 We'll restart our last talk for today, two announcements. First of all, there will be a Q&A session tomorrow, and this is for you. That is, anything you wanted to ask about minimally physics, high temperature superconductivity, S-Y-K model, et cetera, et cetera, this is your time. You can come up with questions from the last week, and the lecturers are supposed to be here, and we'll try to do the best of our ability to answer your questions. And the only thing which will not be allowed is to use this time to show 10 more slides from the lecture, because there was no time during the lecture. But because this is a session for you, it will be also natural if someone from the audience would be the moderator. You will have a volunteer to chair this session. Come on, guys. I'm doing this all the time, and I'm still alive, almost. OK. You think about it, and we'll ask once again. On a more festive note, some of you know that tomorrow is a holiday in Italy, our 15th of August. This wouldn't have any effect on us. We'll proceed as usual. The bar is supposed to be open, but tonight there will be a festivity here in the main building in our cafeteria. There will be a buffet dinner, drinks, or music, and it will go from $7.30 to $11. The price is either two-half mule coupons or 10 euros. So with that, Aaron Kapitolnik, if he finds a clicker for his laptop, will give us second part of his lecture. OK. I'll have to go and click on the computer because one of the previous speakers decided to take it home. OK. So we continue now with this discussion on measurements of time reversal symmetry breaking. In particular, we are looking at optical effects. The last thing we did is we looked at a very simple fundamental experiment that tells us how going through a material that breaks time reversal symmetry is different than any other material that may rotate polarization but does not break time reversal symmetry that is reciprocal. So we saw that and we saw that what we want to do is we want to use the time-reversed state for the beam of light, which is basically light going in the opposite direction, which we can easily achieve using a mirror. OK. So remember, I started the discussion discussing time reversal symmetry breaking, then unconventional super conductors, and then we discussed the way to measure it with optical effects, and now we are back to the, oh, health is coming to get rid of that, and now we'll do this. And it works. OK. OK. So we are back now to unconventional super conductors. We are going to bring things together for this discussion. So we already saw that what started the field, the search for time reversal symmetry breaking in any unsuperconductors actually was, eventually, was not what people looked for. I mean, the high-temperature superconductors were shown not to be any unsuperconductors. Those initial measurements that showed large effect turned to be reciprocal effect. That is, they don't break time reversal symmetry. And the question was, where now? And then what happened is that over time, people have been looking at other types of materials such as superconducting for magnets, heavy fermions, et cetera, which are more general materials for which the gap function has both real and imaginary part, and therefore break time reversal symmetry. We discussed it in the first lecture why that is so. In principle, if you have both real and imaginary part, then this has two options, delta r plus i delta i and delta r minus i delta i. And in a way, these are just two states, just like an icing state, therefore, breaks time reversal symmetry. So as it's shown here, I showed it to you before. If these parts, this F function, where delta naught is the magnitude, both are non-zero, we have time reversal symmetry breaking. And as a counter example, the d-wave of high-tc is a single order parameter, and it does not break time reversal symmetry. OK, so let's get to a situation that does break time reversal symmetry. For example, I have a real and imaginary part for the gap function, and I'm going to use the simplest case. The simplest case is that that has an amplitude upfront, and then the internal structure in the center of mass of the cooper pair has this kx plus minus iky. Now, you'll notice there is single power of the momentum. Single power of the momentum means l equal 1. This particular one is the so-called p plus ip, I mean, multiply it by h bar, then it's px plus minus ipy. And if you remember your expansion, multiple expansion, then px plus minus ipy are cooper pairs moving one way and another way in a circle. Well, it already starts to look like magnetism. You have this charge moving in a circle. Charge moving in a circle is a magnetic moment. It starts to make sense that time reversal symmetry indeed will be broken. So that's basically what we expect. So let's try and use this intuition to calculate what should be the effect. So the way I'm going to do it is just like I did it at the very beginning of the lectures, I used the current to calculate a magnetic moment. Now, the current is the current of the cooper pairs. They are running around with l equal plus 1 running one way, l equal minus 1 running the other way. Therefore, producing a magnetic moment that is either up or down. You all remember the relation between magnetic moment and angular momentum, right? If I know the angular momentum, I just multiply it. Now it's a cooper pair. They will have charge e star and mass m star. Of course, it's 2e and 2m, which I can plug it in. But just in case, we have the relation between the angular momentum and the magnetic moment for a single cooper pair. Now I know the density of the cooper pairs. I can multiply it by the density of the cooper pairs. That's ns. And now I have a magnetization, right? Given by this angular momentum. Remember, again, ns is n over 2, e star is 2e, m star is 2me, right? Very simple. So having a magnetization, I can calculate a current. That's one