 Tako, da bomo pošličiti svoje organizacije, za to, da je časno. Zdaj bomo pošličiti kvantum termodynamikov, in včasno pošličiti obtima, pošličiti termodynamikov. To je vse resulte, ki je obtajeno, v kovalenju v Vasco Cavina in Vittorio Giovannetti z Skola Normale in Pisa, in Alberto Carlini z Piemonte Orientale. Tuk je obtajeno v dve vrvnih vseh, ki se pošličiti kvantum termodynamikov, vseh je odtimil, vseh je, ko je vseh je, kako je vseh je vseh je vseh, tko je to pošličiti, in vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, vseh je, kratičnih sistemov. Ne musite predstaviti spesivnih vsočenju na termalne vse. Zato smo zelo vsočenje kratičnih sistemov, kaj je vse predstavila vamarkovi in master equation. Zato imaš staj, vse imaš differential equation, kaj je vse predstavila vseh ljubiljanj superoperator. Zato imaš vseh občast kratičnih sistemov. And then a very important property of a master equation is the concept of a fixed point. So this is a state, which is invariant under the action of the Ljuvilje. So it is a right eigenvector with zero eigenvalue. And if this is unique, and then if all other eigenvalues have negative real parts, then you exponentially, every solution, every initial state converges exponentially to the fixed point, which is row zero, when time passes. This is a standard property of master equations. And now assume that we can slowly change a parameter of this master equation very slowly, much slower than the characteristic relaxation time of the system. And then what you expect is that the fixed point will change in time, so the gray line will change, and all the solutions will follow adiabatically this fixed point. This is, in some sense, the equivalent of a quasi-static transformation in thermodynamics. But this is a more general setting. And then the question is, what happens if I do exactly the same protocol, so exactly the same shape of the modulation, but faster and faster. At some point, the system will not be able to relax to the fixed point instantaneously, and so there will be a little bit of difference between the true solution and the equilibrium solution. And then what we would like to do is to develop a perturbation theory to compute this difference. The way to define this perturbation theory is to rescale the time variable, instead of going from zero to tau, which is the length of the driving of the system, we go from zero to one with this dimensionless time variable. And then in this new variable the length of the protocol appears only as a multiplicative factor in front of the master equation. And we can now expand as a perturbation series in one over tau, which is the length of the protocol, the solution. And what we expect is that in the limit of tau that goes to infinity, we recover the fixed point, the instantaneous fixed point of the dynamics, and for a finite tau we have all other non-equilibrium corrections. And if you do replace this in the master equation, you can compute explicitly all these corrections in a perturbative way. And so now, for example, this is an example, we have a qubit in a thermal bath, and we change the Hamiltonian along sigma x in our sinusoidal way, just a random example. You have the gray line, which is the fixed point, the equilibrium solution, real solution is the black one, and then you have the first order and second order analytical approximation that we could compute using this perturbation theory approach. So, up to now there was no thermodynamics, thermodynamics enters into the game when you impose that the fixed point is a Gibbs state associated with the instantaneous Hamiltonian of the system. In you do this, the expression is the same, just use the Gibbs state here for all these non-equilibrium corrections. And then you can ask what is the entropy, internal energy, heat, work using standard definitions that you have in open quantum systems. And if you just replace this perturbation series here, you get basically all the quantities that you can obtain at equilibrium when you do a process infinitely slowly, or all the non-equilibrium corrections automatically from this expression here. And for example, these are the first order corrections to the energy, entropy, heat, work, and I don't go into the details, just that we have basically the analytical formulas. And then what we are going to do is to apply this theory to a Carnot cycle. What is a quantum Carnot cycle in this setting? You can assume that you couple the system with a hot bath and you slowly change the Hamiltonian according to an arbitrary modulation. Then you apply a quench, so you multiply all the energy levels by a given constant. And then you apply the time reversal isothermal process. For example, if you compress a piston, you do the opposite expansion. And then you do another quench. This is usually the definition of a Carnot cycle and you can compute with the previous expressions or the power, the efficiency and so on. So for example, the power is for a cyclic process is related to the heat, which is absorbed by the system and you divide by the time that you are coupled with the hot and with the cold bath. So you have this expression where here we trankate the previous perturbation series to first order in one over tau. And so we have this for the power, this for the efficiency. Of course, if you go to an infinite length of the process, you recover the Carnot efficiency and the power goes to zero as usual in equilibrium thermodynamics. But if you instead optimize the power, you get an expression for the efficiency at maximum power. So this expression actually was known in the literature. But what we can do now is you can just replace our analytical expressions