Researchers Observe Four-dimensional Physics in Two Dimensions

Physicists from two research groups have announced techniques that allow materials in four-dimensional space to be studied in two-dimensional systems. Both experiments dealt with the quantum Hall effect, a phenomenon that has been at the root of three Nobel Prizes in physics.

In one experiment, an international team of researchers from Penn State, ETH Zurich in Switzerland, the University of Pittsburgh and the Holon Institute of Technology in Israel demonstrated that the behavior of particles of light can be made to match predictions about the four-dimensional version of the quantum Hall effect in a two-dimensional array of “waveguides.” And in the other, a group from Germany showed that a similar mechanism can be used to make a gas of ultracold atoms exhibit four-dimensional quantum Hall physics as well.

The quantum Hall effect commonly manifests in the boundary layer between two materials, where electrons can only move in two dimensions. A magnetic field perpendicular to the material initially leads to the classical Hall effect, where a current flowing through the material gives rise to a voltage in the perpendicular direction; the larger the magnetic field, the higher the voltage. The magnetic field here generates a force acting at right angles to the direction of motion (the Lorentz force) that deviates the electrons. At very low temperatures and very large magnetic fields, however, quantum mechanics starts playing a role, meaning that the voltage no longer increases continuously but rather jumps in discrete steps.

“When it was theorized that the quantum Hall effect could be observed in four-dimensional space,” says Mikael Rechtsman, assistant professor of physics and an author of the paper from the Penn State team, “it was considered to be of purely theoretical interest because the real world consists of only three spatial dimensions; it was more or less a curiosity. But, we have now shown that four-dimensional quantum Hall physics can be emulated using photons—particles of light—flowing through an intricately structured piece of glass—a waveguide array.”

When an electric charge is sandwiched between two surfaces, the charge behaves effectively like a two-dimensional material. When that material is cooled down to near absolute-zero temperature and subjected to a strong magnetic field, the amount that it can conduct becomes quantized, or fixed to a fundamental constant of nature and cannot change. “Quantization is striking because even if the material is messy—that is, it has a lot of defects—this ‘Hall conductance’ remains exceedingly stable,” says Rechtsman. “This robustness of electron flow—the quantum Hall effect—is universal and can be observed in many different materials under very different conditions.”

This quantization of conductance, first described in two-dimensions, cannot be observed in an ordinary three-dimensional material, but in 2000 it was shown theoretically that a similar quantization could be observed in four spatial dimensions. The experiments were conducted by Oded Zilberberg, a professor at the Institute for Theoretical Physics at ETH Zurich. “At the time, however, that was more like science fiction,” Zilberberg says, “as actually observing something like that in an experiment seemed impossible—after all, physical space only has three dimensions.”

Zilberberg had a clever idea, however, using so-called topological pumps 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, causing 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 researchers can, theoretically, turn a two-dimensional system into a four-dimensional one.

To model this four-dimensional space, the researchers in Rechtsman’s team built waveguide arrays. Each waveguide is essentially a tube that behaves like a wire for light. This tube is inscribed through high-quality glass using a powerful laser. The waveguides were not straight, but rather 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.

Meanwhile, at the Max-Planck-Institute for Quantum Optics in Munich, a research team led by Immanuel Bloch also realized topological pumps using extremely cold atoms trapped in optical lattices made of crossed laser beams. In the experiment, the pumping was affected 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 researchers were able to confirm that the atoms, indeed, behaved according to the topology of the quantum Hall effect in four dimensions. In particular, the team observed the quantized transport phenomena predicted to occur in that case (which are the equivalent of the voltage perpendicular to the direction of the current in the ordinary two-dimensional quantum Hall effect).

“Right now, those experiments are still far from any useful application,” Zilberberg says. But for fundamental research they represent important progress. Physicists can now investigate not just on paper, but also experimentally the effects that phenomena occurring in four (or even more) dimensions can have in our usual three-dimensional world.

“Our observations, taken together with the observations using ultracold atoms, provide the first demonstration of higher-dimensional quantum Hall physics,” says Rechtsman. “But how can understanding and probing higher-dimensional physics have some relevance to science and technology in our three-dimensional world? There are a number of examples where this is the case. For example, quasicrystals—metallic alloys that are crystalline but have no repeating units and are used to coat some non-stick pans—have been shown to have hidden dimensions. Their structures can be understood as projections from higher-dimensional space into the real, three-dimensional world. Furthermore, it is possible that higher-dimensional physics could be used as a design principle for novel photonic devices.”