 Hello? Yes. Okay. I'll fly you for the first couple of seconds just to make sure that everything is okay. Perfect. Thank you. Your name is? Your name? Mahmalko. Good. Thank you, Mahmalko. Okay. Hi, Mahmalko. Can we get started? Okay. Very good. Good evening, everybody. Welcome to the ICTP's Salam Distinguished Lecture Series. For the year 2023. As you know, this is an important event on the academic calendar of ICTP. It has been a tradition now for over 10 years and many very distinguished scientists have come and lectured at ICTP on very diverse topics. And the idea is precisely to subject, I mean to introduce, excuse me, to make the latest developments in diverse fields of science accessible to the ICTP scientific community. And we are very happy that this year in we have Professor Sandu Popescu. I'm sorry. Excuse me. Okay. So this lecture series is supported by the Kuwait Foundation for the advancement of sciences, KFAS. And KFAS and ICTP have extended their long-time partnership for science advance advancement by creating a new joint program of support for scientists from Kuwait and other Arab countries to be engaged in ICTP activities. In addition, KFAS offers support for three Arab students to attend the postgraduate diploma course at ICTP and postdoctoral fellowship to young postdoctoral Arab scientists to continue research in the areas of expertise of ICTP. So we are very thankful to KFAS for our collaboration which has made this important lecture series possible at ICTP. Let me say a few words about Professor Popescu. He's a theoretical physicist from the University of Bristol. His main field of research centers on the conceptual and fundamental aspects of quantum physics, which is the topic of this lecture series. He's a pioneer in quantum information, best known for his work on entanglement and associated phenomena of non-locality which are crucial for our understanding of quantum theory. And as you know, the recent Nobel Prize was also in this field, in this area. He has worked, so it has a long and distinguished history going back many, many years almost to the beginning of quantum mechanics. He has also worked on many other aspects of quantum theory ranging from very fundamental experiments such as the first quantum teleportation experiment, which was one of the landmark experiments in quantum information theory to even patentable commercial applications. So he has really a breadth of view. And another of his interests is the foundations of statistical mechanics. And more recently, his interest has been in the thermodynamics of quantum systems. So we are very happy to have somebody like Sandu Popescu to tell us about these very exciting developments at the frontiers of science. He has, of course, won numerous awards. I will just name a couple. In 2017, he was elected the fellow of the Royal Society. In 2016, he received the Dirac Medal, not the ICTP Dirac Medal, but equally prestigious by the Dirac Medal from the Institute of Physics and many other important awards. He will be giving three lectures on topics related to fundamental aspects of quantum physics. Today, he will be talking about the smallest possible thermal machines and the foundations of thermodynamics. Tomorrow, he will talk about quantum non-locality and beyond at the same time. And on Friday, he will talk about, at 11 o'clock, multiple time states on the flow of time and quantum mechanics. So these are all very interesting and important topics. I just to remind you, usually we hold this series in the week of Salam's birthday, which is on the 29th January, which falls this year on a Sunday. We are having it a little bit ahead of time. So on the last day, we will also announce the recipients of the 2023 Spirit of Salam Award, which recognizes those who like Salam himself have worked tirelessly to promote the development of science and technology around the world with this specific mission to make science available to all. So that will be announced on Friday just before the Friday lecture. Just to remind you that there will be refreshments after the talk in the lobby. And as is the custom at ICTP, the diploma students are requested to have the opportunity to stay on after the talk today to meet with the speaker and the ideas to promote so that they can ask freely questions. Everybody else leaves and only the students can interact with the colloquium speaker for about half an hour. And refreshments will be saved for them even if they talk a little bit longer. So I'm really looking forward to it. I was also very happy that Sandu, in fact, mentioned and I hope he's going to do it. He indicated his preference to give a Blackboard talk, which is really how we had envisioned the Salam lecture so that it's really the involvement of the audience is easier on the Blackboard talk. And I was hoping to be there in person, but unfortunately I'm down the flu, so I would have to watch it on Zoom. So I am looking forward to this talk. The session will be moderated by Professor Rosario Pazzo, afterwards the question and answer session. Okay, thank you very much. It's over to you, Professor Pupescu. Well, first of all, hello everybody. And, well, if Atish still hears me, I wish him all the best and quick getting better from the flu. So I'm really honored to be here in this place that over so many years have done so much to spread educations about physics throughout the world. So it is really one of the, well, probably it was one of the first, and now fortunately one of the many places that are doing this, which is a very, very useful thing to do. So as I was thinking what to do, I thought that three lectures give me a lot of opportunity, but it gave me two different types of opportunities. One of them is to have an interesting subject and have time at leisure to develop it. The other was to give you some introductions in some of the things that I believe are very, very interesting at the moment, and then give you the opportunity to look further. So I decided for the letter, so I will give you three talks that are all of them different, at least apparently, although at the deep level they all contain the same basic idea that is an inquiry in the foundations of quantum mechanics. Now, I don't want to disappoint, but the first lecture, well, I already had prepared. So it will be on PowerPoint. The other two, I will use these blackboards that are wonderful. So today I will talk about the smallest possible thermal machines and the foundations of thermodynamics. So the whole idea came when one of my friends, Hans Briegel, visited me many years ago, and at that time he was interested in the issue of whether or not there could exist non-trivial quantum effects in biological systems. To see, already when people started quantum information and quantum computation, immediately the hope was, since quantum computers can work so much better than classical ones that perhaps our brains could do something like that. Well, this was a bit far-fetched thing, but at least it opened the question that perhaps there are non-trivial effects, quantum effects in biological systems. I insist on the word non-trivial because obviously everything follows the laws of quantum mechanics, but every atom is quantum mechanically, but then every piece of stone has quantum effects. So we're not talking about that. It was more interesting to talk about non-trivial effects in these systems, effects on a larger scale than you would expect. So people generally dismiss this idea, and the reason why they dismiss is because many of the quantum effects get