Seeing four dimensions in two

The four-dimensional expression of an important quantum phenomenon has been observed – by using a two-dimensional model.

The quantum Hall effect (QHE) – a quantum-mechanical outcome arising from electrons confined in a two-dimensional space at very low temperatures and influenced by a strong magnetic field – has been central to at least two Nobel Prizes, in 1985 and 1998.

Modelling the QHE is a popular focus of research because it enables insights into quantum disordered ground states and incompressible quantum fluids. For a long time, however, calculations indicated that the specific wave functions used to describe it were unique to two-dimensional spaces.

However, in 2001 US physicist Shou-Cheng Zhang from Stanford University and colleague Jiangping Hu from China’s Tsinghua University showed that the QHE also arose in a theoretical four-dimensional space.

It was a clever exercise in mathematics, but of little practical use. Until now.

“When it was theorised that the quantum Hall effect could be observed in four-dimensional space,” said Mikael Rechtsman, a physicist at Pennsylvania State University in the US, “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.”

Now, however, Rechtsman and colleagues have succeeded in demonstrating the effect in four dimensions, by creating laser-generated two-dimensional glass “tubes” dubbed wave-guides, and inscribing them in such a way that they evince additional “synthetic” dimensional attributes.

The research is published in the journal Nature.

In two dimensions, the QHE arises when electrons are sandwiched between two surfaces, cooled to just above absolute zero and bathed in a strong magnetic field. The ability of the electrons to conduct charge becomes quantised – fixed to certain constants – and does so regardless of any defects or “mess” within the system.

“This robustness of electron flow – the quantum Hall effect – is universal and can be observed in many different materials under very different conditions,” says Rechtsman.

It cannot happen within three dimensions, however. Four is a different matter.

“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,” explain Rechtsman.

To make the array, the researchers laser inscribed dozens of closely packed tubes through a piece of glass. Each tube effectively acts as a wire along which photons travel.

The tubes were further enhanced by the addition of inscribed synthetic dimensions, resulting in a very complex over all pattern. In effect, the two-dimensional system had a total of four spatial dimensions.

When the researchers measured the behaviour of photons pumped through the waveguide array, they behaved in exactly the way the four-dimensional theoretical QHE model predicted. The result, says Rechtsman, is “the first demonstration of higher-dimensional quantum Hall physics”.

The researchers say their results could eventually have implications for the design of novel photonic devices.