way, I mean, one step towards calculating the optical conductivity. Remember, optical conductivity is going to give me the care effect, right? That was the relation of the diagonal term of the conductivity that is going to give me the term. So I want to calculate current. If I have the magnetization, do you remember how do I calculate current? What do I do? What do I? Very good. So in this very simple calculation, I'm going to take the curl to calculate the magnetization. But before that, I have a problem. Because yes, yes, yes, of the Cooper pair, yes. It's a condensate. They all are in the same state. Obviously, I can have domains. And one domain may have the L equal plus 1. Another domain will have L equal minus 1. But let's say a single domain. I have all the Cooper pairs with one of the L's, plus 1 or minus 1, OK? So in principle, yes, yes. Well, in principle, yes. And what I'm doing here, as you probably know very well, all I want is to have some intuition on getting the calculation of the care effect. This method, in fact, was invented for calculating the moment and therefore the current in helium free. There, the situation is much clearer. But obviously, what Andrei said is need to be taken into account. But let's say that I can keep track on all those Cooper pairs running around in circles. And I add them up. And I get the total magnetization. But there is one thing which, in fact, is not unrelated to this comment that I need to remember. And this thing is that once you see that, even before calculating the current, you can say, hey, you calculated magnetization. Let's go to a magnetometer and measure it. Well, yes, people thought about it. But there is a problem. And the problem is that while at first it indeed may look like an orbital ferromagnet, that is, you have magnetic moments that are all orbital in origin. Add them all together. It's like a ferromagnet, right? But there is a problem because it's a superconductor. And superconductor does not like the magnetization within it and will screen it using Meissner currents. So in principle, we expect m equal to 0 in a single domain infinite sample. However, and I'll come to it in a particular example, if you have surfaces, they are edge states. And if you have domain walls, they are currents along the domain walls. And therefore, this requirement of m equal to 0 may be relaxed there. And indeed, there were searches in some of these materials for such currents. But otherwise, if I simply consider a single domain large sample, then there is not going to be any magnetization. So I cannot really put it in a magnetometer and measure this magnetization. Yes. No. If it's a bulk, if it's a bulk, there is no magnetic field here. So I don't need to have any flux coming in in this system. I'm in the bulk. And in principle, there is not going to be any magnetization anyway. I mean, there are going to be all the magnetization screened. In fact, at any length scale above the coherence length, it's going to be screened. OK. So coming back to that calculation, as we noted, if I have the magnetization and I want to calculate now conductivity in order to obtain the care effect, I start with this magnetization. And I want to take the curl of this magnetization, right? In principle, there is no magnetic field. So the only thing that appears in B, right? B is m plus h. The only thing that will appear is the magnetization. I'm going to take the curl of this magnetization. And now it's very simple. I have this magnetization. Taking the curl is taking the curl of whatever there is here. I'm going from, I mean, ns is simply n over 2. I need to take, in a way, you can re-transform the curl into z. z is the direction of that angular momentum moment. So z cross the gradient of the density. The gradient of the density is related to the gradient in the chemical potential plus the multiplication by the Nd mu. That's a standard way I go from the gradient in the density to the density of states times the gradient in the chemical potential. Remember, this is the full chemical potential. And this is extremely important. And now, OK, what is the chemical potential? You remember Josephson relation? The chemical potential for a Cooper pair is 2e times the voltage across the sample. And then I have that time derivative part of the phase. And the phase have the static part plus the vector potential. So this is the full chemical potential. And I need to take its gradient. So I take this gradient and then there is going to be, once I do that, using this expression, believe me, what you get if you realize that sigma xy for a normal material system is simply e squared over h and 1 over 2d, where d is some thickness of the material. Then you get sigma xy times the z direction and then cross with this piece. OK, any guesses what the result is? Well, actually, not guesses. You should all know what is written here. Well, if I were you, I would say, well, if you really asked what it is, it can either be infinity or 0. Well, because otherwise, it's 0. Why is it 0? Sorry? It's not just E and M. It's E and M plus the fact that this is the London equation. And if you take now this London equation and you see that this is djs dt and that's the electric field, just plug it in here. You get 0. Well, this is what you get, 0. So in principle, we worked all the way, all this, in order to say that, well, in principle, there should not be any care effect for this time reversal symmetry. With this simple, by the way, I could, instead of p plus ip, I could do d plus id, d plus ip. Anything you want, we need the exact same thing. It's just a little bit more complicated. At the end, it's just this angular momentum when, believe me, you get 0. But that's disappointing. So the question is whether there is or not care effect for this kind of time reversal symmetry breaking superconductors. Well, it turned that there is. Because if you remember that this is probably more familiar to