very easily washed out by noise. And people, in order to see quantum effects, they have to work very, very hard in laboratories to isolate the system, to maintain coherence and so on. And people believe that coherence will be lost immediately the moment you go to anything slightly larger than a small molecule. However, we realize that, in fact, all these intuitions were based more or less on the idea that you perturb the system and the system comes back to equilibrium. But living beings are systems that are not at equilibrium. In fact, I don't want to reach thermal equilibrium anytime soon. So they are open systems, open, driven systems far from equilibrium. So with these motivations, we try to think of all kinds of possible effects. So because they are open and driven systems far from equilibrium, by the way, how do we know about this? We know that, for example, when an organism would die, suddenly it will start decomposing. It means that there are always forces that will try to destroy it. But the fact that the organism survives is because it's doing repairing all the time. If you wish, this is like error correction. Error correction that goes on all the time. Now, one of the things that we asked, knowing that our colleagues tried to cool down the system to maintain coherence, was could biological molecules act, for example, as refrigerators? So, you see, the fact that a biological organism can cool itself is obvious. You go to the beach in the middle of the summer, temperature is well above 36 degrees, and nevertheless, your body maintains the temperature. Then you could think, you know, that is at the scale of the full body, but you may think that different organs in the body may have different temperature. Your liver might be warmer than 36, your brain needs to be a little bit cooler. The question is, could we have this effect when we go down to the scale of molecules, say the scale of proteins, because all the chemical reactions that take place in an organism are very, very dependent on the temperature. If your body reaches 43 degrees, that's the end of it. All the chemical reactions going on go wrong, and then the body cannot survive. So, could, for example, a molecule, say a protein, act as a refrigerator? That was one of our questions. So, getting back to this, to the motivation, what do you do when you ask a question like this? So, I was very enthusiastic. And of course, immediately, immediately I made a very big discovery. Not about science, but about myself. The fact that I know nothing about proteins. Well, very little, let's say. So then, what do you do in a situation like this? Well, obviously, a way to approach it, you go to the library, or you go on the internet, get a good book on proteins, start learning about it, and hope for the best. Well, we didn't do that. So, there is another way, which is forget about it. Really forget about it. Well, almost. And go back to the things that you know. It is always good to know a few really basic things. So, the things that I knew, well, I knew a little bit about qubits, about quantum systems with two states. So, then I asked myself, forget the biology. Suppose that I would like to build, say, a refrigerator. Could I build a refrigerator from a few quantum bits, from a few systems with two states? This is interesting because, you know, then you could use them in a quantum computer, to the frigerate part, to stabilize, and things like this. But more important, it is a question of foundations of physics. So, our first question was, yeah, I want something small, like a few qubits or something like that. But when you ask this, you say, okay, if I'm going to go small, let me go to the smallest. Let me ask, what is the smallest possible thermal machine? And this may seem to you that you impose further constraints on yourself, trying not to just build something small, but to build the smallest possible. Well, actually, if you are just in the business of engineering and you want to optimize something, that is indeed hard. But the moment you reach down to really fundamental things, then the more you can simplify, the easier the task is, actually. So, this is a question that we ask. What are the smallest possible thermal machines? Now, there are some things that one needs to see here. First of all, how do you define that something is smaller? How can you say smaller? But it can be smaller as volume, or smaller as mass, or smaller as linear dimension. What do you want? Well, actually, quantum mechanics gives something very basic. In quantum mechanics, I can look at the number of states, because systems are quantified. So, the particular question we ask, what to make thermal machine with the smallest number of quantum states? Other issues. Well, you know, whenever you ask a fundamental question, you realize that there were some issues that whenever you thought, you know, in general terms, you were not aware. So, what we want here, first of all, is that they should be self-contained. What do I mean by this? First of all, that there should be no external source of work. So, you're the refrigerator at home, you plug it to the main, but you get some electric current, but that is produced in, I don't know, some power station. So, your refrigerator uses the power station as well. You should count it. So, you should be a good accountant if you start looking at this. So, we do not allow any external source of work. Or, for example, people doing optics, they cool atoms every single day. That is their business, but they use a laser in order to do that. So, you need to calculate the degrees of freedom of the laser. And then there are three key things. If you are a theoretician, you may want to use unitary transformations. But this means you want to have an entire time evolution, like the time evolution of a quantum gate. But that is a long process, what you want, and that needs to be controlled. When you start an interaction, when you finish an interaction, so you need control. That you should also count. So, we do not allow that. We would like things to evolve under their own Hamiltonian. Okay, so these are the few sort of questions that we would ask. How to construct these very small machines? What is the minimal size? How well do they work? And is there a complementarity between size and efficiency? Okay, so let me start after this very long introduction. Right, so I also didn't know too much about thermal machines. But I know one thing, which is this Feynman's example. So in Feynman's example, he has a box with an ideal gas in it. And in the box, there is a wheel, a paddle wheel, that molecules may come and hit it. So the idea is, we would like this wheel to turn. Because if I can turn a wheel, I can then hook a string to the wheel, put a weight, and I will lift the weight. But this example really doesn't work, because molecules go all around in this thing. Some molecule may hit the paddle wheel and make it turn. Anti-clockwise, another may hit the paddle wheel and make it clockwise. So this wheel will just oscillate here. And the weight will go up and will go down and up and down, and nothing gets done. Okay, so here is a way in which you may want to attempt to solve this problem. And that is the following. You have here this ratchet and pole arrangement. Meaning you have another wheel with teeth. And these teeth are asymmetric. Then you have a spring here and a little stick. And this wheel with teeth is connected to your paddle wheel. So you encounter these sorts of things anytime when you are in a building and there is a door that