So it's a it's a great pleasure to be giving this talk about anions, one of my favorite subjects. I there's a little comment down here. Respect the young because as you'll see, a lot of the great progress in this field was made by people who are mere spring chickens. So the the idea of anions is really, trying to answer a question, what happens when you exchange two identical particles in quantum mechanics? It's a it's an old question.
It goes back to a now famous letter from, say, Chandra Bose to Albert Einstein, written when Bose was 30 years old. The letter says this about 100 years old, 101 years old. Now, I guess this year it says respected sir, I venture in to send you the accompanying article for your perusal and opinion. And he asked Einstein to help him get it published in for physique, which was the leading journal of the time. He says there are complete stranger to you.
I do not feel any hesitation in making this such a request. So what Bose had done is he had derived, using the basic principles of statistical mechanics, the distribution function of how photons will fill modes in a cavity. Now Einstein read this paper. He realized that it was it was not only correct, but it could be, applied to lots of other things, to particles that were not photons.
And in and in this way we developed Bose-Einstein statistics that applies to all particles that are what we call bosons. That includes photons, pions, gluons, phonons, excitons, and of course, the famous Higgs boson. The very next year, when Pauli was 25 years old, he formulated his exclusion principle. This, to remind you, is the principle that says you can only put one fermion in each orbital. Two if you count one spin up and one spin down.
And this principle, of course, is is fundamental to the periodic table and and all of chemistry and everything else in physics as well. Realizing that these particles don't obey the same kind of, statistics as photons, we therefore needed another type of distribution, another type of particle. And this is what we now call Fermi-Dirac statistics, which was derived first by neither Fermi nor Dirac, but it was derived by Pascal Yordan, who is 23 years old at the time.
So, there's kind of a long story about why it is that it wasn't named after Jordan. Jordan wrote his manuscript. He sent it to the journals. I for Zeke, the editor was Max born, who is a well-intentioned but rather forgetful guy. Max put it in his suitcase with the best to ever intention to take it out and read it. But then he forgot about it, and it stayed there for the better part of a year, during which the same result was published by Fermi and Dirac.
So then we had Fermi-Dirac statistics, which applies to in particular electrons, but also to all particles that are fermions, including muons, quarks. And so forth. Now the scientific community is, is is usually pretty good about correcting errors of attribution. Max Born was very clear that he had made a mistake. He was very apologetic about it. He told everyone that he had made this error.
He felt guilty about it for the rest of his life, having robbed Yordan of of credit that he rightly deserved and under most conditions, the scientific community would have renamed Fermi-Dirac statistics into Fermi-Dirac statistics or Yordan from Dirac statistics. But this didn't happen. And the reason it didn't happen is because a few years later, Yordan became very prominent Nazi and and pretty much no one liked him and no one felt the need to do him any favors.
So, you know, there's there's more to this story, which is don't become a Nazi. This is my joke about American politics. So anyway, by 1930, the basics of quantum mechanics were. We're finished. Quantum field theory more or less finished by 1950. And during that time and since that time, you might wonder if people asked if there are other particles out there. There's bosons and there's fermions and is is there something else?
And over and over people came to the same conclusion, which was no. All you have is bosons or fermions and nothing else. And if you open up your favorite quantum mechanics textbook, chances are that's what it says. Lots and lots of quantum mechanics textbooks have, have that answer in it. And they all give the same argument, which is very simple. And I'm going to give that argument right now. It's pretty, pretty easy argument.
You define an operator called the exchange operator which switches the position of two particles. So the exchange operator applied to psi of r1, r2 gives you psi of r2 r1. If you apply this operator twice, you get back to where you started exchanging twice the identity. There's only two square roots of the identity. Therefore there's only two possibilities. If it's a plus one, you call it bosons. If it's a minus one, you call it fermions, and that's all you're allowed to have.
This is a great argument. It's very simple. It's very clear. Unfortunately, it's also wrong. So this was not realized for quite a long time until 1976 with this beautiful paper by the Two Johns in Oslo, John Magna, Linus and Jon Wertheim, who are 28 and 30 years old at the time. Obviously, they're a little older in these photos.
And they pointed out that if we lived in a two plus one dimensional universe, that's two spatial dimensions and one time dimension, then you could have other type of particles as well. And what they envisaged was the idea that if you exchange two particles, say counterclockwise here, the wave function would pick up a phase E to the I theta. Theta equals zero means no phase. That's bosons. Theta was pi into the I. Pi is minus one. That's fermions.