you, the issue of anomalous whole effect, then you get a whole effect because of several types of scattering. One of them is skew scattering. So suppose there are impurities. And here, you do not need magnetic impurities because if it is a time reversal symmetry breaking material, then any impurity is going to cause skew scattering. This was proposed and calculated by Gorio and Luchin immediately after some experiments, of course, our experiments that I'm going to talk about. And it was shown that skew scattering is going to produce a care effect. So OK, that's one way to do it. But then there is another way which is much more relevant for these type of superconductors, these time reversal symmetry breaking superconductors. And in particular, as people learn how to make them better and better with lower residual resistivity and larger triple R ratio, et cetera, et cetera. I'll show you some results of material with triple R, which is the resistance at room temperature divided by the resistance at around 4 Kelvin, which is about 1,000. That's a very clean material. And that is multi-band superconductors. Well, it turns that if you have a multi-band superconductor and any coupling between the bands, it's enough to produce a finite care effect. So this calculation, I mean, here I'm following calculation of Taylor and Catechalian. There are other types of calculations along this line by now. But basically, there are now other indices. One and two are now the band indices. This is the simplest example of two bands in strontium ruthenate, which is a material I'm going to talk about soon, shortly. Then there are three bands in uranium platinum three, which I'll show you results. There are five bands. There are coupling between the bands. We know that. So this is not surprise. So as a toy model, we take two bands. And therefore, initially, there is a different gap on each one of the bands. And there are gaps associated with the coupling. And then one can work through the calculation of the care effect, which is, through this calculation, basically similar to what I showed you. Yes? Yes? Yes. Yes. Yes. Yes. Sorry? Yes. Sorry? I? Yes? Yes. No, the London equation works. But if you want, no. But if you want, you have two inequivalent bands. And it is this, the coupling between the bands that, I mean, it's like you would think that you have one London equation in one band and another London equation in another band. This each will cancel. But then there are coupling. And that does not cancel. If you have Cooper pairs that belong to more than one band, then the London equation is not the simple London equation. Yes? Yes? Yes. There is interband component. And what you said, actually, these are two different approaches that were taken, one by the group of Annette, which looks at single orbitals and by Cathy Callan that looks at the full band. In any event, again, not to go through these calculations, you can look at these two papers as these examples. What one finds is that, again, you go through the machinery. But now there is a finite care effect. And it is proportional to the interband coupling, this epsilon 1, 2. If this is zero, then you do not have a care effect because of the same reason as before. If this is non-zero, you do have it. And it is proportional, if you measure at some frequency omega, it is proportional to the product of the principal gaps. And if the two gaps are, say, of the same size, it's going to be this gap square. Now, I want you to notice a few things. First, gaps in superconductors are of the size of Tc. Now, I'm going to talk about materials that are not necessarily high-temperature superconductors. They are going to have a Tc of order of 1 Kelvin, half a Kelvin, a few Kelvin. Now, I'm going to probe it with light that is H bar omega, which is indivisible. Well, maybe near IR. That's basically what we've been doing. So this means that in terms of temperature, it's a couple of thousands of Kelvin. Now, as it is always the case, if you have such an expansion and you have frequency that is much larger than the gap, there is a reduction basically by the size of the gap square over the frequency square. So this is already alarming. In fact, if I look now, I try to estimate the size of the effect, then for simplicity, I'm going to say that the magnitude of the gaps in the band of 1 orbital is 0, A is delta 1, 1, and delta 2, 2. Let's say they are the same. Give them some amplitude delta 0, which is going to be, of course, of the order of Tc. In Stonse and Ruthenate, it's going to be 2 delta over KTC. In fact, it's very close to 3 and 1 half as expected from BCS. And then if you did not notice, if you go through the calculations, I actually did not point it out to you guys earlier in these lectures. But scale of care effect, in fact, is the fine structure constant. You can see it from the constants that come up front. So I have this. I have this interband coupling, and it's divided by the average indices of refraction for that material. N is of order 1, 2, 3, whatever. N squared minus 1 is also of the order of 1, 2, 3. So this is all, and then epsilon 1, 2 estimates, believe me, is of order of a few. Then I have one good thing, because if I'm looking at the wavelengths divided by interatomic distance, which is what I need to consider if I'm looking at the xy direction, I have some magnification here. But then I have this reduction. And then I have the fine structure constant that is something that reduces the magnitude by 1 over 137. So you put it all together. And if I'm using light at 10 to the 15 hertz, which is what I'm going to use, I'm going to use light at 1.55 micron wavelengths, which is the wavelengths of optical communication. The reason is the cheapest components one can buy are at this wavelength. So when we started these experiments, we used that. And take n equal to the index of refraction of 3, which is typical for these materials in these wavelengths. And