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.