But they pointed out that in fact, other values of theta, any value of theta is really also allowed by quantum mechanics. If you live in two to plus one dimensions. Now from this paper there's a number of things we can conclude. You might be tempted to conclude from this paper that everyone in Oslo is named John. This is this is in fact not correct, I assure you. But a little bit of a coincidence. They both happen to be named John. But there's other things that you should conclude.
One thing you should conclude that there is something wrong with the argument I just gave you. And indeed, there is something wrong. When you define an exchange operator, you need to say how you exchange the particles. So to make that more clear, in two plus one is Shivaji actually mentioned this earlier, but I'll give the argument again in two plus one dimensions.
If you exchange particles counterclockwise and you exchange them counterclockwise again, if you look at the world lines of the particles, the paths in space time, you will notice that the world lines have not it around each other and becomes more clear if you connect up the top to the bottom. And now you have two strands which are knotted, with each other. This is not the same as having not exchange the particles at all. So two exchanges is not equal to the identity.
Now, the reason we got away with saying two exchanges is the same as the identity is because we usually think about three plus one dimensions, and in three plus one dimensions, two plus one and two exchanges actually is equal to the identity that comes from a topological statement that if you're living in a space with a total of four dimensions, four dimensional space, and you have one dimensional strands, you cannot make knots in one dimensional strands. Living in four dimensional space.
If this is not obvious to you, ask me at the end. We can probably make it and make it obvious, but, But it is a true topological statement. Okay. So there's there's some other things we can conclude from this paper here. One thing that we can conclude, which is quite important, is that the scientific community isn't that good at realizing when something important has happened. This paper was more or less completely ignored for the first few years of its life.
It was cited five times in the first five years of its of its life. And three of those citations are by the young Magna Linus himself. So pretty much no one was paying attention to this at all. But a few years later, this person, Frank Wilczek, did take notice and found it very interesting. Now, Frank Wilczek was already very famous for work he did when he was 22 years old in 1973, which would later win him a Nobel Prize. Asymptotic freedom in QCD, which turns out to be very important.
So people were watching what he was doing, and once he got interested, then a lot of other people got interested in this as well. Another thing he did is he he gave a name to these types of particles. He called them anions, particles that have any statistics besides bosons and fermions. We'll check is particularly good at coming up with cute names. But what he was actually concerned with is the famous spin statistics theorem.
To remind you what the spin statistics theorem is, it's a statement that if you have two identical particles and you exchange them, you accumulate some phase. Or if you take one of those particles and you rotate it around its axis by two pi, the phases that you accumulate in those two processes should be the same. For bosons. You get no phase for rotating it, no phase or exchanging.
For fermions you get a minus one for rotating, you get minus one for exchanging, and for anyons there's the same thing hold up. And in fact it does. And that was kind of interesting. He notes in his paper. Although practical applications of these phenomena seem remote, they do have considerable methodological interest and shed some light on the spin statistics connection. So he couldn't imagine how you would ever be concerned with, two plus one dimensional universe.
But it's a nice toy problem to play with. The same year, however, was the discovery of the so-called fractional quantum Hall effect, about which I will say a lot more in a moment, but it's an effect. It's observed an experiment in two dimensional electrons in high magnetic fields and low temperatures hint two dimensional electrons. So how do you get two dimensional electrons? Well, in this was this discovery was made when Horst Stormer was 33 years old.
So to make two dimensional electrons, the way they did it was they sandwich the thin layer of gallium arsenide between layers of aluminum gallium arsenide. And they trapped electrons in this thin purple layer here.
In fact, perhaps the more important discovery, even though the discovery of fractional quantum Hall effect was an important discovery, the more important discovery was made by Horace Stormer several years earlier, in which where he figured out how to make such semiconductor structures without introducing a lot of disorder into the gallium arsenide. This is a trick known as modulation doping. It's used industrially in all sorts of semiconductors.
It was a very profitable patent for Bell Labs the company was working for for a long time in. The patent is now expired, I think. Anyway, in the modern era, there's other ways to make two dimensional electrons, and a really interesting one is the idea of using single atomic layers of carbon. What's known as graphene. Carbon can make a single layer in a little, honeycomb pattern like this, where each of these balls is a carbon carbon atom, all, all stuck together.
It was discovered that you could do that in 2004 by Novoselov and Andrew Geim. Costello was 30 years old at the time. And this is another example of, of, of the theorem that the scientific community isn't very good at understanding when something important has happened. In fact, they had a lot of trouble getting their work published. It took them about a year to get it printed anywhere, and six years later, it already won a Nobel Prize.