Tc of order 1, I expect a care effect of order 50 to 100 nanoradians. OK, I showed you the magnitude for typical ferromagnets. It's 0.1, 0.01 radians, or 10 to the minus 2, some even 10 to the minus 1 radians. So this is 50 to 100 nanoradians. That's a reduction by a factor of a million or 10 to the 7. This is really, really tiny. Nobody ever measured something that tiny in optical rotation. So how do we do that? And you remember also this difficulty of separating reciprocal from non-reciprocal effects. Reciprocal effects may rotate the linear polarization, therefore have a problem. And what we need is really to get rid of all that. So we come back to the definition of a good experiment now to look for time reversal symmetry breaking in these superconductors where the effect is expected to be very, very small. First, we need to reject all reciprocal effects, such as linear birefringence, optical activity that is chirality like in quartz, et cetera. We need to reject it to an unbelievable level, preferably if we can get rid of it by symmetry. That's the best. We need to measure an absolute value. I cannot modulate the material with magnetic field to measure time reversal symmetry breaking, because if I put the magnetic field, I broke time reversal symmetry. The material is very different than it's strictly zero magnetic field. And now I have this new requirement. I need to be able to measure at sensitivity much below, in fact, 100 nanoradians, because I want to track the transition. So obviously, if there is an effect, it's going to start at zero and then builds up. And that's kind of the maximum value that I expect. So this is really very, very tiny effect. So let's recall two things that we talked about. One was how do I create the time reverse state? I use a mirror. And in that case, I have a beam that goes one way through the material compared to the beam that goes the other way through the material. That's in a far-day configuration. The other thing that we recall is the issue of reciprocity. Reciprocity says that if time reversal symmetry is obeyed, then it doesn't matter if the source is here and the detector is here, or the source is here and the detector is here, the result should be exactly the same. The best would have been if I could do these two experiments simultaneously and compare them. So when you do such an experiment in which you have two beams that are doing something and you compare them, you basically create an interferometer. If one of them, of course, you modulate, you create an interferometer. So let me tell you about the interferometer that we are using. For that, I need to remind you of something that I hope you saw in E&M, optics, et cetera, which is something that is called the Saniac effect. In fact, people knew about the Saniac effect already at the end of the 19th century. What is the Saniac effect? Well, take a light source. I'll call it a laser. That's what we use now, but a light source. Let's go through a polarizer. That's going to be important for detection, but otherwise, it doesn't matter exactly. But for this part, let's take a polarizer. I have now a linearly polarized light. And now I'm going through a beam splitter. There are mirrors here that allow me to have through the beam splitter, to have two beams, one going clockwise and the other one going counterclockwise. And remember, the polarization is linear in this. Now, obviously, the two beams go through the exact same optical path. When two beams go through the exact same optical path and then interfere, as they come back through the beam splitter and interfere at the detector, then there is constructive interference and no phase shift between these two beams. So I see a nice constructive interference. I can look at it and say, nothing happens. However, if you take the whole apparatus and you now rotate it with some angular velocity omega, then you broke the time-over-versus symmetry. Because of rotation, remember, rotation breaks time-over-versus symmetry. And I broke it between the clockwise and counterclockwise propagating beams. Why? For example, as I rotate the whole apparatus, I launch the beam through the beam splitter. It takes now the speed of light is constant. It takes now more time for the clockwise propagating beam to reach the beam splitter than the counter propagating beam. This timing difference in time translates into a phase shift here at the detector. And you can show that this phase shift is proportional to the angular velocity by which you rotate the whole apparatus and to the area enclosed by the apparatus. And if you did not know, the second experiment of Mickelson from Mickelson and Morley in which the speed of light was shown to be constant. And in fact, the speed of light could be determined in a better accuracy in that experiment was a Saniac experiment in which he built, this is his paper in Nature from 1925. This is a short paper. I want everybody to try and publish such a short paper in Nature without supplementary information, et cetera. He published it in 25. He built a Saniac loop just like what I showed you here. He built it. It was 2,000 feet long, 1,000 feet. And he evacuated it. And of course, it was supposed to detect ether at the time, which he didn't. But he was able to measure the speed of light with pretty good accuracy. And it was the second of his experiments in that direction. OK, what we are going to do is we are going from this Mickelson experiment, we are going what, about 60, 70 years later, when people started to use fiber optics and realized that instead of using mills, I can actually launch the experiment into a fiber. And I can make a fiber loop and launch it into a fiber and I can rotate that fiber. Now you remember the formula that says that the phase shift that I'm going to detect at the detector is proportional to the area enclosed by the loop. Now you remember how do you make a magnet from a loop of electric