 Okay, very good. You are listed as University of Nottingham USA. Yeah. Yeah, I mean, that's a little bit far west for Nottingham. Yeah, I didn't. The trains would never get there if it was that far. No, they travel even to get to the place where Nottingham is at the moment. Okay. Right, welcome back everybody for the second session today. And it's my pleasure to start off with Kay Brantner. I will, it's very difficult for me to interrupt when I'm not in the room or attract attention when you're in the room. So if you can aim to stop as close as possible to 11 minutes past five UK time, then then there should be plenty of time for questions a couple of minutes over is not a big disaster. 11. So I have 30 minutes to talk. Yeah. Okay, very good. Thank you very much. Okay, brilliant. Yeah, then first of all, I would like to thank the organizers for setting up this beautiful workshop and for giving me the opportunity to speak here. And I would also like to thank the organizers for making it possible to switch my slot with tomorrow, which is the reason why this talk is a little bit misplaced in this session. And I will actually talk about quantum thermodynamics. Nevertheless, I hope you will find that interesting. So I will speak about our journey through permutation invariant quantum. Many body systems that we undertook together with Benjamin Dean and Benjamin Morris, both of whom have a background in mathematics which is why this talk is going to be a little bit mathematical. So the subject matter of this talk are predominantly non interacting open quantum many body systems. And to have something concrete in mind for the beginning we might imagine some letters of two level atoms and these atoms are coupled to some environment and we imagine that the relevant modes that couple to our atoms have a characteristic wavelength. And for the moment this wavelength is supposed to be smaller than the area that our systems occupy. So in such a situation. It is relatively straightforward to write down a microscopic model I mean the system Hamiltonian is really just the sum of the individual to level Hamiltonians, and then there is some interaction Hamiltonian with the environment that we don't even specify further here. The crucial bit is, we assume here that the coupling operators of the environment which are the operators be in this expression for the interaction Hamiltonian do still depend on the position of the atoms in the letters. So that means the system Hamiltonian is obviously invariant under all permutations of two level systems here. So that means the system Hamiltonian commutes with any permutation operator, but the interaction Hamiltonian is not, which is why this is not a permutation invariant system in the way in which I will use this term and the following. So as one can now convert this microscopic model into a quantum master equation by following the standard steps of weak coupling more Markov approximation and in the end rotating wave approximation which is not a problem here because everything is not interacting so the energy levels of the system are actually well spaced. So, and then we get a simple quantum master equation which essentially describes the K of the individual to level atoms. Now the imaginary situation where all the atoms are usually in an excited state. And we would measure the intensity of the radiation that comes out of the system as these atoms decay, you would see a relatively unspectacular exponential decay. And now we can make the situation a little bit more interesting by bringing these items closer together so that the occupying area that is comparable or smaller than the wavelength of the relevant modes of our environment. So in this situation, we can neglect the position dependence of the reservoir coupling operators, so that we can rewrite the interaction Hamiltonian in terms of collective spin operators which are just the sums of the individual raising and lowering operators for the two level atoms. And as a result now also the interaction Hamiltonian with the bath becomes permutation invariant that means it commutes with any permutation operator. And if we now feed this microscopic model in our entire standard machinery, we get a collective master equation where instead of individual Lindblad operators for the two level atoms we have Lindblad operators that act on all of these simultaneously. So this structure has striking consequences that we would now do the same experiment and prepare all the atoms in their excited state. And then monitor the intensity of the emitted radiation we would see that initially not terribly much is happening and then there's a sudden outburst of radiation and this peak scales in height with n squared and the width of this peak scales with one over being the number of atoms. So this phenomenon is known as a super radiance, and it is a consequence of this collective structure of the master equation here. And this in turn follows from the fact. So we'll try to argue that we have a permutation invariant system here. So these kind of effects have over the last years attracted quite a bit of interest in quantum thermodynamics there's a very nice recent experiment on a quantum heat engine driven by super radiance. And then there's a fair amount of theory work so this list is not meant to be complete it's just here to illustrate that these kind of systems are attracting considerable interest and I personally think that they are interesting because they give us a possibility to learn something about the role of collective quantum effects in quantum thermodynamics without without having to deal with all the complications that come from inter particle interactions. And there are interesting effects super radiance being perhaps the prime example in these systems. And therefore I think it's worth studying them. So we now go through this literature. One