No one realized why this was really super interesting, but then all of a sudden everyone realized, yeah, this is super interesting. Anyway, making single atomic layers of carbon is a modern way of, of making to much electron systems, about which you'll hear more, later. Anyway, so in 1982, this effect, fractional quantum Hall effect was discovered. The theory of fractional quantum Hall effect was laid out. Its basic parts by Bob Laughlin. He was 33 years old at the time, actually.
Academically, he was even younger than 33 because he, was forced to join the military because of the draft. And he lost quite a few, a number of his, his young years and not studying physics, which is what you should be doing when you're young. Anyway. The the group of them would win the Nobel Prize in, in 1998. So what about fractional quantum Hall effect was so interesting that it deserves, a Nobel Prize.
Well, the next year, two groups managed to show, theoretically, that the low energy particles that arise in fractional quantum Hall systems really are anions. The people involved in bird hopping. And my thesis advisor, as it turns out. And we'll check. We've already meant Rob Schriever is, well, he was 26 years old when he did his Nobel Prize winning work in 1957, that the BCS theory of superconductivity very important, major breakthrough in physics.
And the graduate student who did all the work then either of us was 23 years old at at the time. So anyway, theoretically, we believe that in these fractional quantum Hall systems, we do have anions running around. So the history of the field just summarizing it. And by 1920s we had bosons and fermions. The first proposal of anions was in 77. By 1984, we believe we actually had an experiment system where anions exist. The theoretical community accepted this almost immediately.
It became gospel among quantum condensed matter physicists. Everyone learns this in graduate school. It's sort of fundamental to a lot of our understanding of of modern condensed matter physics. But as Shivaji said, often theory outruns experiment. It took a very long time between before this statement was, it was confirmed experimentally, before we actually had an experiment where we could show that exchanging two of these particles would give you a phase which is not plus 1 or -1.
So that's what I'm going to talk about. So, before going on, you might ask me, why are you interested in anions in the first place? Well, one reason is because it's a fundamental interest. As physicists, we're always concerned with what kind of things can exist, at least in principle. What are its properties? How can you use it? So it's just fundamentally interesting to begin with.
Another thing is, maybe it's lurking in plain sight, maybe the I mean, there's lots of experimental systems where we don't actually know what's going on or we think we do, but we it's not entirely sure. Maybe there's anions running around in lots of systems and we just haven't realized it yet. There's also a, surprisingly large number of connections to fields like high energy physics, quantum gravity, pure mathematics, and topology, which are also interesting in their own right.
But the field got a huge boost in 1997 by this person, Alexei Khattab, who was 33 years old at the time. Who pointed out that if you ever have a physical system with anions in it, you have a really good way to make a quantum memory, which would be very useful for a quantum computer should you ever build a quantum computer.
This idea to hold then working with Michael Freedman, shortly thereafter, they, they proposed the idea of a so-called topological quantum computer, where all the computations are done by moving, anions around anions of a particular type. Anyway, this idea was, so important that Microsoft invested. I mean, I'm estimating this number, but I think the estimate is probably fairly accurate, over $1 billion so far into trying to produce a quantum computer that runs on this, on this principle.
So the other person who's involved here is Mike Freedman in 1981, when he was 30 years old, he proved an important mathematical theorem that won the Fields Medal. That's like the Nobel Prize of Mathematics. The very same year, he also won the American Rock Climbing Championship, for whatever that's worth. So he's a tough guy to keep up with.
Okay, so what's kind of interesting about the experimental confirmation, which came around, around 20, 20, 35, 36 years after the proposal that we actually have anions is that it wasn't just one experiment that did this. It wasn't just one experimental system, a number of technologies, all matured at roughly the same time. And we had a bunch of experiments all doing showing the same thing.
So the first experiment to come out was a so-called anion collider experiment done by Gwendoline Evans group in Paris, done with, two dimensional electron gases and gallium arsenide semiconductor hetero structures. Then there was the Atom Interference and Interferometer Experiment, done first at at Purdue by Mike Mann. First group in gallium arsenide hetero structures, and later done by the Harvard Group and the University of California, Santa Barbara group by Philip Kim and Andrea Yang.
Done in graphene, carbon, two dimensional electron systems. And then in addition, there was simulation of anions on quantum computers and rudimentary quantum computers. And this had been done by a huge number of groups for people who are familiar with it, the toric code is basically an anion model or the surface code. This is basically the best error correcting quantum code we know of.
