# How many dimensions did you say?

My friends and colleagues from ICFO, Alessio Celi and Maciej Lewenstein (along with O. Boada and J.I. Latorre), have just published a surprising article in Physical Review Letters, which appears in its Synopsis. What is the big deal?  They propose a route to simulate the behaviour of quantum matter in extra dimensions. The idea is extremely simple once it has been understood. But let me start by telling you what is the framework, I mean: what do I mean by simulation.

Consider that you would like to design a new material, which you want to have some nice properties, such as superconductivity, or a given response to magnetic fields… whatever. Most of these properties are given by the quantum behaviour of electrons inside the crystal. The problem is that the behaviour of interacting electrons in a given system is very hard to predict theoretically, using either pencil and paper calculations or huge supercomputers. What to do, then?

When aeronautical engineers design a new airplane, they do some complex calculations. After that, since they can not rely completely on them, they make a model plane and test it in a wind tunnel. They perform a controlled simulation. If the solution to their equations
coincides with the results of the simulation, then they feel confident about the airplane, and the manufacture procedure begins.

Quantum simulators follow the same idea as the wind tunnel and the model airplane. Set up many laser beams, making up a 3D lattice. The lattice spacing will be much larger than in crystalline solids, more than one μm. Now, instead of electrons, we put some ultra-cold atoms. But, I hear you say, atoms are not elementary particles, unlike electrons. There is a nice response to that: anything is an elementary particle until you hit them hard enough! In other words: atoms behave totally like elementary particles if the temperature and the interaction energy is low enough. If they have total spin 1/2, then the atoms are fermionic and behave much like electrons.

So, you have all the elements. Now, let us check a possible design for a material with some concrete properties: set up your optical lattice, put some ultra-cold atoms in there and see. The best part is that if you do not strike oil at the first attempt, you can always change your parameters almost on-the-fly and try again: tune the lasers, heavier atoms… whatever.

Now that we know what a quantum simulator is, let us focus on the novel part: the work of my colleagues. Many speculative theories in physics require the existence of extra-dimensions. If they exist, then their extension must be really small not to appear in ordinary experiments. I do not mean that those theories should be taken seriously, only that we might desire to find out what would be the implications!

Imagine that we prepare our optical lattice and leave our atoms inside. Atoms jump from a cell to the next tunneling through the laser beam. Now, consider atoms that can be in N different internal states, which differ, for example, in the nuclear spin direction. So to speak, N atomic flavours which are nearly indistinguishable. Label the internal atomic states from 1 to $N$, and arrange things so that atoms can only move from state $i$ to state $i+1$ or $i-1$. Now, by tuning up the laser intensities, we can make this movement in internal state to be exactly as movement in any other direction!

The image shows in blue the atoms with flavour 1, and in pink those with flavour 2. An atom at site $d$ can jump up, right, back… but it can also change flavour. And that jump would correspond to a movement in the fourth dimension. Of course, the extension of this fourth dimension is extremely reduced if we have only two flavours. In general, we will not be able to achieve huge sizes, but this is not a problem since, as we stated, the extra dimension, if it exists, must be extremely small.

For example, we can arrange a single atom in a given cell, with a given internal state, and let it evolve freely. After some time, it will be in another cell and with another internal state. This internal state will mark how much it has moved in the extra-dimension.

V.I. Arnold, one of the great masters, once said that mathematics is the part of physics where experiments are cheap. Well, the cost of the mathematical experiment must always be compared to the cost of the real one. Using an expensive supercomputer to follow the behaviour of all the atoms of a stone as it falls to the ground does not seem to be a huge saving. But using ultracold atoms in an optical lattice to simulate 4D space qualifies much better… most of all because we are not aware of any other experimental setup! :)