current? You just wind it and wind it many, many times. And each time you wind it, you increase the magnetic field that you can have in the center of that magnet. Well, the exact same thing is here. Every time that you wind that fiber, you enclose another area A and another area A. And at about one kilometer of a fiber, you can reach that if you use Earth rotation. By the way, that's what, of course, what Mickelson used. He used Earth rotation as the apparatus that rotated that 2,000 by 1,000 feet loop. So what you do is you make a solenoid if you want out of the fiber optic and you can increase the signal. And it's about if you take one kilometer at a diameter of, say, 20 centimeters, believe me, you get about 100 micro radians of phase shift. Now what's the important thing? The important thing is that, of course, first of all, we are going to use it as a magnetometer. We don't need to wind it many times. In fact, we don't want any rotation sensitivity whatsoever. So we are going to make the loop as small as possible. And in fact, we are going to get rid of the loop. But the other thing that we are going to do is simply look at it and see immediately that because I have these two counter-propagating beams that are coming to the detector to interfere, it has these both properties. It has this property of like putting a mirror and having the same beam coming back, but now they are doing it simultaneously. And like really realizing on zagereciprocity within the same apparatus. Okay? Because I have the source and the detector and the source, you can exchange them. These are these two beams that are coming together and are going to interfere at the detector. So the only other component I need in order to make it into an interferometer, remember interferometer, you have two beams that are going to interfere. You take one beam and you modulate it and then you look at the modulation at some detector. This is an interferometer. Well, that's exactly right in the microscope only. You modulate the length, here you modulate the phase and there are ways to do that. I'm not going to go into the details of how to do that, but it's easy. It's very easy to do it. And what you do is you get an apparatus that is immune completely to any reciprocal effect because there are these two beams that are going in the opposite direction interfering at the detector and then that's it. So how do I make it into a magnetometer? Well, I take the fiber. I use scissors of high quality and I cut it. So the first thing that comes to mind is I want to use it as a magnetometer and I have this linearly polarized light running in that loop. Remember, linearly polarized light, any linear polarization is the sum of the two circular polarizations. What I want to do is I want to compare beam going with the magnetization, to compare it with a beam going against the magnetization. So I'm gonna use a single circular polarization. How do I achieve circular polarization? Out of linear polarization? How do you get circular polarization? No, no. What do you do in the lab when you did experiments, that I don't know, first year, second year? Quarter wave plate, that's it. So what I'm gonna do is I'm gonna put two quarter wave plates. If you put them 90 degrees to each other, then you did nothing because there is this linearly polarized light becomes circular polarization goes complete the loop and then in the other direction, complete the loop. Nothing happens, okay? But now I have this cavity here in which I can put my sample. And if I put the sample, then remember, it matters whether the circular polarization going with the magnetization versus against the magnetization. So now I mimicked that mechanical rotation that Saniac first used by using this simple device. And if this does not exist, time reversal symmetry, I mean, this apparatus is completely reciprocal and I can get zero like zero is zero. But then if I do put a material with magnetization, I'm gonna get a phase shift, okay? And this phase shift is a measure of the magnetization or if you want, it's the Faraday, actually, twice the Faraday angle of this material. You can put, and we did, you can put here any material that rotate polarization such as quartz, sugar solution, anything that you can think of that has optical activity, you can put a Polaroid material, you will not see it. The zero stays zero. And this is simply because it's a completely reciprocal apparatus. But if you put in between a material that does break time reversal symmetry, such a magnet, immediately you get a phase shift, okay? So this is at the time of the anions, I told you about it, and that's how we started this research in 1990. This is from my own lab notebook. The reason that I actually did the experiment, despite the fact that I already had students, is that my student that was doing the experiment was part of the Stanford band. And at that year, Stanford made it to the final four, and he was the trumpet guy in the band. And then he was doing the really important experiment, playing for the band, he decided to play with the band. So this is from my own lab notebook in which I calibrated the apparatus. And this is just the simple fiber-optic gyroscope that was in a next-by, next to our laboratory. That's why we used the full one, which allowed us to calibrate it with rotation, and then putting high-TC films that were supposed to give us, remember, 200 milli-radians. And it's zero to where zero can be. Okay. Now, going now fast forward to the years, not of any on superconductivity, but of time reversal symmetry, breaking systems such as tonsum ruthenate, et cetera. The idea was that we needed an even more sensitive apparatus. I gave you the estimate it should be of order, or smaller than 100 nano-radians. So how do we do that? Well, if you look at the loop here, you see that there is a line of symmetry here. I mean, here I'm going through the material, okay? You can see it here. I go through