So more or less, every quantum computing effort in the world is trying to build anion models more or less, and it's been achieved by a number of different groups. By this time. Okay. Of these, the the experiment that I think is the nicest and the easiest to explain is this one, the graphene version. It has some properties which I like very much, and the data is particularly beautiful. So I'm going to show this one to you.
So first I have to explain a little bit about fractional quantum Hall effect. As I explained you need two dimensional electrons minimal amount of disorder and get rid of it all together. That's great. The we put a magnetic field perpendicular to the plane of the sample, and you cool it down to very, very low temperature. It's, 1/10,000 of room temperature is more or less where these experiments are done. The number you want to keep track of is known as the filling fraction.
It's basically the ratio of the density of electrons to the magnetic field made dimensionless by, a flux quantum h bar over over the charge of of of the electron. So this is a dimensionless ratio. And you can change it by changing the magnetic field. When this dimension, this ratio, the filling is approximately a ratio of small integers. So p over q could be one over three, two over five over seven. Then fractional quantum Hall effect can occur.
How do you know when you have fractional quantum Hall effect. Well, you measure something and what you measure is some sort of resistance. So you run current through your sample and you measure a voltage in the same direction as the current. What you measure is zero voltage. Now if you remember for a second that power dissipated is current times voltage in the direction of the current.
So if the voltage in direction is zero, then you have zero power dissipation, which means it's flowing without any loss whatsoever. It's like a superconductor or a superfluid of some sort, dissipation, less flow. And that's kind of interesting. More interesting is what happens if you measure the voltage perpendicular to the current flow. In this case, the ratio of the so-called half voltage, the voltage perpendicular, the current flow divided by the current.
This is known as a high resistance is two pi h bar over e squared. Is the charge an electron times q over p these two integers down here? Exactly. Now when I say exactly, I mean exactly to the precision with which it can be measured, that's about one part and ten to the 10 to 1 part and ten to the 11. That's like measuring the distance from here to California to within a centimeter. It's an extraordinary amount of precision considering this is a sloppy, messy, solid state experiment.
So there is disorder in the sample. You don't know the shape of the sample. Exactly. You stuck electrodes on the sample to measure resistances with, you know, a soldering iron. You know that that there's so many things about about this experiment that you don't control. Precisely. Don't control the magnetic field. Precisely. You don't control the temperature. Precisely. You don't control it. They're vibrations going to the laboratory. There's no light shining on your sample.
There's all sorts of things that you don't control. And yet the result comes out exactly two pi h bar over E squared times Q over p. All right. So that's kind of cool. Here's some real data, taken by George Stormer. And what you have here is a longitudinal resistance down here and the whole resistance up here, every time the longitudinal resistance drops down to zero, you will see that the whole resistance shows a flat plateau, exact quantization.
And these plateaus are labeled by their p over q ratio. So, for example, this one's two over five, this one's one over three and so forth. Each of these is a different fractional quantum Hall state. The one we're going to focus on is the simplest actually the one that was first observed in experiment is a so-called new because one third fractional quantum Hall state, which is the easiest to understand.
So in the new equals one third fractional quantum Hall effect, you start with the ground state. Then you make some excitations. And those excitations are particles that surprisingly have fractional charge. You put in electrons of charge E and the excitations now have charge U over three. You have an emergent particle with a fraction of the charge of an electron. Now how does that happen?
Well, the way you should sort of think about it is that the electrons form a completely uniform soup of, of uniform density electrons. And these particles are the defects of that soup. Okay. It's a pushes a fraction of a charge of the electron away from some region. And that defect becomes the new low energy particle. What's more interesting is that these particles are also anions. When you exchange them, you pick up a phase in the two pi over three. They're neither. Both bosons nor fermions.
But surely you must say these particles really live in three dimensions. Our universe is three dimensional. How can you know? Maybe if you have squashed them down. So they're approximately two dimensional, but they're not really two dimensional, are they? Well, let's look a little more carefully about what we at what we've done. We have our sample like this. We've tried to squeeze our electrons down into our this little blue layer here.
Let's use a little bit of, gratuitous animation and blow up, our, our system here. And the, the potential felt by the electrons is kind of a, you know, a particle in a box, kind of, square. Well, potential. So the electrons are living in here. Blow that up, look at it more closely, and we're putting the electrons in there and so. Well, it's still living in three dimensions.