with quantities, these are non-equilibrium heat exchange and we can go further in this approximation, in this calculation. So this is the ugly expression for the non-equilibrium dissipation and unfortunately it depends on the particular protocol. But we are lucky that the fact that the ratio between the hot and the cold dissipation obejsts this universal scaling law where t-cold and t-hot are the temperatures and alpha is the exponent of the spectral density. This is due to some symmetry property of the cycle and of the master equation. To do this, we get our final first result which is the efficiency at maximum power for a generic spectral density and the nice property of this formula is that you recover all the many previous results that were obtained in the literature for different values of the spectral density like for a flat butt, the most important case is the flat butt in which we recover the Kurtzson-Albor efficiency. And this is a plot of the efficiency at maximum power for different values of alpha and these are numerical simulations that follow quite well the analytical expressions for low efficiency and low power while when you go in the regime of high power, high efficiency, which is far from equilibrium, the perturbation theory doesn't work anymore and so at some point this curve cannot be trust anymore in this region here. And that's the point of the second part of the talk where here we were close to equilibrium and at some point we have to stop but in the second part we just brute force optimise the power and we don't care if you are close or far from equilibrium and we do this using optimal control theory basically. So the general question is I have a d-level quantum system and I have two heat-bats, a cold-bats and a hot-bats and I can do whatever I want. I can change the Hamiltonian, I can couple with the hot with the cold-bats, with both if I want and I have only some limit on the damping rate otherwise the power would be infinite. Then the question is what is the optimal cycle, the optimal strategy to extract the maximum amount of power. So to formalise the problem, we use a master equation approach in which we have the Hamiltonian which now depends on an external set of parameters. This is a vector u of t of controls like I can change the frequency of a laser and so on. And also I assume that I can control the damping constant with the cold and with the hot-bats and then again I can define the heat and the work as before and the optimal control problem is the following. I want to minimise heat dissipation with respect to all possible strategies for fixed initial and final states and fixed length of the protocol. Because if you have a cycle then the work is equal to minus q and you want to maximise the power. I think it could be equivalent but in the end that's maybe simpler to do the calculation but yes, because I would say yes because I want to optimise the power. I don't care if the entropy production what is the entropy production, I just want maximum work and then I want maximum work divided by tau so that's the most direct way to do it I agree that you can change the figure of merit. System can be in every state. Yes, I agree, so you have these problems where there is no universal agreement on what is the definition but these problems are usually when the system has coherence. If you have diagonal terms then you have these extra terms to the work that are... In the end what I'm going to apply is a situation in which you have diagonal states so everything is well defined. Even if, of course, if you have coherences you should be careful about what is the interpretation of this quantity basically. Ok, so how can we do this optimisation? We want to minimise q and then we use a Lagrange multiplier approach because we minimise q to all possible trajectories that are normalised and obey the master equation and the way to do this is to use Lagrange multipliers which is a number for the normalisation and actually an operator to impose the master equation. It's a standard approach to do Lagrange optimisation in control theory and the feature of this Lagrange approach is that as in classical mechanics you can define the analogue of the Hamiltonian also in this case you have the pseudo Hamiltonian which is just nothing to do with the Hamiltonian of the system it's just a mathematical object that is very useful to find optimal solutions and as in classical mechanics you have the so-called... the analogue of the Hamiltonian equations gives the dynamics of rho which is just the master equation this is something that we know but then we have this non-trivial solution for this auxiliary state which is something like a density matrix that evolves according to a different master equation this is like position and momentum if you want and again as in classical mechanics the Hamiltonian is conserved that's a very useful property to find solutions so what is exactly the Pontriagis minimum principle it's a statement about optimal solutions that should satisfy three properties so there are necessary conditions all optimal solutions are such that there exist this auxiliary state that evolves according to this equation just the Hamiltonian equation the pseudo Hamiltonian is minimized by the control field at every time and the Hamiltonian is constant as I said before so these three properties in principle are not enough in general to completely specify the optimal solution but they are very useful and if you are lucky they are also enough to find it and so now one could ask ok we have this nice Hamiltonian which in general in control theory has no physical meaning maybe in this case it could be related with some