the material. That's the further effect. But if the beam that came from here was reflected back, then I would measure the care effect. But what about this beam? Well, this line of symmetry I can really fold. I can fold this part onto this part and have the two beam going in the same fiber. Question is, is this possible? Well, yes, it is possible, because we were using optical communication fibers and optical communication fibers, if you didn't know, now you know that they are polarization maintaining. That is, they are birefringent. And in optical communication, only one of the modes that I don't remember if it's the fast or slow is being used, okay? Think about a birefringent material and you're using only one of the principal axes, okay? So we said, well, we can really use those two axes and make a Saniac loop. With zero area. So that's when you fold it back and I want now these two beams to be in the same fiber. Here is how we do it. So remember, the fiber is birefringent. What I'm gonna do is I'm going to take, let's say, linearly polarized light along the slow axis, which is blue here, okay? So it propagates in the fiber. The fiber has some, I mean, it's relatively long. So anything that leaks into the other direction doesn't matter. It comes out of the fiber. I have now a single quarter wave plate. Remember, I folded this loop. So I have now circularly polarized light in this direction. I reflect it back, then what I have now is through the quarter wave plate. I have the circularly polarized light that in the frame of the fiber is in the opposite direction. And therefore, it comes back 90 degrees. That is in the fast axis. Going in on the slow axis, coming back in the fast axis. The other beam I'm going to launch into the fast axis and I'm gonna come back in the slow axis. So I achieved a Saniac loop with zero, strictly zero area. Okay, so I started from here. That's the cartoon I just showed you. And everything can be done the exact same thing, but now I do it in reflection and I get not twice the farther, but rather twice the care effect. Okay, so it turned that we achieved an apparatus with extremely high sensitivity, very robust because we can get down to very low powers. We can go to low temperatures. This was supposed to be theta. It changed the form, but the most important thing is that by symmetry, the Saniac interferometer measures only non-reciprocal effects. And that's the important thing. It's the fact that it does it by symmetry. I don't need to, people who measure whole effect, for example, to get high sensitivity, if the whole effect is small compared to any perturbation, they measure it with the field up, then with the field down, they subtract, divide by two, they have the whole effect, okay? But if I have to do it here, it is like measuring in one direction one billion and one, and in the other direction one billion, I need it then to subtract one billion and one minus one billion and hope that I don't have anything else which is much larger, which of course I do, okay? But if you do it by symmetry, as this apparatus does it for you, then that's it. So that's the principle and that's what we did and it worked. I'm gonna skip that. That's the way you can actually look. This is a little bit more technical. You can look at it in the notes how you got eliminated. The important thing is that with this apparatus, we were able to get a shot noise limited optical detection above free micro watt. For those of you that do do experiments in dilution refrigerators, free micro watt allows you to go down even to 10, 20 millikelvin. So I can go now to very low temperatures, way much below any TC of the materials that we are going to measure. And as I said before, the most important thing, complete rejection of all reciprocal effects. And it's a beam. I have this quarter wave plate. I can also use a lens as long as the lens is reciprocal. And I'm going to penetrate one optical penetration depth. Okay, that's again technical. So the first thing you wanna do is simple measurement. So this is the Stonzo-Muthinium-03. It's a ferromagnet, has a TC of 150. We measure it now with one of the early, yes. I'll show you. So you see it already here that I have a film of Stonzo-Muthinium-03, I get a very small signal. By the way, anybody guess why do I get a small signal? The beam size, by the way, is about 10 microns. Why do I get a small signal? I mean, it's a ferromagnet. I expect, and I showed you at the beginning that I get for this material at low temperatures, I get about 10 milli-radians. Why do I have only a few micro-radians? Domains, it breaks into domains. Some of the domains are up, some are down. In principle, I should get zero. However, I will not because of random statistics, of Gaussian statistics. So I get a finite number. So if I do this experiment many times, as I did here, okay, so you see sometimes it's positive, sometimes it's negative, that is the residual. If you do Gaussian statistics, sometimes of these domains give you up, give you down. Sometimes it's very close to zero. You asked about resolution. Well, you see this goes way below a one micro-radian. And I can track the transition, okay? So I can do statistics of that. And if I do statistics, and this is actually showing that indeed the beam size is much larger than the domain size, the domains are here viewed using Lorentz microscopy for those that know what it is. And what you find, you can do statistics of how many and find what is the natural effect simply by doing domain statistics. But you can also cool it in magnetic field. Therefore, it's gonna cool down with all the domains aligned and you'll get the full number which is here it's only down to 90 Kelvin. You get two milli-radians at 90 Kelvin, okay? By the time you are at four Kelvin, it's about 10 milli-radians. So it all works, but it also tells me how do I wanna do the experiments of these time reversal symmetry breaking superconductive? Well, that's