Maybe they're sort of confined a little bit in, in the well but they're still really living in three dimensions aren't they. Well think about that more carefully. Remember that they form discrete eigenstates in the z direction in that well, and they occupy some of the different eigenstates. At low temperature they all get frozen down into the lowest eigenstate. And you remove any ability for them to change their wavefunction in the z direction. Their z direction is completely frozen.
There's no freedom to change anything in the z direction, and so they can only move in there in the x and y direction, they become strictly two dimensional objects. Okay, so people might be thinking something else. Another objection. But surely these aren't fundamental particles, not like an electron is a fundamental particle, is it? Well, you know, maybe nothing is.
You know, that we think of an electron as being a fundamental particle because on the energy scales available to our experiment, we have not seen it break up into other things. We have not seen it emerged from other things. But that just means the energy scales available to us. It looks fundamental. It's the same thing here in this low energy, low temperature system.
If you were a low temperature person living in this two dimensional low temperature quantum, well, you would swear that the particles, the fundamental particles are charged E over three. And it's only when you got yourself out of that two dimensional, layer and could go up to higher energies that you would notice. Oh, actually, it's the electrons that are running around and the E over three is just emergent.
We're always in the business of describing physical, physical systems at the relevant scale for the experiments. We can do. All right. So these were the this is sort of the history of the story. And the experiment I'm going to explain is this one here. So to explain this experiment and the remaining 15 minutes is I'm going to need to tell you a couple things more about fractional quantum Hall effect, but not much. So the first thing I have to tell you about is quantum Hall edge states.
So here I have drawn the blue region and the white region, the blue regions where there are electrons. My fractional quantum Hall effect. And then the white region is outside of the sample. That's a vacuum. I know that there's going to be an electric field. Well, minus the electric field is going to point into this, into the sample. So there's an electric force holding the electrons in. How do I know it's there. Well if it wasn't there the electrons would leak out and they're not leaking out.
So there's an electric field there. So and then there's a magnetic field perpendicular to the sample. And I know from basic and I'm that, whenever I have a crossed electric and magnetic field, there's a drift velocity have any charge. And you can, you can calculate the drift velocity just by finding the reference frame in which the Lorentz force e plus v cross b is turns out to be zero. So if I put a charge in particular one of these particles on the edge, it will drift along like this.
And just because of the x b effect, now we're going to use that to our advantage to transport these particles around our system. So here's the the geometry of the sample we're going to use. We're going to take you know the the blue region again is quantum Hall fluid. And we're going to pinch it down in some region in what experimentalists call a quantum point contact. It's a point contact.
And then they put the word quantum because they like the word quantum. So anyway, so if you put this you send these charges in along the edge and they, they kind of move along the edge bump, bump, bump, bump, bump like that. And most of them go through. But occasionally you'll discover that one of them comes along and then jumps across the narrow neck and gets back reflected instead.
Okay, so we should think of this constriction as being a half silvered mirror to send some through and reflect some back. Just as a side comment, the first measurement of the fractional charge of these particles was done with a single point contact like this. You measure that when the, the current coming back at you and you measure some total current.
But you notice that the noise in that current is indicating that the charges are coming back to you in units of E over three, rather than in units of E. And this experiment was done in the 90s by several groups. And it's now not not controversial that this works as, as as described. All right. This is the experiment I'm going to describe. It was proposed in 97 by our first speaker, Shivaji, and, his friends Claudio Sherman, Denise Reed, Steve Constant and Sheldon Wen.
The idea is we're going to have two of these, point contact, one of them called T1 and one of them, called T2. This is going to act as a beam splitter and a mirror. And if you remember your optics, this is basically a fabric parallel interferometer. So the idea is that a particle can come along like this. It will split into two partial waves. One partial wave jumps across, the other partial wave goes on and is reflected around the cavity. And then they're interfere and go on their way.
Okay. Now the, if you count both of those partial waves, the wave function of the particle coming back at you is the sum of the part that went across T1 here and the part that went across T2 here, but the part that went around T2 picks up an additional phase for having gone around the cavity. Okay. Then if you want to know the current coming back, you have to square the wave function the usual way.
You know, probabilities are squares of amplitudes gives you t1 squared plus t2 squared plus two t1 t2 cosine phi. Assuming t1 and t2 are real for simplicity okay. So this is basically Fabry Perot interferometry physics. And the thing we're going to be interested in is this phase Phi down here. So there we're going to try to change that phase Phi and measure the change in the backscattered current.