thermodynamic quantity and this is the case and it is strongly related to the maximum power of the cycle because if we take the power which is just minus the heat divided by tau we take the variation of this power we get a term which is proportional to the variation of the heat and a term proportional to the variation of tau then we use our Hamiltonian which is linked to this Lagrange optimization and what we find is that optimal solution should have p equal to minus k so when you are at maximum power then the optimal solution has a power which is just minus k this is this conserved quantity of the control problem so this tells us that we have a clear optimization procedure we just take the set of all possible values of this conserved quantity that are consistent with the master equation and the Hamiltonian equations we check if the power is equal to minus k and if this is the case then we know that this is the optimal power solution otherwise we should try a larger value of k but as I am going to show for your qubit the minimum k is already a good solution but before going to the qubit we may ask what is the general cycle for a d level quantum system in general what is the kind of transformation that is optimal and if you first optimize the damping rates you find that it turns out that the best strategy is a bang-bang type control you couple a maximum rate with the hot bath or you couple a maximum rate with the cold bath there are no intermediate situations plus what is the optimal control for the Hamiltonian it is a regular driving along isothermal transformations but you can have also here some discontinuity some bang-bang effects and these are exactly the equivalent of adiabatic transformation in a Carnot engine so we know that finite time Carnot cycles are optimal solution and we have to optimize over these cycles so this is the general theory now I apply this to the qubit I don't know maybe I have 5 minutes ok then let's apply this to the qubit for the qubit we assume that we can couple the system with two baths of cold and hot temperature that just tends to thermalize the system to the Gibbs state this is a thermalizing dissipator and moreover we assume that we can modulate externally the gap of a two level system by simply rising or lowering the energy level of the excited state the ground state is at zero energy so as I said here we consider diagonal qubit we have also the diagonal auxiliary system in terms of a single variable, q we have this pseudo Hamiltonian that we can compute and we have these pseudo Hamilton equations that just give the master equation for the state and the kind of master equation for the auxiliary state then we have just to solve these equations it's not a physical state actually it's a zero trace object and these are the solutions that we get that's the optimal isothermal, cold isothermal process optimal hot isothermal process for a given value of kappa in the UP plane, what does it mean this is the probability of excitation that's the state and this is the energy so by definition you can compute the heat as the integral as this area below the curve and this is the heat dissipated in the hot isothermal in the cold isothermal, this is the heat absorbed in the hot isothermal but as I said before you can switch between one isothermal process to the other with a quench and now you can do a Carnot cycle basically and actually our control theory approach tells us exactly what are the optimal points where we should apply the quench so not every Carnot cycle is optimal only one Carnot cycle is optimal for a given value of kappa so we have a one to one correspondence between kappa and Carnot cycles Carnot cycle with the minimum value of kappa and we hope that this is the maximum power heat engine what happens when you go to the minimum value is that this Carnot cycle basically collapses to a very small Carnot cycle where the isothermal process is basically a constant Hamiltonian process and then you can say that this is an infinitesimal Otto cycle and this is exactly the optimal control result you have to go to one particular state with a given value of optimal p star for a given temperature ratio here and then around this state, this quantum state you switch the energy level between two particular values of u and you do this exactly as in a standard Otto cycle I change the energy level I thermalize a little bit I change the energy level I thermalize in the opposite path and if I do this I obtain exactly the maximum power of the heat cycle and for which we have this curve and we also have the optimal asymptotic the asymptotic expansion in the limit of high power and also an upper bound and it will never be violated and we can also compute the efficiency at maximum power which is the blue line here and the green line is the Kurtzson-Alborn efficiency we are very close and this is the full solution for the qubit so to summarize I have presented two different approaches to study optimal driving of thermal machines the first approach is based on a perturbation theory which is good when you have a master equation in which you slowly change some parameters of the master equation and in this case you get many universal results but they are not good at maximum power actually they are not good at far from equilibrium and the second part instead we studied optimal driving of a thermal machine using a control theory approach and I've shown you that optimal cycles are finite time carno cycles that the maximum power is linked to this conserved quantity of the control problem, Kappa and using this formalism we could derive the full solution for a two level system close or out of equilibrium Thank you very much