YOU are running through an open field with the wind in your hair. Or you are diving into the ocean, feeling the cool water surround you. At moments like these we feel free, liberated. Few of us stop to consider the truth – that we are trapped in an invisible prison.
Up-down, left-right, forward-back: these are the three dimensions in which we eat and breathe, make friends and grow old. As prisons go, it could be worse. Then again, we have never known anything else. Despite some imaginary claims to the contrary, no one has ever really experienced a higher dimension.
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But now, in some of the world’s most sophisticated labs, we are building our own synthetic extra dimensions. The concept is so far removed from our experience that it is hard to imagine what they could be like. We have, however, already seen the ghostly effects of four-dimensional space touch on our own and wired up electric circuits with an extra dimension. It is unlikely to stop there. Now we have got the hang of it, there is talk of creating five, six or even more dimensions, and even suggestions that exotica such as new particles might lurk in the extra-dimensional wilderness.
This is a frontier that we are barred from exploring directly. We are forced instead to look for the subtle imprints that extra dimensions make on the three dimensions we are confined to. Even so, we could be about to extend the boundaries of reality in ways that come close to the limits of our descriptive powers.
Talk of extra dimensions might sound a bit mystical, but spatial dimensions have a clear definition. They are a way of describing our possible range of movement. In normal space, you only need three of them – usually labelled x, y and z. True, time is sometimes referred to as the fourth dimension, and physics tells us that it is married to space in the union known as space-time. But that is as far as we conventionally go. Even the majority of physicists seem to be resigned to just three dimensions. If they were seriously expecting more, they might not have chosen their labels from the end of the alphabet. Our struggle to grasp extra dimensions is nicely captured in Edwin A. Abbott’s 1884 novella Flatland, which set out to criticise the small-mindedness of Victorian England by portraying a flat world inhabited by two-dimensional shapes. When the square-shaped narrator is visited by a sphere, he has great difficulty believing in the existence of a third dimension. All he can perceive of the visitor is the shape created by its intersection with his familiar two dimensions – a circle. Likewise, when the narrator has a dream in which he visits a one-dimensional world, Lineland, the locals reject his tales of the second dimension: all they can see are the dots he casts on their narrow path.
“Physicists seem resigned to just three dimensions. If they were expecting more, they might not have chosen their labels from the end of the alphabet”
Life on the edge
The story of synthetic dimensions also begins in a flatland, in materials that are wafer-thin and therefore, to all intents and purposes, two dimensional. If you apply a magnetic field to such a wafer, it makes all the electrons inside it want to move in tiny circles. And that is just what happens – except at the edges, where there isn’t enough space and the electrons’ trajectories are chopped off into semicircles. But instead of stopping in their tracks, these electrons zip along the edge, forming a conducting periphery. This is called the quantum Hall effect, and it creates a material that is electrically insulating in the middle but conducting on the sides.
This unusual duality depends on the one-dimensional edge feeling the effects of a higher dimension. To see how this works, imagine a one-dimensional line, much like Lineland in the novella, with electrons sitting on it. If you apply a magnetic field to this, the electrons can’t move in circles; that isn’t possible in one dimension, so they remain fixed in place. If this line is the edge of a wafer, however, the electrons can skip through the two-dimensional plane. This edge conductivity is known as a topological state.
If a one-dimensional line can pull neat tricks when it feels the imprint of another dimension, can higher dimensions do the same? The answer is yes. In 2008, decades after the original discovery of the quantum Hall effect, physicists discovered a similar phenomenon in which the electrons on a 2D surface skip through the 3D innards of a material. Like all topological states, these materials have a useful characteristic. It turns out that any impurities on the surface won’t impede an electron’s progress because it can always skip through a higher dimension. This makes them good electrical conductors. Some physicists think they will be useful in designing superfast quantum computers.
Long before this, in 2001, the late theorist Shou-Cheng Zhang and his colleague Jiangping Hu, then both at Stanford University in California, dared to consider a wild progression. Would it be possible to create a four-dimensional analogue of the quantum Hall effect, one in which a regular three-dimensional material felt the imprint of a fourth dimension? They ended up . But it seemed destined to remain theoretical – it was tough to picture how this kind of maths could be made real.
Lately, though, a few physicists have given it a go, including at the University of Birmingham, UK. “Trying to understand higher-dimensional physics is like crossing into a different universe. You don’t know what’s waiting beyond that frontier,” she says. She wanted to see what new physics might be there, so set out to try to make Zhang’s ideas a reality.
To see how, let’s briefly return to Flatland once more. In the story, the sphere finally persuades the square of the existence of the third dimension by bobbing up and down, and thereby changing the size of his intersection with the square’s plane of vision. He starts as a dot when he and the plane are just in contact, becomes a big circle when his equator passes through the plane, and reverts to a dot when he is all the way through (pictured, far right). The late theorist David Thouless developed a real analogue of this process in the 1980s. It is called a topological pump and entails changing the distance between particles in an array in such a way that it looks like a higher dimensional object is being “pumped” through them.
In 2018, Immanuel Bloch at the Max Planck Institute of Quantum Optics in Germany, together with Price and others, created a lattice of atoms held in place by lasers. By tweaking the lasers, they could deform the lattice to generate the ghostly shimmer of a four-dimensional object. It was a real-world example of what the square had experienced with the sphere in Flatland – and, together with a , it was . “We were a bit spooked when we wrote the papers,” says co-author Oded Zilberberg at the Swiss Federal Institute of Technology in Zurich. “We thought people might think we’re dealing with science fiction.”