And the way we change Phi is by actually changing the shape of the cavity slightly by pushing on the edge. Oh, this is the particle going around the cavity. There it goes. So we're interested in the phase of the particle going around the cavity. That shows up here as phi. And we're going to change the this cosine phi by changing the shape of the cavity. So the the phase accumulated by the particle going around the cavity changes.
So you do that with an electrode that sort of pushes the the electrons out of the way and changes the shape of the cavity. So this is this is all changed with an electrode and it will change cosine phi. And so the current you measure backscattered is going to oscillate sinusoidal. It's just like taking a fabric interferometer. You have to marry your half silver mirror and and a solid mirror and you just move them back and forth and you'll see interference fringes.
Okay. Now the interesting part of this experiment is what happens if you add an anion to the center of the cavity. Well, if one particle has now gone around another particle, it picks up a braiding phase two pi over three braiding braiding phase. So without the particle, if you see the black curve with the particle you'll see a shifted curve. The blue curve okay. And if I add another particle to the middle this curve shifts again by another two pi over three.
That's what we're going to try to see. Okay. So it sounds like an easy experiment right now. So it was about 15 years of effort trying to make this experiment work. And people eventually came to the conclusion that it's actually a very hard experiment. It might even be impossible. So the reason it is hard is sort of it's sort of a conflict between two two issues. For any finite temperature, there's a coherence length beyond which you don't see any interference.
So where that comes from, is it the phase can be, stated as e to the I length times a wave vector. Now the wave vector is a function of energy. So you expand this k0 plus decayed e times d. So depending on the energy of the incoming particle, you get a different phase. But at any finite temperature, say even 30 Millikelvin D is big enough so that the the changes in this term end up scrambling the phase completely. And the only way you cannot scramble the phase is if you make L very small.
So this is going to force you to do the experiment on a very small sample on the micron scale, even at 30 millikelvin. I mean, if you go to zero temperature, zero zero temperature, you could do it in a much bigger sample. But, you know, 30 Millikelvin is about the limit of what you can do experimentally. However, there's a conflict with that, which is that adding a single electric charge of E or a three to a micron sized object is a very strong perturbation.
It changes the position of all the edge states. And then you're measuring something completely different. Once you once you add the U over three. So you're not seeing the change just from the statistics of the particle. You're seeing that the change from the Coulomb interaction of the particle with the particles running around. So this is problematic. And then on top of that, even at 30 millikelvin there's significant thermal noise from various sources that you have to wrestle with.
So all of these things were addressed by Mike Manfred's group in 2020. Using gallium arsenide have a structure with lots and lots of tricks to get around these problems, and it was done successfully and a beautiful tour de force experiment. But that's not the experiment I'm going to describe. I'm going to describe this experiment, which, which was done more recently by the Harvard group. Thomas Work Meister is a graduate student who's, the, the lead author.
And the reason I well, one reason I like it is because the data is really beautiful. And the other reason I like it is because it invokes some of the things that we had mentioned in this earlier paper, from 2006. So the idea of the experiment is we're going to do exactly that. Explain we're not going to change any edge voltage. We're not going to change the shape of the interferometer. We're just going to wait.
So you just sit in your experiment and you measure some current back scattering and you would think, okay, just I'm not changing anything. The current back scattering should be exactly the same. It should just not change at all. But it does change. It sort of jumps around after a half a minute it jumps up to this level, and then another half a million jumps up to this blue level and jumps back down to this green level. It's jumping all over the place. It looks like it's a noisy sample.
And typically what you do with noisy samples is you throw them out. But then if you look at this for a little longer, you realize that actually it's only jumping between three different levels the green level, the blue level and the purple level. So let's plot those three levels over here.
And then once you've accumulated data to find what these three levels are, then you change the shape of the interferometer and you trace out three curves which are three sinusoidal curves shifted by two pi over three. This is exactly these three curves here. What you're seeing is you're seeing telegraph noise as one particle is jumping in and out of the are the interferometer are two particles are jumping in and out of the interferometer. And they are.
The blue curve will be when you have one for seven particles in the interferometer the purple will be two, five, eight. And the green will be zero, three, six and so forth. And it's jumping back and forth between them. But at any number of of particles in the interferometer, you're on one of these three sinusoidal curves. So how do we address these problems. Well, this one we got rid of the thermal noise by making lemonade out of lemons, I guess. So we used it to our advantage.