will notice that it almost exclusively not quite but almost exclusively focuses on ensembles of two level atoms, or effective spin systems. So the question that we were asking initially is, how can we actually describe such ensembles of non interacting permutation invariant quantum systems. If they have a more complicated structure yes for instance we might assume that we have three level atoms or four level atoms and then this level structure might not match a quantum spin event. It's not a priori obvious at least what we would have to do. So, and this is what we have tried to investigate and one might think that all of that is basically textbook knowledge and to some extent is, and it turns out that the high energy physicists and mathematicians have done a lot of work to put together the group theoretical methods that make it possible to describe the system so elegantly. And we have tried to put together these methods, and to see how we can use them in quantum thermodynamics, or what we can learn from these tools in this thermodynamic context. And go through the mathematics a little bit. I will only scratch the surface really of all the group theoretical background. And I will in certain places sacrifice mathematical rigor in favor of keeping the notation light. But if you find that interesting then please have a look at the paper and we, we made considerable efforts to try and explain these things as pedagogical as possible. The first ingredient we need is the algebra S U D. And for practical purpose this is just the set of permission D by D matrices with finishing trace. Now because the set is also vector space, we can expand any element in the set in some given basis. And it turns out a useful basis is so called Katan basis, which consists of diagonal generators and off diagonal generators. So the diagonal generators as the name suggests a diagonal matrices with finishing trace, they are a mission, and they obviously commute with each other. Fine. And then we have this off diagonal generators, which can always be chosen such that the commutator between any diagonal generator and any off diagonal generator returns a scalar multiple of this off diagonal generator. And the scale on front maybe zero. But if it's not zero that basically means that the off diagonal generators are later operators on this spectrum that is generated by these diagonal generators. So any what does this mean in terms of SU two so in spin language. It means we can write any trace less operator on a spin space as a linear combination of Sigma Z and Sigma plus minus. So Sigma Z half years our only diagonal generator and then Sigma plus minus are the off diagonal generators. So far so good. So now we can try and make this a little bit more complicated by assuming that we have an identical copies of this algebra. Now so now we have n particles, and each particle is described by operators drawn from this algebra. And then we can construct collective generators. In this way, simply by extending each individual generator to the entire Hilbert space by multiplying with a lot of identities and then adding them up. These quantities are permutation invariant obviously, and they just correspond in SU two to the collective spin operator SC and the collective raising and lowering operators as plus minus. So this is really just a generalization of what is well known from from spin physics, or more generally angular momentum physics. Operators now these collective generators satisfy the same commutation relations as their single particle correspondence. And therefore, they form a representation of this Lee algebra on a higher dimensional space. And this particular type of representation is called the tensor product representation of SU D. So that's basically what I wanted to say about these generators and now we can go on and use this language to construct our master equation and that goes as follows. We now assume that our system Hamiltonian so we can assume that we are in the basis where the system Hamiltonian is diagonal. We can expand our system Hamiltonian in diagonal generators, and then assume that the interaction with the bath is mediated by the off diagonal generators, and some bath coupling operators which we do not need to specify any further. Also here now, obviously everything commutes with any permutation operators that this is a permutation invariant system. So we can convert this microscopic model into master equation again and to this end with the weak coupling and Bob Markov approximations and then we also want to apply a rotating wave approximation to get an implant equation out. And to this end we have to decompose these off diagonal generators in the individual frequency components that means we write every new now as a sum of components, and each of these components is a letter operator with respect to the system Hamiltonian. And if we do that we end up with this master equation down here where the gamma mu new form a Hermitian matrix which is positive semi definite so that this is a proper linear equation. There is some lunch a lamp shift which I have absorbed here into this effective Hamiltonian and the lamp shift is also diagonal commutes with system Hamiltonian and therefore can be expanded the diagonal generators. So now we have constructed this thing and now the question is of course what can we do with it and how can we analyze this equation. And it turns out to this and the so called sure vitality is a very useful tool. This is a statement from representation theory which has far reaching consequences. As far as we're concerned here tells us the following as we consider a product Hilbert space h n which is composed of n identical copies of some finite dimensional Hilbert space. And now the statement is this Hilbert space has an autonormal basis in which all permutation invariant operators, ie all operators that