09:02:11 is. 09:02:11 Can you hear me okay yes we can hear Yes, okay, cool and you can see this as well. And you can see the. 09:02:26 And so good, and I'm going to pick up so thank you to the talk by Thomas before which was of course a lot about spins, and I'm going to also talk about spins but in a quite different context so it would be not very much groups and all that it will be 09:02:37 quite a different approach so it's sort of really motivated by some discussions I've had with people who work on magnetism and to want to understand, new materials that they are using for hard drives that have quite different properties from the ones 09:02:51 that they have been seeing in the past, and they are trying to explain them and they don't have the tools, and so we as quantum physicists, at least is a quantum physicist and I think many in the audience. 09:03:03 I think we have the tools to help them. And we have to see where we can, we can use them. 09:03:08 So, the question that that we, I want to solve here is the following. So in magnetism there's something called the energy equation, that has been used by that community for the last 15 years really to model everything in magnetic materials and but with 09:03:25 the new materials they are now needing for hard drives, they're hitting limits for some materials at all times. And they're trying to extend their energy equation in all sorts of directions but they don't have to sort of understanding of some of the aspects 09:03:36 of course they have understanding other aspects and sort of do something that isn't actually maintaining the fluctuation dissipation relation which is a little bit of a lot. 09:03:48 And so our question was simply, can we systematically go beyond the fundamental logical allergy equation, and in particular can be derived at energy equation as a special case of a general three dimensional open system Hamiltonian, and then can be used 09:04:03 our open system model and suitable coupling function to explore the dynamics beyond the allergy equation. And of course the answer is yes. 09:04:12 So let me first. For those of you who don't know what the energy equation is flooded like up. So this is it. It describes spins procession in a magnetic field. 09:04:24 And, and obviously if there was no environment it would just simply continue processing. And we, there's also of course, in a magnetic material will have many spins but today I will not talk about the neighboring spins actually at all. 09:04:46 I will be more interested in these two parts so these two parts, is to give the damping fundamental logical dumping term and a stochastic field. So these are usually plugged in by hand and then people solve the dynamics of many students interacting with 09:04:53 these ingredients. 09:04:54 But in in the recent past as I explained papers are now coming out, where they have to include further terms to that term that I just showed you, because otherwise the dynamics wouldn't fit anymore so the experiments require this and they simply add those 09:05:08 things in by hand by additional, including additional parameters. 09:05:13 So we want to take a different approach that is very much rooted in sort of open quantum systems and familiar to many of you. 09:05:21 So we are going to construct the system by FINRA Tanya, and that one will look like this. So here we have a system Hamiltonian that describes spins interacting with each other, and also an external field, we have here boss Hamiltonian of oscillators, 09:05:36 one actually for each spin in this case. And here I'm putting in the linear interaction between those spin and and and above, excellent practice. 09:05:46 And so this is sort of the simplest model one can come up with really to describe the, the dumping of spins in, in this magnet. And it's very nice because one can then take the equations of motion of course for the spins and also for the oscillators in 09:06:00 the bath. And what one gets when one puts this all together, is this equation. So this is the Heisenberg picture of the spin operators, so it's obviously quantum. 09:06:11 And then we have s course be the usual, and then we have these additional two terms that nicely now capture the buffs impact on to the spin dynamics so that's exactly the part that is usually plugged in by hand. 09:06:23 And here we have it. 09:06:24 Just like in the cut out and again, but this time for spirits. 09:06:28 So obviously these two terms arise because of coupling the original coupling between our spins an hour, an hour also the oscillator positions. And so, obviously they're going to fulfill the fluctuation dissipation relation by a construction, and obviously 09:06:43 they're going to do that for any coupling function. So here's the fluctuation dissipation relation in the quantum case, and then there's also the classical limit of it will be important so that will be the high temperature limit of disco Tongans discussed 09:06:57 it in a minute. So this is always going to be true and that's important for the magnetism community because they don't always get that right. 