only moves one way to allow you to exit and not to come back in after hours. So what happens here is that the wheel turns and if it turns clockwise, it just pushes the little stick up and then the little stick falls behind it. If the wheel would like to move anticlockwise, it cannot because it hits the little stick. That's all. So that means that in such an arrangement you would hope now that whenever the paddle wheel is pushed anticlockwise, it doesn't move. When it's pushed clockwise, it moves and you are able to lift something and do work. Good idea, but it really doesn't work. It has a problem. And here's the problem. Imagine the tooth, imagine the little stick. The wheel goes normally. Clockwise the stick falls behind it. But it doesn't stay there. It's oscillating. Yes, it's just a harmonic oscillator, a stick on a spring. So after a few turns, now the sticks start oscillating so much that at a certain moment it goes higher up than the size of a tooth and now the wheel can move anticlockwise. OK, but now you find out a way of doing it. All you need is to cool this movement. So you put your little stick in another box with an ideal Gazette temperature T2 smaller than T1. So it cools down this vertical movement of the stick. So now the arrangement really works because now only movement of the ratchet clockwise will go. The stick is cooled. It stays down after every single tooth and the wheel cannot move anticlockwise. Now you can really put a string with a weight on it and it will be lifted. But there is a penalty here because now not the entire energy that you will take from this box will go into turning the wheel and lifting a weight because part of it is dissipated in the other box that needs to cool it. That is the important process here. So there are a few things to learn. First of all, that you really need two different temperature bars and you need a constraint. The constraint between the little stick there and the wheel. When the stick is down, the wheel cannot go in one direction. Once you implement these constraints, they are at the basis of every single thermal machine that you want. Okay, so here is what we did. So here is for the simplest or the smallest possible refrigerator I wanted to also cool a very small system. So here I have a system, a two-state system, say an atom and two energy states that is immersed here in a bath at temperature T1. So what happens in such a situation? Well, the system spends a lot of time on the ground state then because of collision with external molecules in the bath may get excited, comes out to the excited state, may dis-excite and go down and so on. So overall it has some probability of being on the lower energy state, some smaller probability being in the higher one corresponding to this temperature T1. To cool it means that you would like it to spend more time at the lower energy. So that is what you want to do. So here is one first design which is bring next to it another system with exactly the same energy level difference. So put both of them at the same temperature T1 and add an interaction. What is the interaction? The interaction is that it is possible for your system to go down in energy and push the energy up in this external qubit. So then you say, ah, good, I've done it because now I open another channel for my system to lose energy. Previously it could lose energy just to the molecules in the medium around it but now it can give energy to the machine. Now the bad thing is, well, I don't know whether it's bad, the bad for the designer of this thermal machine is that Hamiltonians are hermits. The fact that the Hamiltonian is hermits means that once you have this interaction you also have the reverse one. If that would not be the case nature would look completely different. But this is the basic fact about nature is that all transformations, well let me say transformation, there are different ways of saying transformation concerning formation and so on but basically the Hamiltonians that drive the evolution have this property. Once a transformation is possible the reverse is equally possible. All right, so it doesn't really work because the one, the green transformation cools the system passing the energy to the machine but then the machine goes down and then it hits my system. So as I open the channel for the system to lose energy I also open a new channel for my system to gain energy. So it's not a good idea. All right, but now that you understand this here comes another idea. You see, what was here the thing is that the bad transition which couples them the red one that required my system to be ground and the other system to be hopped to be at the excited state. If I find the mechanism of removing my machine from the excited state for example taking this system by hand and moving it down then there is nothing to push energy into my system. And I can do this if I put my machine at a lower temperature T2 so it means that my machine interacting with its own environment will now be brought down to the ground state. And now this constraint will mean while I'm able to pass energy from the system to the machine the machine is no longer there to give it back. Does it work? Yes, this works. It works perfectly but it is not exactly what you wanted because you see this means that I have my machine at a lower temperature than the system that I want to cool. This is not a refrigerator. This is an ice cube that you plug into your drink. It works but you need the ice cube to start with. But we already understand how all these constraints go. So once we understand we go to the next one. So here's the next one. This is my system here. It is a system with two states energy difference E1. My machine is built out of two other systems each of them with two states. So for the time being let me just put all of them at the same temperature and let's see what happens. I would like now to make a coupling between these three systems of the following way. The side systems which is my system and this part of the machine could both go down in energy and push the energy to the middle. Okay? That is the good one and also my system goes down in energy accompanied by the third. The bad one is the red transition which exists there because the Hamiltonian is her mission in which the middle qubit goes down in energy and pushes up my system and this ancillary thing up. So that is the transition that I would like to suppress. Okay? How can I suppress it? Well, there are two ways actually to suppress. You see the constraint for the bad transition which is like this. This one goes down and these two go up. One way would be to remove the middle one from the excited state down. That again I can do if I put this intermediate one not a temperature T1 as my system but at a lower temperature. That was the ice cube. I don't want to do that. But I can prevent this triple thing middle one going down the sideways going up. If this one, the third, is not on the ground state. If it is not there, this transition will not occur. And I can do that if I take the temperature here to be larger than T1. So just by warming up a part I'm able to deal to implement this constraint so that only the green transition will take place and the red one will not. In general, what do I want here? Well, all I want in fact I can tinker with both temperatures of the middle one and of the side one but that is the idea. I have these transitions from my system that is excited ground and the other one excited so the two side ones going down the middle one going up and backwards and I