another type of experiment. Let me, because of time, let me move to some measurements. And I'm gonna start with two examples of different type before I move to another more broader subject of heavy fermions. So in fact, what prompted us to go back to the apparatus and ask how can we make it more sensitive? How can we measure much smaller effects? Was the fact that stontium ruthenate was discovered and was predicted to be a superconductor that breaks time reversal symmetry. But now in the normal way that is P plus IP. I'll show you why. Well, first of all, if you didn't know stontium ruthenate that is stontium II ruthenium 04. This is not stontium ruthenium 03 which is the cubic material. This is the layer material. It's isostructural to the first Berners and Müller high-temperature superconductor, isostructural to lanthanum II copper 04. At that time, I mean for high Tc, it was docked with barium, the first paper. And then bent structure calculations as well as measurements using R-pass and using quantum oscillations determined that there are three bends. And that's the picture of the bends cutting the Fermi surface. And there are these bend alpha, beta and gamma. And in the Z direction, it seems to be cylindrical. So it's really quasi two-dimensional material as was determined from these measurements. So what is then stontium II ruthenium 04? It's a quasi two-dimensional material predicted to be strongly correlated Fermi liquid. Why? Because Andy McKenzie measured the temperature dependence of the resistivity down to very low temperatures before Tc. He showed that T square is restored at low temperatures which is a signature of Fermi liquid and quantum oscillations were shown to exist which of course helped to determine the bend structure of this material. Now also, well, there were some other measurements, I'm not going into all the details, which showed that it's consistent with the spin triplet and extractions of Fermi liquid parameters for this material show that it looks very much like helium-3. Helium-3 has phases, actually one phase, the helium-3A, that breaks time reversal symmetry in a way that was predicted also for this material. So what do we know? We know that this crystal structure is like that. And then soon after the discovery, Tc was measured as a function of the residual resistivity that is the purity of the material and it was shown that Tc plunges down as a function of increasing the disorder very much like magnetic impurities in a BCS normal superconductor, conventional superconductor. This is a signature of unconventional superconductivity if you remember this argument of averaging the per wave function over the Fermi surface. And spin triplet pairing was shown to be the most reasonable for pairing when Ishida showed that the spin susceptibility does not change as it should have been for an S wave superconductor. An S wave superconductor, as the cooper pairs condense, you have more and more pairs and they do not contribute to spin susceptibility because the cooper pairs are spin zero, S equals zero. So that's why you have reduction in spin susceptibility but now it was shown not to change through Tc as signature of S equal to one. In addition, people did phase sensitive measurements to determine whether, I mean, what is the symmetry if it is indeed a spin triplet, spin triplet then you expect that if you go in one direction that is you connect, I'm not sure if you guys remember the phase sensitive measurements that were used to determine the symmetry, order parameter symmetry in high Tc but these were edge symmetry here. It was opposite direction because this is what will tell you whether you have the same or opposite sign on both sides where the same, and basically you see that the critical current as a function of small magnetic field is zero in this configuration, it's maximum in this configuration very consistent with spin triplet. However, time over symmetry breaking needed to be shown. Spin triplet does not mean time over symmetry breaking but even with these measurements it led Maurice Rice and Manfred Sigrist as well as Baskaran to analyze the spin triplet evidence, the four fold symmetry of this material and then adding to it a recoupling. I don't have it here but if you look at the specific heat, the specific heat look very much like a BCS specific heat, really a signature of recoupling and then assuming weak spin orbit coupling they came up with a particular symmetry prediction for this material. Because of time I'm going to skip all these, you can go over to see how they did it. This is the table of all possible symmetries when you have spin triplet. This one for example is consistent with the B phase of helium free but this one, only this one, the P plus IP is consistent with the A phase of helium free and that is the symmetry. So let me show it to you here. This is a cartoon from Yoshima Eno who is the discoverer of this material. And then we said that, what are the things we said? We said that it is spin triplet. Okay, spin triplet means that the L should be one, three, et cetera. So it was predicted by these people to be L equal one. So that's P plus IP, okay. And then remember the two dimensionality, okay. So you put these together, so if I have LZ equal to one I can have plus one, zero and minus one but if it's quasi two dimensionality I put the electrons in the plane then it immediately select only the P plus IP and I have either the Cooper pair rotating one way or the other the spin triplet on the other hand does not need for each Cooper pair it's a random direction. So the average is going to be zero but the angular momentum is in either plus or minus. The degeneracy is two, just like an Ising model time reversal symmetry is broken. And if you remember the first lecture we said that all people can use muon to detect such time reversal symmetry breaking. And indeed this was done by Gamluck et al and they indeed found