 Lunch break and welcome to this afternoon's session. We have three talks two in presence and one online and We are happy to welcome Liliana. We give a first talk Hello, yes, you can hear me. So thank you very much for coming and I would like to thank especially to the organizers who persisted in organizing this a nice activity in person and Who were really very helpful in finding all kinds of last-moment solutions? To enable us to to come and to be here and I'm going to talk about the implementation of a thermal machine operations in QE systems and the formal Description of that on how to optimize how to describe the the performance and a way to optimize it So first problem Well something yes and This is a work with has been done in collaboration with all these colleagues a Pablo abuso from Barcelona Vivek Vandari who was here in Trieste when we started the work and now he moved to Rochester Pablo terreno Alonso from Buenos Aires PhD student in my group Sarofatcio from here from Trieste Felix from from open from Berlin Martí Perornolovic from Geneva and Fabio Tadej from Pisa and This study is partially motivated by a some nice recent experiments on thermal transport in qubits. This is an example where a the thermal Conductance in a cubit place between two reservoirs at the different temperatures these two reservoirs are made up of By the surrounding circuit in different configurations with and without resonators which show very impressive features of coherent transport through the qubit and also by this other type of configurations where qubits are manipulated in order to exchange energy and In this case to implement a Maxwell-Demond operations The our goal is to implement thermal machine operations here and as I said before to Develop a formalism in order to have control also on the performance what come to our mind when I Mentioned thermal machines are these type of cycles that we Study in high school But I'm not going to speak about exactly this kind of cycles which operate quasi statically so They basically operate at in the regime of zero power But on these other type of operations where there is some mechanism which is a a permanently Operating in order to overcome in this case the Gravity the influence of the gravity but in our case it will be some similar operation in which we inject work in order to a To a Refrigerate or to extract a heat from a cold reservoir against a thermal bias or The other the reverse operation in which the heat Flowing from a hot to a cold reservoir can be used in order to generate work in a similar way as these Archimedes a pump operating in their in reverse And To Present these ideas. I find it useful to first of all introduce the concept of pumping and quantum pumping Which was a very popular a subject? Already 20 years ago in the context of electron systems. This is a Very well studied device where we have an electron quantum dot which is a Affected by These two a voltage gates these two voltage gates control the Degree of contact with the neighboring reservoirs in a way that for instance this gate opens Electrons come inside this gate close while this other one opens. So the electrons in inside a flow towards the The right lead. So this is a Possible protocol for this operation with alternating voltages with a face lag and the net result after a cycle is a net a Charge flow from the left reservoir to the right reservoir, which is measured here So remember these plots this is basically a measurement of the electron Charge from the left to the right in a cycle as a a function of the face a difference Imposed in the to a gate voltages The question is if we can achieve a similar operation, but with heat Instead of charge and with a qubit instead of a quantum dot and The well the answer is yes, and I'll show you how this works in a while and we can think up in The qubit in terms of these qubits implemented in these circuits in these superconducting circuits in which We represent the the surrounding circuit as a gas of Many modes In our case because they will play the role of a reservoir we have this Hamiltonian for the qubit and These parameters this qubit the the parameters entering in the qubit Hamiltonian can be operated in time and This is the coupling between the qubit Sorry the qubit and the the circuit or the reservoir Which in which enters here some Pauli matrix so this toe is some Pauli matrix that can be x or set Or maybe y but I will consider x or set and this type of coupling can be a Architecture by the design of the circuit So now consider that the qubit is coupled to two of these surrounding circuits in principle for the for the moment at the same temperature and Imagine that we implement the coupling with a different Pauli matrices To the left and to the right so in in our case we choose a sigma x in the coupling to the left and sigma set in the coupling to the right and now let's a perform this type of cycle so implementing this this kind of a of a different coupling means that if we We prepare the state of the qubit aligned in the x direction It couples to the left and if it is in the set direction It couples to the right and then we can implement this cycle when we change So this is a as a function of time the cycle in the bx and b set a plane and Let's assume that at a given time we couple to the left reservoir and a the qubit a allows the reception so there is some exchange a Of energy which is a Received by the qubit from the reservoir now we make the qubit evolve Introducing also a set component component and increasing the modulus increasing the level separation here in this way in this protocol, we will increase the Amount of energy stored in the qubit and at some point we couple only to the a right Connecting with this sigma set when it is perfectly aligned in the set a direction So the energy is delivered from the qubit to the right reservoir at the end of the day as in the case in a similar way as in the case of a charge we will have After a cycle and net a pumping of heat from the left reservoir to the right reservoir Of course during this process we are implementing time-dependent a protocols So we are also generating some dissipation into the system a So this is pure pumping, but we can also make a similar game Introducing