want to make sure that this backward transition doesn't work. That means I will have a constraint on the populations. I want more population when the side ones are up and the middle one is down which will say that more times that will happen than the condition when the side ones are down and the middle one is up. So once I implement this condition my refrigerator will work. You recognize the pieces in fact. The pieces of this refrigerator are pretty simple. You see what does it mean that I take energy from here remember the side ones go down and the middle one goes up. That will have to be in the undissipated somewhere in its environment at temperature T2. Well, go to your refrigerator put your hand behind it and you'll feel it warm. That is the spiral behind your refrigerator and it will have to be dissipated in the room. T3, well, if you go to this one is the hottest one. If you go to your refrigerator at home you will not see that because it has an engine that compresses something. If on the other hand you had a very old refrigerator you will see that all you need it doesn't have any engine in it just some arrangement with a gas that you can warm it up and you just put a resistor that makes it hot or you can put a flame there so that is the one that drives the action here is where you the energy is taken from there it's also pushed out into the environment and on the way it drags energy from your system from here that is how it works. Okay, what else? What can we do? Well, this was at the refrigerator when I discussed about Feynman's example I discussed about an engine that does work Why did I go from that to the refrigerator? And the reason is very simple it's because I was silly. You see, when I thought of this I knew that work the difference between work and heat is that work is delivering energy in an organized way while heat is delivering it into a disorganized way but when you go to small systems what is organized and what is disorganized you don't know the difference so it's very difficult to define what is work when you go down to to small systems and that was what I wanted to avoid Why was I silly? Because nobody told me that the system must be small the system could be big I wanted the engine to be small but you know whenever you discover something you always discover it into a more complicated way and then you simplify Nevertheless, this leaves opens the question of what is work at that level So what is work? But if you would look in papers discussing about work for quantum systems you would see a large range of definitions for it just because people had to somehow try to understand the difference between transfer of energy in an organized form or a disorganized form It was good that I didn't know about these papers because then I would have been confused by them Sometimes it's good not to know too much because you could then think for yourself Otherwise you need a supplementary effort to try to throw away many things before starting to think So I try to do the simplest thing We know that we do work to try to lift a weight That's very simple So all I need is a machine that has two thermal baths a hot one, TH a cold one, TC and a weight If I succeed to lift the weight I'm done My machine is doing work So here is how I implement it My system which previously I took a system with two levels A system with many levels Like this weight that actually goes from down to up and it just goes and goes So let me take something like a harmonic oscillator a system with many energy states And all I want I want a machine that will take this system and push it up, up, up, up So what is my machine? Exactly as it was before My machine is made out of the same two qubits two system, each of them with two energy levels with exactly the same interaction as I did in the case of the refrigerator The side ones go down the middle one goes up or the middle one goes down and the sideways go up Why is this not a refrigerator? Because I run it into a different regime Now I want the middle one to go down to push the side ones up So I want to put more energy into the middle one Previously I put more energy into the side one So that's all I let the system go and now these two parts of the system T2 and T3 I drag energy from T2 some energy is passed into T3 and it's in its environment and my system will go up Exactly the same principle as before which all I need I need that the probability that T that the middle one is excited should be larger than the probability than the side one is excited That's all So that would mean that if I look at the Boltzmannian distribution E2 minus E2 over KT 2 must be larger than E2 minus E3 over KT3 So basically all I need in this case is T2 larger than T3 It will always work Okay So I built my machines What else? Once you have them you start asking questions How good is a machine like this? How efficient it is? Now if you still remember your thermodynamics or in fact this is what I discussed in Feynman's example you see that you cannot transfer all the energy from the hot gas into lifting the weight you need to dissipate part of it to cool that little stick The interesting thing that I did not tell you at that moment is that by doing better engineering you cannot decrease as much as you want this transfer There is a finite amount of energy that needs to be dissipated Sorry That is the great result of Carnot So even the most efficient machine will need to waste some energy That is one of the basic laws of thermodynamics and everything that goes with it The fact that we are aging the fact that things are becoming more and more disorganized in the universe and so on follow from that But then you add Okay So suppose you have the best designed system and it reaches the Carnot efficiency Now it doesn't mean that every engine reaches that thing Of course you can do a bad design And in fact almost always you do a bad design So you are very far from reaching even that goal of the Carnot thing Now sometimes you may do a bad design simply because simply because you are silly and you don't know how to do a good design Well people that design it design refrigerators or machines are not silly but they are grappling with other problems the type of materials they have you cannot have everything done at perfection Here there is a third thing If I want the smallest machine I mean I put to myself some supplementary constraints I cannot design any machine I want I really want a small one So one question is Are these things efficient? A normal machine like the machine that Salikarno imagined that reaches the maximum efficiency It's a classical thing It goes to an infinite number of states If I only allow my machine to have four different states How well can I do it? It is conceivable that my machine will not work so well because and we are used in quantum mechanics there are all these complementarity relations For example you can measure position but not momentum and if you measure position better you lose your information about momentum and vice versa So there is always in many instances in quantum mechanics there is a trade-off So is here a trade-off that when I increase the number of states I can get with my efficiency better and better and when I decrease my number of states my efficiency will become worse and will hit a limit This is very natural to be so Very natural but the question What is it in reality? Well, what is it in reality? That's a good question First let me tell you what is this system that is the best system The best system you would imagine that you take