commute with any permutation operator a simultaneously block diagonal. And moreover, so this is implied by this expression here and moreover we can now label these blocks systematically. And this label lambda here is a compound label consists of D numbers. And each of these set of numbers represents a decomposition of n, I eat a number of particles into D I eat a number of levels integers. And furthermore require that this. The sequence is not increasing and that every integer is positive or not negative they can be zero. Fine. So and the old lambda now is a block in the representation of this operator O, and it acts on some subspace of our product Hilbert space. So, and calculate the dimensions of these individual blocks and there's a formula for this dimension and what I mean formula itself is not so important the important thing is that it exists so one consistently calculate the dimensions of these blocks, and the amount of their multiplicities that means the number of times a given block occurs in this decomposition. And to make that a little bit more transparent we may apply this theory into SU two so again to angular momentum systems. And here, our compound index lambda has only two components that because the sum is fixed that effectively translates into one degree of freedom. So we can define a quantum number J as the difference between these two indices divided by two. So J can take values between zero and and half of N is even. And then, if if N is odd, then you have this half integer sequence. Fine. And now you can for this situation construct the shore basis that block diagonalizes the permutation variant operators explicitly. There's nothing else but the usual databases here. So the J is of course, our usual total angular momentum quantum number, and then you need an additional label for these states which is the magnetic quantum number. And then one can go and calculate from the formulas that I've just shown you to block multiple duplicities and to block dimensions. And of course that returns the result that we already knew before. So just a little graphical illustration of what all of that means here. So that's supposed to be a matrix representation of the permutation invariant operator in this basis. And you see you have a series of blocks, and the DJ is the dimension of the block J, and the MJ is the multiplicity of block with a given Jane. Fine. So now that we know that. We can further decompose this master equation. Yeah, the effect of Hamiltonian and all the link that operators are permutation invariant and therefore they can be decomposed into these into these sure why blocks. And if our initial state is also permutation invariant, then we can also decompose it in this way. The block structure is preserved and therefore the state will remain block diagonal at any time T. And in particular this implies that also the stationary state of this master equation which is not unique, and generally not a formal state even if you have a formal environment. We'll have this block diagonal structure. You can prove an even stronger statement. That's not part of the literature as far as I'm aware at least so that's one of the little contributions that we made here. So if we now consider this master equation that I just showed you, and we require some cryptic condition here, which I don't have time to explain in detail just so much. This condition basically means that we want our bath to couple to sufficiently many transitions in our atoms, so that there are no invariant subspaces other than those implied by permutation invariance. That means for instance if you have four level atom and you connect only two of these states pair wise so that you have two disconnected pairs, then this condition would be violated. You can introduce a third transition, so that this entire network is connected, then this condition is generally met. That's basically what it means. So now this condition one can show that the stationary state of this master equation will always have this block diagonal form. So it is initially permutation invariant or not. That means all the off diagonal blocks in this profile representation will go to zero on the dynamics generated by this master equation. But then you have some pre factors in this decomposition and this pre factors depend on the initial condition. If the initial state is permutation invariant, then they are one, but otherwise they can be anything. You can show that if your rate matrix satisfies beta balance, then the stationary states inside every lambda block will be thermal states. Cool. So now that we have that we can have a closer look at these stationary states. So now we assume we have a thermal environment and that means we have a stationary state of this form. If we want to now calculate the expectation value of some permutation invariant observable. We just have to multiply these two things and take a trace. And it turns out in the end, all that you need to know is the probability to occupy the certain lambda block which is set by the initial condition. And then the average value of the lambda block of the observable O with the corresponding thermal state in this block. And now you can for instance go and calculate the internal energy and it turns out that you can write this as a weighted sum over the derivative with respect to beta of the lock of the partition function. In this individual blocks so this is not the partition function of the entire system which would be trivial. This is just the partition function in one of these irreducible blocks. And it's, I mean this quantities generally not straightforward to come by but there's another trick to do that. And one can calculate the C lambda without having to actually calculate the shore basis so without actually having to generate this block diagonal form. This can be done by by Wild's character formula. I