09:07:05 So, from our point of view, this is going to be true for any coupling function that we can choose in the beginning, so now it starts to be a question of what coupling function should choose. 09:07:16 So, let's go with. So the coupling function of course goes here, and it's a function that tells us how much does the spin cover to a bath mode at frequency on the go. 09:07:26 And so here's the first choice, the first choice is a linear coupling function is C, omega and ignore this and it's simply proportionate to omega. 09:07:37 And here's a picture of it so this pink line shows us the coupling function over the frequency. 09:07:44 And I've put it into the scales of the bowl for law more frequency the level of frequency is the one in the external field, this is the dominant frequency and so that's this yellow region here is around that frequency so in this range, we have a linear. 09:08:00 In fact, always linear coupling function, and good sorta kinda that corresponds to his coupling function is a no memory canon right this derivative of the delta function I believe you know all of this. 09:08:16 So, the point of choosing this linear coupling function is that when the plug that into our general spin dynamics equation. 09:08:24 This of course simplifies a lot this kernel here that takes into account the memory. Now simplifies to just simply give this term, this derivative of x with respect to t and and in front of it is this Gilda damping parameter. 09:08:40 So in other words, I've just shown you how to derive the energy equation that's been used for the last 50 years from a quantum system bath Hamiltonian, with a linear coupling function linear and frequency and then you get it as one of your results. 09:08:54 So, this makes the connection but of course. 09:08:57 The great thing is we can, we are going to be able to go further than this, obviously, and because we want to go beyond the allergy machine. 09:09:05 Before I do that, let me just tell you that what people in magnetism currently do is they simulate the dynamics of many millions of spins that interact with each other, classical spins, and that's called optimistic simulations and what they do is they 09:09:19 assume energy coupling so they assume the allergy equation. And they also assume the high temperature limit so they assume, instead of this power spectrum that belongs to this coupling function. 09:09:31 They assume the high temperature limit where this cotangent simplifies and actually becomes frequency independent, so that will be this pink line here, that's what people do. 09:09:42 And as I said, many of the simulations do not fit anymore to the experimental results. So the question is what is wrong. And there are many things that could be wrong right so there's. 09:09:54 It could be that it's classical simulations, and they should be quantum. It could be that it shouldn't be allergy coupling. and that's what we're going to discuss next. 09:10:03 So what happens if we take into account a different coupling. So, let's look at an Orion see uncoupling so here I've plotted. 09:10:12 This is the plot. This is a plot for the door I'm seeing coupling function of this form with three parameters that allow us to talk, different types of complaints so alpha is the strength, omega zero gives us a peek so we're here assuming that there's 09:10:26 a sort of peak frequency at which our spin couples to the environment the strongest and after that it decays again which is physically intuitive. And then there's also with gamma, which tells us how how broad is the peak. 09:10:39 And as you can see this is not linear. So when this red curve is is actually sort of peaked in the region of the normal frequency we haven't really truly low ransom coupling function. 09:10:52 And we will see memory effects so here's the colonel that corresponds to this function and we have no memory effects that will play a role in the dynamics. 09:11:01 And likewise we will have a power spectrum that belongs to this fee. And here it is so the main point is to see that it has a peek in the relevant frequency range, and that compares very, you know it's quite different from what the standard energy equation 09:11:17 would assume standard energy equation with classical nice will be here. And with quantum noise at 200 Kevin's also there but if you go to slower smaller temperature, then there's a difference between the two. 09:11:29 But you see that this wraps up this is even different even, even. 09:11:36 Okay, now the point of taking such Lawrenson is on the one hand side it's physically nice to have something peaked secondly it's very nice to solve a lot of things because it can analytic yourself many of the integrals you need, and. 09:11:48 And thirdly, it gives us three parameters with which we can toggle between the allergy regime and and the non allergy machine. So in particular, we can be covered allergy expressions if we take an omega zero. 09:12:01 the peak being very far outside this relevant regime of frequencies. 