something from the hot bath you take some quantity of heat you will do some work and you will dump into the cold one some quantity of heat QC and what it was discovered is that the ratio between the work and the energy that you take from the hot bath is not one but it is one minus the ratio of the temperature So that is the thing that you aim for So how well can we do? How much is it degraded? Does it degrade when the number of states becomes small? And the answer is no Surprising as it is even if you restrict the number of states the design can go towards the extreme You can do a machine that is as good as possible even if you are forced to do the simplest possible machine which already tells you a basic thing about nature that the simplest thermal machine can reach the maximum efficiency Now actually the example is so easy that I can give the proof in such a short talk and it goes like this You see the design constraint this is the only design constraint is that the energy difference between two of these levels E1 plus the energy difference here E3 equals E2 That is the basic constraint apart from the fact that here I have two state systems When is a system at maximum efficiency? Well, this was known already in classical thermodynamics You really have to drive things slowly You don't want to mess up with a thing If you have a piston you don't want to create shock waves if you move the piston too fast and everything has to run very smoothly And in fact in such a situation you are as likely to go one way as you are likely to go the other way Ideally when you hit Carnot limit in fact nothing gets done you are the most efficient when you do nothing but you need to wait That is the extreme procrastination So you have to make this machine to be so moving one way the other way which simply means that the population of being here and the other one down must be equal with the population and being this one up and that one down because those populations are the things that drive the machine So all it means is that these two functions E2-E3 over KT3 must be equal to E2-E2 over KT2 because they tell me about the populations of driving one way, driving the other way This expression I can simplify because all I need is the exponents to be the same So E3 over T3 must be equal to E2 over T2 That's all Now there is something else that you like This tells me what are the design requirements on the energy levels They don't tell me how much work I do That depends on how many transitions occur per unit of time That depends on everything else How well I couple the systems to their own environment How strong is the coupling in between them and so on But all I know, I know that when one system goes down this goes up and that goes up Things come in pieces So the ratio of heat and work is the same as the ratio of the energy separation because that is the value of one piece that is transferred at that time So the ratio between work Q2 and Q3 is the one, is the ratio of the energies and you do a little bit of algebra here just to divide things and you find out that actually your system can reach the Carnot efficiency Okay, given this, let me go back and ask what really is work I know now how I can do a machine I know that this machine is efficient so this machine can be used as a basis for defining what work is So let me look more carefully into it This is my machine that takes me the system up, up, up, up, up But let me just focus on two energy levels What would happen if I would only have two of them? Like this Well, what happens is that when my machine is running it just pushes this middle system higher up How much? Well, it will have to push it towards the upper level so it will produce inversion Now, if you think of that now it's something very simple Up to now, you were thinking that work is to put something organized But here is an example I take this and I lift it So originally it was at ground state and zero probability at the higher state Now it is at the upper state and not down I created a population inversion So here's the thing until now you knew about work in one of the basic things in physics You learn the very first year when you really start learning physics On the other hand, population inversion is some, you know, tiny effect People dealing with lasers deal with it but you don't really encounter in many other things And what we claim is that work and creation population inversion is one and the same thing Uniquely, what means doing work is doing a population inversion Okay, so in the end when you start building this you start building from some funny motivation and the signature that you get something interesting is that it allows you to ask questions that you would have never thought of them before in the direction that you never thought before So with all this machinery let me ask what do thermal machines actually do So here is my thermal machine This is the system and this is the machine The machine has two pieces, two qubits two level systems And let me look at the interaction So the interaction is like this It takes ground state, excited and ground two down, one up into the middle one going down the side one going up into EGE So let me look at the machine Let me look The machine has four states The two pieces of the machine could be ground They two could be excited or one excited one ground or one grounded one excited They are four states But in this interaction two of them actually come Now let me give them a name to these two combined states I'm now going to look not at the individual but at the combined states of the machine So the full state of the machine And I will call this as an excited state and this as a ground state Why is this excited on that ground? The energy of this is E2 The energy of this is E3 But E2 is larger than E3 So this is like a lower energy state I call ground an excited And the difference in energy between this state and that state is E2 minus E3 which is nothing else than E1 So what I have here I have one system that has two states ground and excited that interacts with another system with two states excited on ground at the same energy level I will call this system of the two states I will call them a virtual qubit So here is now what the machine is You are used with this machine that has two pieces in the hot bath a piece in the cold bath doing something here and there and within the machine transferring energy from one side to the other But what you see is simply when you simplify when you look at the core of it that the machine is just a two state system that is put in very simple direct thermal contact with the other You don't see this when the machine is apparently working and doing but it is actually very static it's just something static put in ordinary thermal equilibrium like you you put an object in a bath of water ordinary thermal equilibrium nothing else Now I know at what temperature this is this is at the temperature T1 this is my system what about this system it's a virtual qubit we live at some virtual temperature and it will have why does it live at a virtual temperature you can associate a temperature to any system of two levels simply by looking at the ratio of population on the excited state and the ground state and they would have a distribution you can always write the distribution as being e to the minus e1 which is the difference in energies and it is indeed e1 we discuss here because the difference between capital E and ground which is e to minus e3 is nothing else than e1 so the energy level here is e1 and then looking at