don't really want to go into the details. The only thing that's important is this character formula gives you a means of calculating these partial partition functions without having to calculate the shore basis. You can write it down explicitly for SU2 and then you get a well known result and then you can also write it down for SU3 and as you see these expressions expand rapidly but you have explicit expressions in principle for any number of levels. So now we have everything together now we can actually start calculating things. Once you have these these partial partition functions you can calculate more things than internal energy. It's just the quantity that I focus here on for the sake of illustration. So let's do that, because I wanted to show at least one plot. So here we have done a little calculation that could not have been done with a spin ensemble. So we consider a quantum Otto cycle. We have an ensemble of three level atoms, and we initially prepare the system because we have to set the initial condition yeah, even if we have some fixed driving protocol the stationary state of this master system is not unique so everything will depend on the initial condition. We assume that the entire sample is initially formalize at some inverse temperature beta not. And then we choose a single particle Hamiltonian which has this funny structure here. So this is essentially, it keeps the overall level splitting between the ground state and the highest excited state constant, and this parameter delta changes the related position of the intermediate level. And now you can run this ensemble through an auto cycle that means you couple it to a hot temperature bath. And this is now done through this collective master equation as the system does not go to an equilibrium state. So we can inject heat in this way, and then we can compress this entire spectrum by a certain compression ratio copper, and in this way extract work from the system. Then you couple to hot cold bath and you extract heat from the system, and then you have to expand again to close the cycle, but because you have cool the system down before that costs now less work, then you have extracted in the second stroke. So that's an auto cycle and the advantage of the auto cycle is that you can calculate everything just by taking differences between internal energies, assuming that the relaxation process in the thermal strokes is complete so we wait a long time. And we also assume that these mechanical processes where we compress and expand the spectrum essentially instantaneous as an idealized model. And it's really just here for the sake of illustration. And now one can make some plots. So this is now for seven particles. All of this has been analytically calculated. And what you see here is the network output so that's w out minus w in as a function of this parameter delta. And for comparison the dashed lines show exactly the same quantity for an ensemble of for not permutation invariant ensemble where every individual atom really formalizes. And what we see as well. We can apparently extract more work with these permutation invariant ensembles then with the standard case where every atom is essentially independent. And also we find that these curves are monotonically increasing with Delta that means we get the maximum work output in a situation where the excited state is degenerated and the ground state is non degenerate. One illustration of things that one cannot do with this formalism, and there's plenty of more things that can now be explored. And that brings me to my conclusions it turns out this entire mechanism is very useful to do asymptotic analysis. For example, I mean we have, we've quite a few of those. If you prepare the system initially in a formal state, some temperature beta not, and then you formalize it through this collective master don't formalize it but you bring it to a new steady state. Now we assume that the beta not is very large so the initial temperatures very low, and the data of our bath is very high, so if I am is very very small to the final temperatures very high, it's that means, then in the in the limit where n is very large, the internal energy of the stationary states scales like square root n. So that's a non extensive quantity which shows you that this is really not an equilibrium state and more. Universal yeah for any number of levels and you have a pre factor which depends on the dimension of the underlying symmetry group. I for practical purpose the number of levels but the scaling is universal, and you can find several of such examples and this funny plot down here which I don't really want to explain full detail, basically shows a difference in free energy in the relaxation process. So you can actually calculate other things than energy for six dimensional system so you can really go to relatively high dimensions here. Future work. I mean, this would be now hopefully relatively straightforward to use this formalism to analyze heat transport problems because none of the mathematical results really hinges on the fact that you have a single variable and can couple to several paths, one could try and do the same thing with with time dependent driving. Floquil and blood equations, for instance, and then of course at some point would be desirable to talk about interactions and the principle that's doable because most of what I showed you just hinges on the fact that you have permutation problems. But then once you have interactions usually have a dense spectrum and then rotating wave approximation and becomes questionable and it's no longer really clear. If we can use the sleep that equations presumably not. But it would be an interesting thing to think about. I think I'm in time right. Yes, you are brilliant. Very, very punctual. Thank you. Do we have any questions for K. Once again encourage students to be