a temperature bias between the two reservoirs and in this case We will have something very similar to this archaic midis pump operation because the pumping will enable to Working as a refrigeration refrigerator, sorry The the pumping will enable to Inject energy from the cold to the a hot reservoir. So this is a more or less the picture we have in mind and Now I'm going to Tell you which are the Theoretical tools we develop to study this in more a precise way So the theoretical challenges are the following we are we are treating a problem Which is all the time out of equilibrium is So there is no quasi static evolution at any time of the cycle and We have to take properly taking into account the effect simultaneous effect of the temperature bias and the driving and Our aim is to a provide a proper description of the dissipative effects as well as the heat work conversion mechanisms, which are the the ones which Are the essence of the thermal machine operation Well, this was a proposed in this work already published just before the original date of this conference and and The the geometric idea comes because in the description of the pumping and the heat work conversion Mechanism Concepts like the very face are really very Useful and important This is the the general setup So in our case, I'm going to tell you the case of the qubit here Place between two reservoirs one hot one cold the these two have a different Temperatures and It will be operated by So that the theory is formulated by N a driving parameters and I'll focus on the case where these parameters change in time very slowly because of that. I'll Mention I'll talk about adiabatic Driving and also in I'll focus on the case of Small bias temperature bias. So this delta T is on also small parameter So this is the operational regime I'll focus small velocities for the change of the time dependent parameters small temperature bias, but I in principle for this general Formalism we cannot make any assumption about the degree of coupling between the the driven system and the and the reservoirs so the energy balance Is as follows we have some a heat flux into the left reservoir some heat flux into the right reservoirs the addition of these two will be equal to the Total a power delivered by all the external sources But I'm going to distinguish from this Heat currents some component which is Which is exactly the same but with different sign In the two reservoirs. So this is the the current that goes from one reservoir and is injected into the The other one on top of that. There is a dissipative a component of the heat current and We can distinguish a two operational And two operational modes for this system for this device one in which as a consequence of the driving As a consequence of the driving we of course have a dissipation But we can use a part of the heat transported from the hot reservoir into the cold one to generate some Useful work. This is the heat engine operation and the reverse operation is the refrigerator in which we inject a some Power and we extract a heat from the cold reservoir and inject it on the hot reservoir in addition to the this Dispensive flow which should be smaller than the other effects in order to have something to have the this operation in in reality From the formal point of view the strategy is to implement a linear response Treatment in the velocities and in the in the Temperature bias and To analyze the geometric properties in the Space of the parameters of the driving parameters. I'm not going to provide the details So these are the two References this in this reference we implemented this adiabatic linear response and we in the work in this work we combined with the Latviger theory in order to also include the thermal bias as a Velocity in this formalism and in this way we could treat these two objects together but what I'm What is important? I would like to tell you is that the In the these calculations for the heat Fluxes and for the powers using these These expansions the coefficients that appear the transport coefficients that appear in these expansions can be collected in this Object that has the structure of the of a tensor and we Call it the thermal geometric tensor as it is interesting that the if we Focus on a zero a temperature and we eliminate the reservoirs this this tensor contains only the this Asymmetric component at this asymmetric component is exactly the one that Wolfgang Belsig presented early today, but in our case we will have this also this Symmetric component and it is precisely the symmetric component which is associated to the dissipation While the other one is associated to the use useful work The structure Given these Expansions for the network over a cycle and the transported component of the heat is the following We have a first term which accounts for the Dispation here enters the symmetric a part of the of the tensor and is bilinear in these velocities while we have this additional a term which is proportional which can be expressed in terms of a line Integral and is proportional to the temperature bias and This process is precisely the interesting one the one that describes the heat work conversion As for the charge and we have this first component this first component here is the same line integral as here with different sign and Describes the They pumped a heat so you see that this This component of the transported heat is not proportional to delta T So this is something that exists even if we don't have any Temperature difference while this other one is the usual thermal conductance and is the purpose is proportional to Delta T in the heat engine Operations of these expressions are valid for both heat engine and refrigerator operation in the heat engine what we want is to extract useful work and the way to achieve that is to Gain with this term is this term gains against dissipation. So we need here some negative contribution so negative in this sign conversion means that the work is extracted from the system and This is a cheap Precisely when we have pumping from the cold to the from the sorry the hot to the cold Reservoir the other operation Implies that we need to extract so to refrigerate we need to extract heat from the Cold reservoir. This is done Thanks to