the two distribution you find a T virtual and a little bit of algebra tells you immediately what T virtual is which is a funny number and I leave it to you to do the tiny algebra but if you look at this number you see something interesting this number although you calculated it by having two pieces the temperature T to 1 at T is not the average in fact it could be smaller than the smallest larger than the largest if it is smaller than the smallest and you put in contact with your system well the system will cool because you put it in contact with something that is colder like your ice cube if you put it in contact with something that is hotter well that is a heat pump something that takes a system and just increases the temperature by using two baths of much smaller temperature and what would you like to be the temperature TV if you would like to make here a population inversion well you would like TV to be negative ok so understanding and this formula simply has everything from minus infinity to plus infinity depending on exactly what temperature you put and how you couple the two different energy so let me go now to the final thing what really is going on I had here my system which is in contact with its own environment a big bath at T1 I have the middle piece which is in contact with its own environment at T2 I have here the third piece or the second piece of the machine in contact with T3 now let me look at this two thermal baths the thermal baths are this beautiful Boltzmannian distribution E to the minus energy over KT which is an absolutely beautiful state beautiful but it's boring it's boring the same ratio between energy levels but the machine is one piece of the machine that interacts with this bath another piece of the machine that interacts with that bath but we just learned I have to look at the machine as a single object it's a composed object but it's one single system then let me look at the two baths as a composed thing the baths are not interacting but they are you know it's like two quantum systems that are not interacting but I could define the energy levels with these things and then you find out that although each of them separately was Boltzmannian when you look at the energy level of the common system it's not Boltzmannian at all there are different energy levels of virtual levels which are levels of the two baths at different temperatures and your machine just interacts with one of these virtual things by design so what you saw that the machine puts one hand into one bath another hand into another bath it does it by designing such a way that it actually interacts just with some of the two level systems in the bath that are exactly at the temperature that you want and then they are in ordinary thermal equilibrium with that the machine is in ordinary thermal equilibrium with your system which is in the ordinary thermal equilibrium well, thermal contact I would say not equilibrium with your environment so nothing else happens when you simplify to this level everything is ordinary thermal contact like putting bodies of two temperatures in fact the external temperature is T-virtually given by the thermal bath with which the machine is in contact and the design of the machine and the environment of your system these two are your two temperatures if T-virtual here is smaller than T-environment then because of this thermal contact will lower that state somewhere in between T-environment and T-virtual if this is hotter it will warm that up if this is negative will produce here inversion when is the machine the most efficient ah this is interesting the machine is the most efficient when T-virtual equals T-environment otherwise energy will be dissipated if you put in contact two things at two different temperatures so but wait a minute if I would have a machine that has more levels here they would be in contact with different other things from the bath which would have different temperatures but my system is in contact just with this same thing so if I have a machine with many levels that cannot be very efficient that will never reach Carnot so originally we thought that the Carnot thing works only when I have an infinite number of states and if I'm limited in states I will not be able now I realize that in fact if I have the machine too many states I cannot reach Carnot and it is only the smallest machine that can do it so how does a macroscopic one that does go towards the Carnot limit is because basically it decouples most of its states and at the end of the day only acts as a two level engine you know the initial part of the piston the final part of the range of the piston and so on so I will conclude here but what is interesting in my view is that this gives a completely completely different approach to thermodynamics you can now ask all kind of things what is the structure of this combined bath sometimes I can make a refrigerator by using work like you know that is what all the refrigerators do but if I use work I'm depleting levels from this combined bath that are at negative negative and virtual temperature but if I only want to use it as a refrigerator it would be enough to get something that is positive but small depleting other things from the bath it is very interesting I'm not as efficient as I want I destroy the bath too much if I convert it into work and then use the work after the refrigerator there are all kind of other things that immediately emerge from this but ok this was just an introduction to the subject so thank you very much thank you very much questions so I start with the questions so in principle you can also consider this few level system in which transition are classical rates quantum mechanics is not required so in that case can you have a rate model such that you get the same efficiency ok well so the question was what about the rate model when I really want to look at all the details how the transition takes place well so let me tell you what I did not show here just to be clear I did not do this analysis because to see the time evolution that depends on many many other factors that depends on how strong the interaction is between your system and the environment or between the machine and the environment and you know that if you for example decouple the spiral at the back of the refrigerator from the room say you put a polystyrene piece of shielding you better run out from that because it will explode over hits very many things and that's a complicated thing what is interesting however is that when you try to ask that some of the questions that you may ask for example what is the maximum efficiency they are independent of the rate so you may have something that goes faster or smaller but what is the maximum that is independent of this so you may have a classical few-state system few-state engine or a quantum I may have a classical one but what I see and I don't have time to explain this here when I try to design it to go towards the Carnot limit I will see that all the intermediate states decouple basically somehow thermodynamics already knew about some of the things of quantum mechanics even if it was not obvious ok questions going back to the beginning so at the end in biology what are the smallest refrigerators ok yes obviously I was waiting for that question what happens in biology yes we went back to the biology and we have a tiny model which is not quantum in which indeed an enzyme can pull part of itself that's a funny thing if I may take a moment to say you see the way in which all biological processes are based on