But what about the conflict between the size of the the device and the Coulomb interaction? Well, here what they did was they screened the Coulomb interaction by slapping down a metal plate very close to the two dimensional electron gas that you're interested in, that if you put a metal right near you, two dimensional electron gas, and every time you have a charge in the two dimensional electron gas, you have a mirror charge in the in the metal plate.
So instead of having coulombic interactions between E over three and were three over here you have dipolar interactions, much weaker dipolar interactions between this pair and this pair. Now doing that in gallium arsenide was really a very difficult trick because while the gallium arsenide the quantum wells are 100 nanometers to begin with and the, the the gallery, you know, the gallium arsenide world needs a cap. And then the metal plane can only be so close.
But with graphene, it's super easy to do because graphene is only an atom thick. And you can plunkett right down on top of a piece of metal within a couple of angstroms, so you can screen the the Coulomb interaction extremely effectively. And that's why some of these new graphene experiments are so, so nice. Anyway, that more or less, ends the story. After about 36 years, we can finally, put a checkmark next to the experimental confirmation of.
Okay, we can finally put a checkmark next to the experimental confirmation of of anion statistics. And I will thank you for listening. Just in time for. That. Yeah, actually. So, so they customize era three. Looks like the charge on a quark. So there's a legend that when, first Starman and Dan Sui were taking the first data on fractional quantum Hall effect, they, you know, they were served the way, you know, you you scan the magnetic field slowly, and and you see these plateaus form.
They saw this plateau toe form at, you know, three times the other plateau that they'd seen. And Dan Sui immediately said, oh, quarks, you know, and it was it was completely a joke. But he realized immediately that the quantized one third would be consistent with a one third particle. It's not quarks, you know, the quarks are bound with enormous, enormous energy orders of magnitude, higher than anything in these experiments.
But, but nonetheless, you know, it has that odd similarity that there are other fractional with quantum Hall, states where, where the, the charged particles are, you know, you're five or E over seven or any number like that. So three is, is sort of the, the minimal odd number on one. But yeah, it's, it's, it's a little bit. Yeah. It is a complicated combination of effects. So the question is what why are the width of the plateaus, what they are.
So there's a theorem which says that if you had no disorder at all, there would be no plateaus anymore. So you need some amount of disorder. And it actually depends on not only the amount of disorder, the tendency to, grow, you know, initially grow wider as you reduce the disorder. But it also depends on the type of disorder, the the range of disorder.
And in when, you know, quantum Hall effect, because of the precision, the effect is used for metrology, you know, for setting resistance standards. You know, if you want to ask how do you define an ARM really accurately? You do it this way. Use quantum Hall effect, and they use very special, samples with a particular type of disorder, which is known to give a, wide plateaus. So it's actually a combination of things that that goes into the width of the plateau.
But it has to be sufficiently clean. But then the details of the disorder actually matter to. Yeah. So okay, so the question is why do you need, the integer ratios to be small. It's, it's only comes from the statement that as you get higher integers, the gaps tend to get smaller. And this is going to have to be the case because otherwise you're going to have a double staircase. You know, where there's a different quantum Hall effect that at each epsilon you change the magnetic field.
So the the as you get to a cleaner and cleaner samples you, you a lots of more fractions. Do start emerging between other old ones. But the, the ones with the lower denominators are the ones that, that emerge first. Yeah. So, the question is about the rationality or irrationality of, of, of these, of these, these effects.
So the so I wouldn't say this, this is, it's not a physical constant where we're measuring, it's, you know, we're measuring a number when we're measuring a number, the I guess I would say that the, you know, we are measuring a third of an electron to very high precision in, in some ways, although, to be honest, the, the experimental measurement that tells you directly that you're measuring, you know, the charge on these things is one third if you are is the unless
you are saying that the Hall resistance itself, which is very easy to measure, is evidence that the charge is fractionalized and theoretically you might make the connection. But if you want a direct measurement of the of the charge on that particle, which you can do by noise measurements, or you can even these days, they can do it actually by by using an electron very sensitive electron meter. And you scan over the sample and you say, oh, there's a bump and its charge is about E over three.
But those experiments are not accurate. Apart in ten to the ten, those experiments are accurate to say 5%, something like that. So it's consistent but it's not highly, highly accurate. The way the the resistance experiment is. Yeah okay. It's a very good question. So there was actually there was in the early days of fractional quantum Hall effect, it was believed to be a theorem that all denominators had to be odd.