Despite Zilberberg’s enthusiasm, it is hard to point to what, or where, the fourth dimension is in these experiments. It could be seen as an illusion cast on the positions of the atoms when their behaviour is viewed over time. Freeze the system at any instant and there is little sign that anything special is happening. “Our experiments weren’t 4D enough,” says Price.
Far better than conjuring the impression of a fourth dimension is actually building one – so that is what Price did next. To understand how it works, again picture a scenario in two dimensions. Start with a grid on a sheet of paper. Now redraw all the points on that grid in a row, and connect them up with squiggly lines – don’t worry about crossing them – so that they are connected with their original neighbours. What you have just drawn, topologically speaking, is a two-dimensional grid in one dimension (see “Two dimensions in one”, left). Now, replace the points with electrical components, and the lines for wires, and you have a situation like the quantum Hall effect, where electrons can skip through a higher dimension to get to where they want to go.
“With the shackles of traditional dimensions cast off, things could quickly get wilder”
Expect fireworks
Earlier this year, based on a concept of Price’s, Yidong Chong at Nanyang Technological University in Singapore and his colleagues expanded this kind of circuit to include components not just in a row, but multiple rows and layers. Applying a voltage across the edges of the stack had no effect: it didn’t conduct. But when the researchers applied a voltage to the components that would have marked the edge of the four-dimensional grid, had they not been rewired into a three-dimensional space, . And unlike the previous experiments, the effect wasn’t time-dependent. “It’s a permanently four-dimensional lattice,” says Price.
With the shackles of traditional dimensions cast off, things could quickly get wilder. Price, Zilberberg and others say that topological pumps could manifest the quantum Hall effect in six dimensions. Theorist Motohiko Ezawa at the University of Tokyo in Japan says that electric circuits have the potential to manifest as many dimensions as experimentalists have the patience to wire up.
But as we build experiments that are governed by more than four dimensions, the behaviour we observe from our limited three-dimensional vantage point is no longer going to be easy to make sense of. It won’t be as simple as the Flatlanders discerning a sphere as a circle of changing width. Instead, we can expect a firework display of bewildering effects.
In 2018, for instance, Seiji Sugawa at Kyoto University in Japan and his colleagues laser-cooled a cloud of rubidium atoms in such a way that its internal states obeyed the mathematical rules of five dimensions. The signal that trickled out had the weird hallmarks of a magnetic monopole, an exotic object that, unlike any normal magnet, has only a north or south pole, not both. Zhang calculated that it might even be possible to , which behaves like a massless electron. It isn’t clear whether these could be harnessed for a practical application though.
There is a bigger question before we get to that: are these synthetic dimensions real? In the case of Price’s electric circuit, the extra dimension certainly has tangible effects. Yet there is a difference between it and the three dimensions we normally experience. We can still see the interconnects that collectively manifest the extra dimension: it is as if the fourth dimension is somehow wrapped up within the three dimensions of regular space. Experiments like this ask us to pretend that the interconnects – or the lasers, or whatever else is responsible for sustaining the extra-dimensional physics – aren’t there at all, like stagehands at the theatre.
“There is a more powerful way to build a synthetic dimension, one that relies upon quantum rules”
at Princeton University raises another issue. If we take Price’s four-dimensional electrical circuit, he says, the primary flow of the electrons is governed by the synthetic dimension, but their natural interactions – for example, the mutual repulsion that results from their charge – may well still be governed by the three normal dimensions. As a result, making our own dimensions might not be a reliable way to study the finer points of extra-dimensional physics. “That’s my worry,” says Ryu.
Still, the story might not end there, according to Ryu. There is a more powerful way to build a synthetic dimension, one that relies on quantum particles such as atoms having energies that go up in discrete steps. You can think of these energy states as being like a ladder, which the particle can hop up or down. This might feel like a trick because the particle itself doesn’t actually move – its extra dimension is contained within a fixed position in three dimensions. But this is exactly the point. This type of extra dimension is completely independent of the regular three. Magnetic fields could be used to tune the way each particle interacts with its neighbours – whichever dimension they are in – and this would potentially allay Ryu’s concern about unnatural behaviour. “This is my dream,” says Price.
The dream is perhaps not far from coming true. In 2015, of physicists created synthetic dimensions of the quantum energy-ladder variety out of atoms, making systems of two dimensions in total. That is still two dimensions short of the other implementations, but Price believes that this strategy could ultimately provide a way to explore bona fide extra-dimensional physics.
If we pull that off, then Ryu is hoping there will be new applications, such as making it easier to link up the quantum bits, or qubits, that form the basis of emerging quantum computers. The fact that topological states are protected from impurities and other sources of disruption suggests that those states could deliver high volumes of data without fear of signal loss.
Studying the behaviour of synthetic dimensions could also help us understand the possible role of extra dimensions in fundamental physics. Thanks to Ryu and others, the theory behind topological states is so well mapped out that there exists a “periodic table” of possible extra-dimensional behaviours, dictated by how much underlying symmetry there is in the system. According to Ryu, it may be no coincidence that there are 10 classes of symmetry in the table and 10 dimensions in string theory – the most famous extra-dimensional idea in physics. “There’s certainly mathematical connections between them,” he says.
We are, it seems, only just realising that we have been living in our own Flatland. “It opens our minds,” says Zilberberg. “We can explore things like this for real.”
