Here I provide more detailed (and, hopefully, pedagogical) explanations of some of the quantum-mechanical concepts used in other parts of the website.
Wavefunction
In quantum mechanics, particles typically do not have well-defined position (as well as other quantities, such as momentum and energy). Instead, there are “blurred” – there is a wavefunction, which determines the probability of finding a particle in a given place. There is, however, a catch. Instead of ordinary, real number, the values of the wavefunctions are a complex number. While this is not how you typically represent a wavefunction, you can think of it as if to each point in space you attach an arrow (see (a) below). The arrow can have any length and any direction in certain 2D plane (in a picture below, all arrows are horizontal, but can point in different horizontal direction). The length (“magnitude”) determines the probability of finding the particle in a given place. The direction (“phase”) determines how two complex numbers are added to each other. A general rule for that is shown in (b) below. If two complex numbers have the same magnitude at a given point, the result of addition depends of the phase – the resulting magnitude can be twice as large (the same phase) or zero (opposite phases) or anything in between (other phases).
Energy levels and band structure
Similarly to “blurred” position, the quantum particles usually have “blurred” energy. There are, however, special wavefunctions, called eigenstates, which have fixed value of energy. They are special, because they do not change in time, and therefore are a nice starting point for the description of the system. Eigenstates usually cannot have just any energy. For example, an electron in an atom can only occupy certain energy levels. For simplicity, we can approximate an atom as having two energy levels, called the ground state and the excited state (although, in reality, an atom has many excited states).
When two atoms form a molecule, the electrons can “hop” from one atom to the other, and, as a consequence, the energy levels change (instead of two levels, the molecule will have four, each at different energy). Similarly, if many atoms combine into a crystal, the energy levels also change – but now there are so many of them that it is hard, or even impossible, to distinguish them. Instead of single levels, we rather talk about bands – regions where the energy is allowed, and gaps – regions where it is not allowed.
The bands have more structure: the energy depends on the momentum. One can imagine the wavefunctions of electrons in crystals as waves, and momentum vector tells us in which direction the wave travels and how fast it oscillates in space (in contrast to frequency, which is related to energy and tells us how fast the wave oscillates in time). The band structure tells us what is the energy of the electron in a certain band and with certain momentum.
The band structure has tremendous importance. For example, it explains why some materials conduct electricity (a band is only partially filled by electrons) and some don’t (a band is completely filled and nothing can change until we apply enough energy to move some electrons to an another, empty band). But there are more subtle properties of bands, such as a Chern number. It is quite an abstract, mathematical quantity, which becomes of physical importance, because bands in which it is not zero exhibit many interesting physical effects. If, in addition, such bands are nearly flat (compared to the surrounding gaps), they are even more interesting. For example, sometimes they are capable of hosting topological orders called fractional quantum Hall states.
Bosons, fermions and anyons
The wavefunction of many particles is similar to the wavefunction of a single particles, but the dimensionality of the underlying space is enhanced. We need three numbers x,y,z to pinpoint where is a single particle is in 3D space, but we need six numbers (x,y,z for each particle) to describe a position of two particles. Therefore, a two-particle wavefunction “lives” in a six-dimensional space.
Particles of the same kind (e.g. two electrons) are indistinguishable. This means that the probability of finding first particle at point A and second particle at point B is the same as for finding the first particle at B and the second in A. That is, if we exchange the positions of two particles, the wavefunction can only change up to a phase – that is, the arrow (see wavefunction ) can only change its direction, but not length.
If the change is a property of a given type of particles , then it should not depend on the details of the path. We could deform the path a little, and the result should still be the same. This puts a further constraint on what the change should be. In three dimensions, we can continuously deform a clockwise exchange path into an anticlockwise one. That is, the clockwise and anticlockwise exchange should have the same effect. This is only possible if:
- The exchange does not change anything. This occurs for bosons, e.g. photons.
- It flips the arrow by 180 degrees (changes to the opposite phase, i.e. multiplies the wavefunction by -1). This occurs for fermions, e.g. electrons.
All the known elementary particles are either bosons or fermions. The difference between them manifests itself via the “exclusion principle”. Two bosons can occupy the same point in space. Two fermions cannot. This is because exchanging two particles located at the same position does nothing at all, yet the phase is flipped. The only case when a complex number is equal to itself times -1 (i.e. the arrow is the same as arrow flipped by 180 degrees) is when it is equal 0 (arrow of zero length, i.e. no arrow at all). This means that the probability of finding two particles at the same place is equal zero, i.e. such situation is impossible.
However, if we imagine a two-dimensional universe, the clockwise and anticlockwise exchanges cannot be transformed into each other and can be regarded as distinct. Then, the exchange phase can take any value, and we have a third group of particles: anyons. Anyons are, in some sense, an intermediate case between bosons and fermions, for example, they can obey more complicated exclusion principles (like, two but not more anyons can be located at the same point in space).