09:12:07 So it looks them like this so this is an actual renewal prevention coupling function in the relevant range there is a peak. And this one is also in Lawrence in the blue one, but the relevant for the peak frequencies well outside of the relevant range 09:12:22 and therefore you can approximate it very well with the energy equation. And so when your material the specific material interested in has a peek frequency over here, you can work with the allergy equation and that would be fine. 09:12:35 If your material has a different frequency here that couples to dispense then your energy equation will be fine. 09:12:42 So it will depend on the material, what you're going to get in terms of the dynamics, and we think this is going to make quite a difference to the simulations, they will arrive. 09:12:53 So here's some. 09:12:56 Maybe I'll skip that. 09:12:57 So here's some short time dynamics pictures, they're very simple pictures were actually in terms of complexity there a bit complex but what they show is in the short, the short time dynamics of a single classical spin vector, okay, Justice in the spin 09:13:12 factor, and no exchange nothing and become this with different continents now. So, integrating the dynamics with different dynamic equations. So in particular, we can take the standard energy equation of classical noise or the standard energy equation 09:13:29 with quantum noise or the Laurentian with this one here the blue one is the Red Sea and set by the Red Sea and said it's meant to be close to the energy set, and indeed we see that all these curves are very similar to each other. 09:13:46 So that's just a sanity check that our logic is working. 09:13:50 Now what you can see here is the red curve here is, I should say all of this is the SF component, apart from the Queen curve that's the next component. 09:14:01 So this bread curve here is when you switch on a different Laurentian with a peek in the relevant range range, and as you can now see there's a real difference in the dynamics. 09:14:13 So it starts immediately and at all times and it will also influenced everything else. 09:14:19 So memory effects clearly make a difference, very quickly, and that was what I showed you here was for large spin so often, of course in magnetism they take many spins and consider them as a single spin of the larger dimension. 09:14:35 And that was this picture, but if you really go to the optimistic level of individual spins one half spin one house, then here's the picture, of course it all was a bit more but it's also taking it at a lower temperature, again you see a difference in 09:14:50 in the blue curve, which is like an allergy curve and the red curves which is not like the allergy curve. 09:14:57 So you see the differences in memory, obviously. 09:15:02 Right. 09:15:07 I skipped it to the main point I want to make is that deliverance and Catholics really allow us to systematically study how memory and current, and how the memory kind of affects the short and long time dynamics and that obviously always with guaranteed 09:15:20 fluctuation dissipation relation. 09:15:24 Let's have a look at a calibration curves. So, here is this is a curve in time, but this time we have an ensemble average over many such individual spins and processing and time being done by the environment with different dynamic equations so we again 09:15:43 have the classical energy equation. And we also have our Lorenz encounter with quantum with the quantum bath for quantum statistics and tolerance in Canada with a full content bath that has memory here. 09:15:58 And so what you can see is that whenever you have something energy like this is the this is the equalization curve. Okay. But if you have something with memory, you get to the to roughly to the same equilibrium state much faster by a by a factor of three, 09:16:33 the power spectrum of course, that acts on the blue curve, for example, is broad is over here there's a lot of noise and power spectrum of this red curve is very concentrated so in other words there's much less kicking and twisting on our numbers we can 09:16:35 said, of the time. And if any of you know how this works. Please can you tell me. At the end of this talk, why does it so because I don't know why this is so it just simply know it's that simple argument that I know and that is simply that 09:16:52 equilibrate faster. 09:16:53 This is my simple explanation if someone can formulate this in very nice mathematical terms, I'll be happy to hear that. So let me move forward. 09:17:02 So we see this much quicker calibration is very important for magnetism people because they want to write a bit of information really quickly. In a magnetic hard drive, they want to have quick time of equity ratio afterwards. 