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Optically Imaging the Fourth Dimension

Photonics Handbook
A visual representation of the fourth dimension has been observed in two independent experiments. The theoretical groundwork for these experiments was laid by researchers at ETH Zürich.

Platform for study of higher dimensional topological physics, ETH Zurich.

A physical phenomenon in four spatial dimensions was realized in two experiments: with light in waveguides (winding tubes) and by using cold atoms (orange spheres) in optical lattices. Courtesy of ETH Zürich.

In both experiments, the quantum Hall effect was applied to four-dimensional systems. Typically, this effect manifests in the boundary layer between two materials, in which electrons can only move in two dimensions. A magnetic field perpendicular to the material initially leads to the classical Hall effect: A current flowing through the material gives rise to a voltage in the perpendicular direction. The larger the magnetic field, the higher the voltage. With very low temperatures and very large magnetic fields, however, quantum mechanics starts to play a role, which means that the voltage no longer increases continuously, but rather jumps in discrete steps.

The quantum Hall effect can also be understood as a topological phenomenon. Around 20 years ago, it was shown mathematically that analogous topological effects should also occur in four spatial dimensions.

“At the time, however, that was more like science fiction, as actually observing something like that in an experiment seemed impossible,” said researcher Oded Zilberberg. “After all, physical space has only three dimensions.”

Zilberberg hypothesized that by using topological pumps, it should be possible to add a virtual dimension to both of the real dimensions of the quantum Hall effect. A topological pump works by modulating a specific control parameter of the physical system, which causes its quantum state to change in a characteristic way over time. The end result then looks as though the system had been moving in an additional spatial dimension. In this way one can, theoretically, turn a two-dimensional system into a four-dimensional one.

That this can also work in practice has now been shown in two independent experiments. A team of physicists at Penn State University and including a group at the University of Pittsburgh has realized Zilberberg’s hypothesis by burning a two-dimensional array of waveguides into a 15-cm-long glass block using laser beams. The waveguides were not straight; instead, they meandered through the glass in a snake-like fashion so that the distances between them varied along the glass block. Depending on those distances, light waves moving through the waveguides could jump more or less easily to a neighboring waveguide.

The varying couplings between the waveguides acted as topological pumps and thus doubled the number of dimensions of the experiment from two to four. The researchers could now literally “see” the expected four-dimensional quantum Hall effect by feeding light into the waveguides at one end of the glass block and recording what came out at the other end with a video camera. In this way, for instance, the characteristic edge states of the four-dimensional quantum Hall effect, in which light should emerge only from the waveguides at the edge of the lattice, became directly visible.

A second team, from the Max Planck Institute for Quantum Optics, used extremely cold atoms trapped in optical lattices made of crossed laser beams to realize topological pumps. In this team’s experiment, the pumping was accomplished by periodically varying the properties of the split lattice wells in which the atoms were trapped. By measuring the resulting two-dimensional motion of atoms in the lattice, the team was able to confirm that the atoms behaved according to the topology of the quantum Hall effect in four dimensions. Specifically, the Max Planck team was able to directly observe the quantized transport phenomena predicted to occur (this is the equivalent of the voltage perpendicular to the direction of the current in the ordinary two-dimensional quantum Hall effect).

Although these experiments do not currently have practical application, they provide a platform for fundamental research into higher-dimensional topological physics. Physicists can now investigate not just on paper, but also experimentally, the effects that phenomena occurring in four (or even more) dimensions could have in our three-dimensional world.

The research was published in Nature (doi:10.1038/nature25011).

GLOSSARY
optical materials
Materials that, by virtue of their optical characteristics (i.e. refractive index, dispersion, etc.), are used in optical elements. See crystal; glass; plastic lens.
Research & TechnologyeducationEuropeAmericaslaserslight sourcesopticswaveguidestopological pumpquantum Hall effectquantum simulationmaterialsoptical materialstopological insulatorsTech Pulse

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