Why do we care about anyons? First, they are simply peculiar physical objects, interesting from the point of fundamental research. But also, there is a more complicated type of anyons called non-Abelian anyons. Their characteristic feature is that they can “remember” the exchanges done in the past. Hence, we would be able to store and process quantum information using only the exchanges – that is, one could build a quantum computer which works by exchanging particles. What is nice about this idea is that all working designs of quantum computers are susceptible to noise and imperfections. But a topological quantum computer, the one built using anyons, has certain resistance to noise built in. It does not matter how exactly do we move the particles as long as we exchange a particular pair of them. Any noise that affects the trajectory of particle motion, but does not change its topology (which particles are exchanged or encircled – encircling corresponds to two exchanges) would have no effect on the result.
How do we create anyons? Of course, our universe is three-dimensional. Even if we manage to confine electrons to move in two dimensions only (experimental physicists can do that), they still are fermions. There is, however, a way of creating “fake”, non-elementary particles which are anyons. One can do this in topological orders.
Topological orders
Systems of many particles can order themselves in presence of interaction. An example is ferromagnetism. The electrons can be characterized not only by their position, energy or momentum, but also spin. A crude, cartoon picture of spin is that an electron can rotate around its own axis clockwise and anticlockwise, which can be represented by a up- or down-pointing arrow. To avoid confusion with the other arrows – the ones representing complex numbers in a wavefunction – I will draw the spin-representing arrows in a circle. Like any quantum property, the spin can be “blurred”, but for now let us forget about this quantum character and consider the spin to be either 100% up or 100% down.
Now, imagine a different kind of crystal than the ones I mentioned before: one in which a single electron sits at each atom, and is not allowed to hop. Instead, it can change its spin due to the interaction with neighboring spins. In particular, a ferromagnetic interaction makes the spins prefer to be aligned with each other. Thus, the true state of the system is all spins pointing up or all spins pointing down. This is why a magnet produces a magnetic field. Also, this is why you can flip the poles of a magnet by applying a strong enough magnetic field – this means flipping all the spins from “up” to “down” or vice versa.
But there are more exotic forms of ordering, such as the topological orders. One example is the Z2 spin liquid. It is are predicted to exist in frustrated antiferromagnets. An antiferromagnetic interaction makes the spin prefer to align in opposite direction. If one spin is up, the other should be down. But if we arrange the spins on a lattice that contains triangular motifs, such as kagome lattice (see the picture above), then one cannot satisfy this condition for all spins. This is called “frustration”
Consider now a quantum case, in which the spins are “blurred”. In the case of spin, the wavefunction “lives” in a more abstract space than described before. One can describe it as a sum of different arrangements of up and down spins multiplied by complex numbers. For simplicity, we can consider real numbers instead of complex numbers (in the “arrow language”, this would mean horizontal arrows only, pointing right or left for negative and positive numbers, respectively). For example, for two spins we can form a so-called singlet state: spin up spin down minus spin down spin up. To form a spin liquid wavefunction, we have to cover the lattice with singlets in all possible way and add up these coverings, all multiplied by the same number.
Anyons can be created on the top of the spin liquid “background” by modifying it slightly. There are two kinds of anyons in a Z2 spin liquid: spinons and visons. Spinons are spins unpaired in singlets. To create them, one has to break one singlet (say, now both spins point up), and then move the unpaired spins by rearranging the singlets. Visons, are a bit harder to understand. The idea is to draw a line (“string”) crossing some bonds. If there is an even number of singlets crossing the line, we multiply this configuration of singlets by +1, and if the number is odd, by -1. The visons are located at the ends of the line.
Spinons and visons are mutual anyons. That is, if they were just bosons or fermions, encircling a vison by a spinon (which is equivalent to exchanging them twice) would have no effect on the wavefunction (one exchange could flip plus to minus and minus to plus, but the second exchange would undo this change). But instead, a minus sign arises in such a situation. One can imagine breaking one singlet into two spinons (green dots in the “braiding” picture), and then moving one of them around the vison, until it is again next to the other spinon and both spinons can recombine back into singlets. Moving a spinon rearranges the singlets. In fact, it changes the number of singlets crossing the string. Singlet configurations that had minus in front, now have plus, and vice versa.
Z2 spin liquids are just one type of topological order. There are also, for example, the fractional quantum Hall states, a “liquid of electrons”, arising from the interaction between electrons in two dimensions in high magnetic field or in “topological bands”. Little “dips” and “hills” in the density of this liquid are anyons (this was, by the way, recently confirmed by two experiments).
In the project, we expect that both kinds of topological orders can be realized in atom arrays using different means.