09:17:17 Okay. 09:17:19 Now this is my small spin. 09:17:21 And here I'm discussing the difference between applying classical noise. So the one the high temperature limit noise or quantum nice, so the food court handguns. 09:17:32 And what you see here is that the classical noise with a standard energy equation gives you this value of the organization. At one Calvin. 09:17:42 But if you include in your dynamics quantum noise. And obviously, even at zero temperature. 09:17:48 It's still wiggling, right, you still have quantum fluctuations, and they make your spin decay, stronger, even at lower temperature, you have already in the one monetization value here. 09:18:02 And that's true for all the different interventions, sort of quantum noise really dominance at this low temperatures. 09:18:11 And that's also an effect that we believe is actually affecting the experiments. 09:18:16 But to what degree we will see. 09:18:19 Okay. Here I'm showing the monetization as a function of temperature so this is the super standard curve that you learn in school. This is your Boltzmann distribution for a classical spin and you take the average, and then you get this curve, which are 09:18:33 the black dots and very nicely. Our dynamics that we solved all layer on top of it for all the different choices so different coupling functions different bass classical quantum. 09:18:44 So that's again the sanity check. Now what we get for a small spin, where we look at a smaller temperature range. 09:18:54 And then we get deviation. So when we go towards small temperatures, these curves here are the ones that are integrated with quantum noise. 09:19:04 And as I already explained compromise of course means that even at low temperature is still fluctuations on. So our spin we're not actually equilibrate two to one. 09:19:15 I mean, it will not fully aligned because we still have patients that kick out of spin. 09:19:21 And so we get this much lower monetization. 09:19:24 And we think that there's there's some truth in this to to compare this to experiments I cannot be more detailed at this point. 09:19:32 But this is a very promising that we can see some differences here that arise from the quantum effects from the above. 09:19:42 Okay, I think I come to the conclusions and often questions I hope I'm perfectly in time so let's see sort of conclusions where we have based on system baton baton and they have derived this three dimensional versatile spin dynamics equation which is 09:19:57 this one. 09:19:59 So this is very much in logic like the cadaver legged model and the symbols on one. I think one of the interesting differences is it's three dimensional and you really get this cross product, which generates dynamics that is not captured in your, in your 09:20:14 standard symbols and model in this way so you don't have a cost productive. 09:20:19 So you recover the spindles all in fact if you take, remember at the beginning I had this coupling function, and actually the company function as a tensor, because I'm coupling a three dimensional spend two or three dimensional positional parameters so 09:20:32 choose our coupling tensor in different ways. And what I've done so far is to take an identity matrix. So, each dimension covers in the same way, the spin Bowser model is recovered if you just take one element in here or just one row column or something. 09:20:48 So it makes it lower dimensional but obviously you can study a lot of different situations or three dimensional ISO tropic or nicer tropic and all sorts of things. 09:20:58 Depending on the company intensive shape. 09:21:01 So in that sense as versatile. 09:21:04 I think it is not going to be useful just for magnetism is but also for other rotational Brownian motion. So I'm thinking of for example experiments where you have, in light levitated nanobots that you can play around with you you can twist and turn them 09:21:20 in different ways down the go Brownian motion and they will also be described by some type of corporate structure and any memory that the boss in in impacts onto the dynamics will also be described by such a colonel and so on. 09:21:32 So I think there will be other applications outside of magnetism. For this equation. 09:21:41 But within the magnetism benchmark. And what we've tried to do is to match it against what current micro magnetic and optimistic simulations do. And they standard Lee assume classical spin vectors, energy coupling with no memory and classical white noise. 09:21:58 So, so far we've taken the two steps of including the memory effects, and we've seen that this makes a difference to their calibration time quite significant difference. 09:22:08 And we've also included quantum noise in the bath, and we've seen that this reduces the steady state monetization. 09:22:14 Obviously, the next thing is to take quantum spin factors and we're doing that, of course, as well. This is medically. 