catalysis without the fact that the procedure that we call catalysis that is what makes life possible so imagine that you have oh this is really a wonderful model so imagine you have a molecule here some molecule and that is in interaction with other things surrounding it that may come and may hit it and you hope to arrange for it to break ok the bounds in the molecule are very strong so there is some rate but it's not very easy to break it on the other hand imagine that this little this molecule lands on say a piece of metal here that is like your catalytic converter now what happens is that there are all kind of atoms here and the atoms in the molecule they have charges, positive and negative and even when it lands there are some dipole-dipole interactions here or something more so that means that the simple fact that the molecule just sits now on something it is in fact tensed a little bit due to the interaction it's not a chemical reaction it's just an electrostatic attraction that puts a little bit tension and deforms a little bit this thing and then when it gets hit by other things well because it's not in its best state it breaks easier and like this the rate of say breaking of a molecule is enhanced when it goes and attaches to something else that is catalysis basically and that happens in all your cars and so on but if you look here this molecule has some funny shape and it sits like a potato on the table and it just touches it in a few points now biology is very interesting biology has some very big molecules some very big molecules these are enzymes in a certain place they have they have a shape like this that is very similar to the thing that you want all this is a very naive way of telling you what catalysis is and there are so many interesting things so sometimes the molecule may come in here exactly like a key in a lock but now there are more boxes in between this so it is more tensed and now when some external thing come and hit it it breaks easier and there are orders of magnitude of enhancing the rate of reaction from here to here and many more from here to there everything works okay and the important thing here is the fact that the shape of this which is called an active site of the molecule really matches the shape of the molecule that you want okay but imagine now that you that there is water around whatever at some temperature when things start at a given temperature they oscillate a little bit so let me take this out from here so just see here this thing oscillates a little bit if the temperature is higher it oscillates more so let me look at the rate of reaction so this is temperature this is the rate of reaction the rate of the reaction increases why because I have my molecule here that would like to enter and if everything is colder everything moves slow so it takes a long time for this to enter a long time for this to go if you increase the temperature it moves faster it goes in it goes out it goes in it goes out the reaction increases but at a certain temperature it plummets why? because this thing now moves so much and the active size gets so deformed that it simply cannot enter and even if it enters it doesn't fit well so this is one of the reasons why things are so sensitive in temperature if your temperature goes 43-44-45 degrees that's it so it would be very useful if somehow the enzyme could cool this active size so that it does not get deformed and the question is can it do it well that needs to have something active like a refrigerator you really need either to pump work into it or to put in connection with too much here I allowed to have work so how can I give energy to this molecule well there are enzymes that also have another active site here in which you can plug a battery they are not such big active sites the battery is you know there are some universal things like ATP there are molecules that can give energy in two places so you may have something in which you plug something energetic and that for example may produce a big transformation at this active center now in general when you squeeze it then the the vibration become faster so you may have a situation where if I just model this thing like a harmonic oscillator just say the vibration of this something like that and you have some probability being the lower state another and another then when you close you may have this whole thing steeper so you may excite a lot and so on but then at the difference of the energy level is bigger so as you increase it you will raise its temperature but because the temperature of the environment is the same T in the end it will if it stays like that for some time may reach the temperature of the environment which now means many more molecules here because the difference to the higher states is smaller and now comes the thing but this is not good for me although and this is the normal distribution again for that temperature but now if I open a diabetically slowly I would each this thing the original one but now with much bigger population so there is a mechanism to cool a molecule does it exist in nature I have no idea this is a theoretical proposal give it to a biologist will tell ah this is biology free thing show me that it exists well I don't have a lab so okay but but here is what a sort of thing that you could play with so that is a possibility there are other effects in biology which are not related to this but I told you something coming from thermodynamics there are many things known for photosynthesis photosynthesis there you have there are experiments that show quantum coherence at a scale much much larger than you would have ever imagined so that that is a real thing that has been known already for maybe ten years thanks for the interesting talk I just have a question regarding one of the motivations at the beginning on non-trivial quantum effects because from what I understood after the talk those procedures you described can all be taken into account with classical rate equations of probabilities of what state the system is in and also the baths are classical heat baths and the qubits are that I can understand them as classical bits with just two states with probabilities so I'm wondering whether there are any non-trivial quantum effects involved and how they might affect our discussion of like using classical thermodynamic theory okay so so indeed my talk was very much with a classical flavor the reality is very far from being classical so one of the things is that the basic idea was that you may have discrete states which classical you don't have if you try to do a classical a real classical model not a simulation of a system with discrete states then it will not work so already quantum mechanics is from the very beginning embedded in the fact that you have systems with discrete states this is a highly non-trivial thing this is in fact the whole basis of quantization now that is one thing another thing relates there are many interesting purely quantum effects related here one of them well I've been asked previously can I look at the rate and I told you that if I put for example a piece of polystyrene in the back of my refrigerator at home is not a good idea okay because I need to let it cool to the roll so you would think that the best thing classical to do is to try to in fact dissipate easier to put for example you could put it in water or to have more humidity to carry the energy from it and the more you couple them the better will work quantum mechanics doesn't go like that you run into a zeno effect you couple it to the