that there is a change in the muon relaxation. They found extra relaxation below one and a half Kelvin which is the TC of this material. They determined that the internal fields were of order of about half a gauss. Now this is really the limit of what muSR can do and determine internal magnetic fields. So it already shows you that the effect is tiny. The effect is tiny in muSR. It's therefore expected to be tiny also in the optical regime. This is our measurements. So who asked the sensitivity? Here it is. This is measurement of the care angle using that Saniac apparatus. The way we do it we measure in an environment double mu metal which therefore the environment is below three milli gauss, not Tesla, three milli gauss. You cool the material and then you measure as you warm it up. Yes there is scatter but as you can see below one and a half Kelvin there is a finite care effect if you fit it to some order parameter then you see that at zero temperature it will be of the order of 100 nano radians as we estimated. But I can track the transition. I mean this is 20 nano radians. I can track the transition. This shows you the sensitivity of the apparatus, yes. Excellent question. Obviously I would expect, and there are unfortunately you don't have it here, but at zero field this was exactly, this was the maximum that we could measure. So that's, I'm showing it here. We had cool downs in which you see nothing. We had cool downs in which you see one direction and then another and well we did not do as many cool downs as we did for the ferromagnet. It's a little bit more tedious measurement but overall not always you get this large effect. But this means that I wanna make sure that this large effect is the maximum. How do I do that? Just like in the ferromagnet I'm gonna cool it in a field because magnetic field couples to the time-reversed symmetry or the parameter and therefore I'm gonna see what I get. And this is it. Here we cooled it in plus 100 Gauss. Then turn the field down but just before turning the field down we measured these two points. Okay. So these and these two points were measured on top of some background because when you turn on a magnetic field the optical fiber and the components give you some background. So you subtract it. That's why here it's delta theta care. The previous one was full nothing else delta care. And then you can cool it in negative magnetic field say minus 50 Gauss. You see it's the same and you see that we get to at about half a Kelvin we get to the exact same size of the effect. Okay. So this really tells you it's a genuine effect. Now I wanna pause in a minute and talk about again it will be related to domains and will be related to this issue of, did you say peace? Oh I thought you said peace. Okay. So, and that's the thing that if you take a finite size ferromagnet you probably know that we expect edge currents. You take this finite size material you expect edge currents. Pictorially it's very easy to understand. If I have these loops and I add them all together of course if it's infinite I get just the magnetization nothing at the edge but if I have the edge then you see that I can add up all these to get an edge current. Okay. That's a cartoon that basically shows you why I expect edge currents. But of course edge currents will be canceled like any other thing by Meissner. So you need to compare the extent of the edge currents with the extent of the Meissner the edge currents scale is going to be the coherence length. The Meissner scale is the penetration depth as you remember I hope. So if they are not the same then there should be some something left. And people try to measure it. The group of Catherine Moller among other groups try to measure it and they saw nothing. Okay. Here is the second paper the more detailed paper. Basically this is the result. They didn't find any edge currents which of course means that something needs to be understood. Okay. So let me summarize Tonsi-Ruthanate. We saw the time of the symmetry is broken. Maybe it's broken not because of P plus IP. There are other possibilities including non-unitary type of order parameters. The signal onset at TC it looks like I can fit to it an order parameter that is proportional to TC minus T that's what you expect. Order parameter is expected to be proportional to the gap square therefore T minus TC. The domain size based on the beam size that we were using seems to be relatively large domains because only on occasions we saw these cancellations et cetera. And based on the fact that we cooled in a field and find the same type of effect we can determine that fluxes are not responsible for the effect. We also did measurements as a function of power, optical power just to make sure that we don't have some local heating and therefore some effects like that and the world. So the only thing that is still missing for Stonzium-Ruthinate is the absence of edge currents. And I should say despite the fact that this was the motivation for these measurements and the fact that was the first measurement that showed a finite care effect of this magnitude the story of Stonzium-Ruthinium-O4 is not finished yet. I think somebody wants me to finish because he's hungry and he wants to go for dinner. So I, should I stop? Yes, and then we'll ask questions. Okay, who wants to stay two more minutes to hear about that? Okay, only one, we'll talk later. So I will stop here, these are going to be online, so. But this could be used for other type of time over symmetry breaking superconductors such as superconductor ferromagnet bilayer now you induce the time of a symmetry breaking using the ferromagnet, you create now a system with P wave superconductivity in an artificial way and we could detect it. And of course I didn't have time to talk about these in heavy fermions, but these are going to be in the notes which I'm going to distribute. So you will be able to display it for yourselves. Thank you.