09:22:33 So comparing the differences of a three dimensional quantum vector obeys all of these things with it with a classical version can be interesting. The reason why I focused on classical spend matters is because currently the optimistic simulations, only 09:22:39 do classical spin back to us but millions of them. So if we can first tell them where you should be including some memory effects and you should be including some quantum effects that's a step that they can easily take if they need to go for full quantum 09:22:52 simulation. 09:22:54 That's a completely different animal so we will have to see what is a reasonable step forwards in this direction. 09:23:02 And there are many open questions. So the prediction of the time scale of exploration as I said I have called the steady state we have a clue we know what to do here this we go in the direction of the main force Hamiltonian, and and strong coupling so 09:23:18 but I'm also talking with people who actually do these optimistic simulations there, they are now starting to implement our equation, instead of the energy equation to see what the difference is. 09:23:27 And it's very interesting to see that. 09:23:30 As I said, there will be a discussion in the future about the difference between when you integrate with quantum spins, or with classical spins. 09:23:37 What we've not touched on at all yet is that of course I've assumed that all my bass notes are at the same temperature, obviously you can also take your classmates at different temperatures, that's realistic in the magnetic material because they actually 09:23:50 interact with electrons and protons, and they can be a different temperatures the electrons the forms and even different modes can have different temperatures. 09:23:59 So once you start doing that, you will be able to talk a lot about all sorts of transport of heat and and so on. 09:24:06 And that's, Yeah, very interesting and relevant. 09:24:10 I briefly mentioned that you can also discuss more nicer topic coupling tenses so this coupling cancer here can go from 3d to one day to everything in between. 09:24:19 and this may be very relevant for cinematic layers so if you have just a small, thin layer would be more like a two dimensional material, and then this will be relevant. 09:24:28 And then in the end what I really want to do is to bring all of this, but I've just discussed, together with a strong coupling some dynamics framework, because I believe it is exactly the tool that that will experience. 09:24:40 Give us many analytical predictions for what we find numerically. 09:24:45 With this I end. So these are the people who would get to work. So Corona is a PhD student Simon is also an accident. And these people are working with me at the moment on all sorts of extensions to this. 09:24:56 These are the two optimistic models who I work with, and I thank you very much for your attention. 09:25:03 Amen. Thank you very much for interesting talk, 09:25:10 you have any questions 09:25:16 came in, maybe I'll ask again and let people think a bit more. 09:25:21 I'm just curious so in classical models in principle, one can do like a no first principle simulations right so you can take your best to be set of some as as being so some good data legged and don't do any approximation just saw consulting and dynamics 09:25:51 as you said like one consultant like so many questions, but I just want to if you try to do this, this is what we did, yes this is what you want for coffee and you will not release it so you did like direct say I missed it, so it's cool it's fully non 09:25:55 microbial every everything so there's no approximation here, so you're not trying to derive any Colonel so whatever you just really did simulations, I see the kind of Oh, you mean, oh, I understand No You mean, some sort of microscopic principles of this 09:26:11 interacts with this and so on you build your content together or your coupling function together and then you've got the next step. No, we did abbreviate this we simply postulated a public function initially right so it was too late for example this will 09:26:25 mention happening function that allows us to model it peaked coupling strength and. And then, the point is that you. 09:26:36 I mean, of course, there's a lot of value to doing it microscopically, but on the other hand, you have millions of power me to choose when it. 09:26:45 In the end, only one is to know your past couples and so if we can start at this point and say my golf is coupling with this romancing or I can choose different things which I can separately discuss what is a reasonable assumption. 09:26:59 But once I have a Lawrenson. 09:27:03 I can fully solved it medically with all the memory all everything. 09:27:10 Okay. Okay. 09:27:13 It's over. 09:27:13 Thank you. 09:27:14 So any, any more questions 09:27:21 still have a couple more 09:27:29 questions. 09:27:31 If there are no questions let's send the game. 09:27:36 Thank you.