environment and it stops working so there are many many purely quantum effects that the moment you start looking more carefully here you will see them working other quantum effects well I have these three pieces my three systems do they get entangled the question is not very simple because of course they get entangled but they will get entangled also with environments around them and if you just look at my system and the two pieces in the machine they might very well be non entangled well it turns out that something it's interesting here whenever you go towards Carnot efficiency you become non entangled you cannot maintain entanglement and there is an entire space of states that are surrounding the Carnot point that are not entangled they really must be totally disconnected at the Carnot point on the other hand if you want to actually have a machine working not one that is just doing nothing and you want to increase the rate this is one question you know increase the power of the machine at a certain moment this must get entangled there are very many things that you can discover on the way there is a question up there yes please thank you for the enlightenment professor please go back to the biological molecule the enzymes substrate interaction I want to know how how is the activation energy barrier affected how is the virtual temperature TV affected by the activation energy barrier is there any relationship here whether there is a relationship yes for sure whether I know it depends on the level of detail so clearly the more you increase the temperature the more this thing gets deformed ok now what is the relation between how deformed it is and what is the rate of reaction well you would expect to be something like tunneling so you would expect things to decrease very fast to decrease exponentially fast when you are outside of the range ok the situation is complicated I don't want to enter the detail this was an extreme I think if I would tell this to a real biologist will laugh me out of the room first of all this thing doesn't really need to match the exact form it would be much better if it would match the desired form after breaking it would lower its energy that this is just a cartoon but you imagine that this molecule here has some positive charges in some location some negative charges in other locations and so on which will want to match with some pattern with some pattern here so some parts of this are more sensitive parts of this are less sensitive it is a very complicated system but one needs to start somewhere generally the fact that the rate of reaction has this form that is a standard thing last question and then we have the diploma hi can you give us some insight of what happens when these two pieces of the environment get entangled and not with the environment what happens then look I think that what is known at present is the fact that they must get entangled one can prove that if they do not get entangled you cannot achieve a rate higher than something what is the the deep reason for that I don't think it is clear at the moment or at least it was not yet absolutely clear some years ago ok on the other hand one can prove that at Carnot they are definitely they must be unentangled which again is something very interesting entanglement prevents them to reach the Carnot the Carnot efficiency I think that much more work in this area is needed to say so ok let me try to zoom out of this question whenever I don't know an exact answer I try at least to to give a general thing there has been a lot of work in quantum thermodynamics lately but that work involves some other questions they involve the second law and things like this not the working of the machines themselves so and that work is extremely interesting and gives you very interesting connection with information and so on but I think that one needs to couple that with actually understanding the interaction part of the thing so those two areas of investigation were not pursued with equal intensity this one at the moment is more on a back burner although I believe that it is essential and to couple that with the advances in the information part of quantum thermodynamics is essential so that is more or less where the field is okay so let's thank professor Popesco thank you sorry no thank you for a very nice talk I had a question related to what you just said I mean the Carnot efficiency normally did you formulate in terms of entropy and reversible processes where the entropy doesn't change but which is valid only if you have it's like a statistical odd system so I wanted to ask this what is the connection with the second law of thermodynamics okay you also mentioned like fine grain entropy it's quite different from the coarse grain entropy yes and that's related to entanglement I understand you can comment on that indeed where should I stand so that you will see me well the idea is first of all you saw one example the final example is how the entropy is conserved and let me get here so the basic idea was that I have my system here this is my system and here is the thermal machine and I view the thermal machine as the virtual qubit just the two states that play that interact with my system my system was in interaction with the thermal bath but the thermal bath had many many levels and since it is in interaction with just the thermal bath you know it will basically resonantly interact with two levels here and they are at the temperature of the environment on the other hand if I look at the composite bath which is simply you know the direct product of some states of bath of the hot bath and some other states of the cold bath so there are all kind of levels but all these levels are at various various populations and some of them are at the same the same energy difference and couple resonantly other are not and the arrangement arrange that what the thermal machine did to couple with something at a virtual temperature now once you did this all you need is to look at these four things one system, another another, another so what does the Carnot thing do in the end you want to couple this temperature with a T environment here you forget about everything else forget about everything else and you only have these four things and as long as they are not at the same temperature Te with a virtual temperature that is generated then you will lose entropy ok now what kind of entropy if you want to go more into details because you are right now people are talking about many different types of entropy at least well at least as far as the information is is concerned there are many second laws and so on so there are all kind of other details this particular model of interaction did not look into absolute detail of how this one gets in fact entangled with many many other many other levels here and there are events that try to to uniformize this bath and so on so to be able to answer your question one would need to look in more detail in the actual structure of the bath the structure of the bath itself that is the key element when you want to look at thermodynamics whenever we say it is very simple are just two baths two different temperatures that is not enough in order to be able to answer this it depends very much on the level of degeneracies associated with each of these energies in the bath and so on so as talking about open questions I believe that is one of the very fruitful ways of continuing to ask questions that is all I can tell at this moment okay so now we can thank Professor Bobetsko again thank you thank you thank you