And that actually comes from the fact that the underlying, particle electrons that you're putting in is a fermion. And so it's a little bit of a complicated, connection. But from the fermionic statistics, the statement is that you would need to have an odd nominator, and three is the smallest odd denominator or higher than one. The what that one would gives you the integer quantum Hall effect, in which there is no fractional ization.
That turned out not to be true, actually, that people have measured even denominator fractions, and that comes from a more subtle effect where the electrons pair into bosons like a superconductor, and then that condenses so you can have even denominators too. It could be that the first one we measured was at one half, but it just turns out that the one half plateau is is weaker and a little bit harder to see.
So they have been seen, but only in only weak, more weakly and, and, and more high, you know, key and cleaner samples. So if they can they can exist. Even denominators can exist. But some things, you know, people have, have observed something like 80 or 100 different fractions and fractional quantum Hall experiments, of which all but a very few have our denominators. So there's there's a community in the world who wants to build, quantum computers out of anions.
Now, initially, Microsoft was the home of this. Sorry, I should repeat the question. The question is, is where do you see this being applied to in, technology? And where do you see this being applied in fundamental physics? So in technology, there is this community that wants to, use anions to build quantum computers. And they initially started exploring fractional quantum Hall effect very intensively. And that's why people did this 15 years of experiments of, successful experiments actually.
And they at some point after doing this for about eight years, Microsoft said, okay, that's not the way to do it. We're going to do something else. And they're still thinking about, so anion based quantum computers or Majorana based quantum computers are very similar. But they switched the platform to, using superconductor semiconductor structures. It's not quantum Hall effect anymore. So it has a lot of similarities. But no, it's not exactly the same.
So that's something that they're really pushing very hard right now. And that could be a real technology. Although it's not fractional quantum modeling, although there are some people in the world, myself included, who would love to see fractional quantum Hall effect come back into the into the quantum computing game. And I still think that that's, you know, not an insane possibility, for fundamental physics. So in some ways, I have to ask maybe what do you mean by fundamental to begin with?
But I would say that this this is fundamental physics as fundamental as anything else you will come up with. And, you know, seeing this, this principle that particles don't need to be, don't need to be bosons or fermions is a really fundamental advance.
And and that is, is, you know, what I would call fundamental physics, I, I probably should have I mean, as I went further in, I mean, one to some extent, it's it's not, not fair because I think a lot of the modern modern work is, is much more frequently done in collaborations than it used to be. And so there will almost always be a student on the paper, a postdoc and a senior, you know, senior faculty member or several and so forth. And, then the question arises, you know, whose work was it?
What's this? You know, is it really the graduate student who came up with the great idea? I mean, sometimes it actually is. And and the, you know, the faculty member is just the one who raised the money to pay the graduate student. But other times it's, you know, it's it's a more of a collaboration. So I think it becomes harder to, to say whether the ideas are still coming from, are coming from the young people.
I my guess is that, in fact, a lot of the ideas still are coming from the from the young people. It's hard to prove that Nobel Prizes have become fractional. Yeah. Yeah, exactly. Yeah. So actually, there's a couple of things that I, that I can say that that where it's, it's not, it's not an irrelevant connection.
So the underlying theory of anion models or its so-called topological quantum field theory is that you mentioned earlier where you throw out all space and all you care about is whether the one thing went around another. And topological quantum field theories were actually cooked up by, by string theorists in the, in the 1980s, more or less.
And they were thinking about theories of quantum gravity in, in if you go down one dimension to a, two plus one dimensional universe instead of a three plus one dimensional universe, it is known that the gravity is very different. In two plus one dimensions, it becomes completely topological. And, you know, a lot of the structure goes away. And these kind of theories actually do describe universes in in lower dimensions.
It's, it's beyond my pay grade to say whether any of that survives in our, three plus one dimensional universe or not, but it's definitely interesting to study. Yeah. What are the questions? What are the statistics of anions analogous to Bose-Einstein and fermions? You can write down a distribution function for anions statistics, which is somewhere between boson and fermion as well.
There's another description of statistics that also arises which is which is interesting, which is, to ask the question, you have some Hilbert space and, you ask how big it is. And then when you add a particle to it, how much smaller did it get? You know, how many fewer orbitals are allowed for the next particle that comes in. So for bosons, if I put a particle in the next particle I put in, it has exactly the same in the options for fermions.
If I put a particle in, the next particle has a left, has fewer options with anions, it's somewhere in between that you put two particles in, and then you reduce the number of options by one for example. So it does I mean always seems to interpolate between the the two possibilities. Okay. I think another question is that we just didn't find. So let's think okay.