more) dimensions can have in our everyday three-dimensional world.

An example of this are quasicrystals in metal alloys. In three spatial dimensions these do not have a periodic structure, but if you look at them in virtual higher dimensions, they again show regular patterns. And finally there is string theory, according to which higher spatial dimensions should be “compactified” in such a way that in the end our normal three-dimensional world emerges.