brave. My, my main question would be. Someone who who who has no real intuition for the sort of quantum ideas that are coming up here. Do any of these ideas transfer over to sort of a classical framework that if we can you imagine an analog of these sort of things in if we were dealing with a molecular like a biophysics system with something classical transitions between states. Is it very reliant on this, this peculiar state that you've got in the middle, but you can only get in a quantum setting or was that something you can recreate. Okay, I mean, the mathematical statements are in the end statements about matrices. From a physical point of view, I would have doubt that there is any analog in a classical Markov jump process because this is phenomenal like super radians hinge on the fact that it's possible to have coherence between different. That's what that's what I was thinking. Yeah, so that's presumably I mean that there is probably some analog with classical wave phenomena that wouldn't surprise me. But I'm not aware of anything specific. But it's always a bit dangerous to claim that something is G9 quantum. I mean, if one tries hard enough, it's usually possible to find a classical analog. Right, right. But what you're saying is that the the the analysis you've done at least is very embedded in the coherence and someone is very important. And the motivating question came really from from the super radians phenomenon and the fact that it is almost always really done with two level systems. It would just be interesting to see it can write down a similar formalism for for more complicated systems and it turns out we can. And much of the theory is borrowed from high energy physics really. Yes. Anybody, anybody else. This is the the challenge of moving moving from a quantum day. Yeah, I realize I ended up in the wrong session but that's my fault. The good news is that all the talks are recorded so that anyone I believe so that they'll be made available after the session. So, digging then a bit into the future research. Perhaps you can elaborate a bit on what you expect interactions to do or what's the interesting consequence for heat transport. I mean that there are a couple of papers now that this, I mean this, this phenomenon of super radians also translates into very large heat currents that scale with the number of particles square. And that is known in general the other one, one has this kind of effect can food has a paper on this and there was also a recent paper from another Japanese group. So, very large, very large as a function of the given the number of particles. Yeah, I mean you can presumably, presumably an experiment is never that many particles doing or is it or is it a very big number of particles doing it. No, I mean, I mean you can have a considerable considerable number of particles like I mean certainly on the order of 10s. But that's that that's not that's not macroscopic. Still, no, no, it's not macroscopic. But I mean if you heat current scales with a minute then if it's 10 particles that already makes a difference. So I mean, and this, the hope would be that one could analyze these phenomena that have been described previously now from this group theoretical perspective and perhaps learn something new. And independent driving with these floquette link blood equations that are well established for few body systems should essentially be applicable here as well and then one can do like this, this multi level mazes and then things like that interactions interactions are tricky. I mean principle one can write interactions into these Hamiltonians. And the mathematical structure still holds as long as everything is permutation invariant but one could imagine some sort of all to all interaction. The problem is more the link blood part. Because we usually mean to get the link blood equation in the end one has to apply rotating wave approximation, and that hinges on the fact or on the assumption that levels are well separated. So the spectrum is sparse. So that fast oscillating terms can be canceled. And if you have interactions then usually have a dense spectrum. And then this is no longer really possible. But would you, you would still work in some kind of approximation where you're talking about energy levels, you wouldn't be doing density functionals or anything, anything like that. No, no, no, no, not at this level. No, we've got a couple of questions from the audience so there. You could unmute. Thank you. Thank you for a nice talk. I have a question so this phenomenon of this square scaling within. You had to build your system with identical units have you tried have you looked at what happens when you have some disorder if the units are not identical. There is a little bit of deviation from this exact degeneracy. What happens then to this effect. Okay, I mean, it, it critically hinges on on the fact that the systems permutation invariance I would. I would think that if the individual units of your system are really different than the behavior will change qualitatively. But then if there's a little bit of disorder that can be treated perturbatively. I think it should not change anything qualitatively I mean. This has been observed in experiments and the experiments are never perfect. Yeah, so there's always a little bit of disorder. So I would, I mean we haven't looked into that if that was the question but I would assume that a little bit of disorder. Doesn't do qualitatively very much. At least to the physics. But I would be afraid that the mathematical apparatus explodes. Thank you. One other question. The person has lowered their hand. I don't know if it was the same question. I can't remember who it was. Okay, well, any other questions of course follow up via email or via the direct message in the chat. Let us thank Kay again for kicking off the afternoon session. Let us.