this pumping component which is negative here and then here in the heat war conversion Term appears as a positive Contribution so I have to inject some extra work in order to get this This operation So interestingly this part so this this part in green that is the one identified as the heat work conversion term Has formally some similar structure as the one we get in Carnot cycles but now We have not only this component, but also a dissipative a one and this is Evaluated over The whole cycle performing a line Integral This is a as I mentioned before associated with the Antisymmetric part of the thermal a geometric tensor and with all these components of the turn the tensor we can build up at this very a field and Explicitly express this component as a very face The same something very similar happens in the case of the of the charge Pumping in the case of charge pumping. You can also express the pump charge as this Something that has the form of a of a very face. So This is just to highlight that the pumping of heat is somehow similar In The adiabatic regime as a pumping of charge which has been widely investigated Well In passing I mentioned that there are also other Interesting pumping a mechanism which are nowadays studied in the literature and these are related with the others components of the thermal geometric tensor if they have some a Antisymmetric component and this is pumping a between yes between the different driving forces not between Different reservoirs and so I'll show you the results for the examples of the driven qubit the way to solve it was by Recourse assuming a weak coupling between the qubit and the and the reservoirs I'm writing here the The references, but I'm not going to present a Technical details is basically to work with the suitably defined master equations Which in which we have some a frozen component the frozen component means that we freeze the Hamiltonian at a given time and Then we add some adiabatic correction. The adiabatic correction is precisely proportional to the velocities Associated to the rate of change of the driving Parameters From there we can compute the currents and we can compute in particular the the heat the pump heat So I said when I showed the charge pump I I told you remember this picture and you see that we have here the same picture for the For the heat so this is the pump heat is Shown here in a black and in red. We are showing the dissipation the way to calculate this this is by solving these equations Defining properly defining this vector field and Given these protocols these different protocols with different phase lags we calculated the average over the a Period for the heat current so I told you the How to get this a pumping and this pumping guarantees the operation of the thermal machine And I presented a geometric framework to describe that The point is that we can take advantage of this a geometric framework Also to control and to optimize the the performance and to optimize the performance We need to control the dissipation and the nice thing is that dissipation can also be described by a Geometric concept so I the dissipation was bilinear in the velocities and what appears here is the a symmetric component of the of the tensor and This integral has the structure of so the symmetric component has the structure of a metric in some strange space defined by the parameters and this Quantity can be related to the length in this between two points in this strange space with this metric And it can be shown that the dissipation is Lower bounded and this bound is saturated by a lens which corresponds Which corresponds to the To the the connection So sorry by a protocol which corresponds to a lens Connecting two points at which the heat production is constant at every time So something like moving at the constant velocity, but now generating heat In the at the constant rate and For instance in the case of the heat engine Operation the same equations I show before can now be rephrased in this way so we have here this a Heatwork conversion, which I expressed before in terms of a very Carbatch of a very face Which can be a by records to Stokes a theorem be expressed in terms of a of an area with a strange Carbatch are defined by the very Carbatch are and The dissipation as the square of the lens in this space with the strange metric So if we want to get the maximum power so the power is the total work divided by the duration the the period of the cycle the Efficiency is the work divided by the heat With if we want to optimize this for instance the the power We get expressions like this that the maximum power can be expressed as a ratio between an area and a lens in this space with a special metric and This Area defined Wait with the thermal with the with the with the very a Carbatch are so this is a Problem so the the the maximum power corresponds to the optimal ratio ratio between this a quantity We know that in Euclidean space. This is this a circle, but In an arbitrary space is something That this is still an open problem in mathematics. This is a list of a mathematical works devoted to investigate that problem But it is still useful if we build a Protocols to see which are the results. So I'm finishing now This is an example of a protocol that optimizes this is the Again Very Carbatch are represented in the parameter space to different protocols and we can show that this kind of a elliptical protocols are the ones we Optimize in this a problem this ratio So this this is all and I would like to thank you very much for your attention Thanks for talks any questions Maybe a bit provocative question Well, we understand that of the geometric phases very face is purely classical object And we also understand that adiabatic limit is not necessarily semi-classic limit So where is the quantum component in your in your theory? What is quantum the qubit? It's only qubit, which is quantum the qubit and the Hamiltonian. Yeah, of course the very face. This is the very face can be defined for Quantum as well as a Classical system so very first is not a mark for quantum And but In this case it's useful to analyze the so it appears naturally and In the structure of the problem So the quantum is the is is the system itself and the fact that the it's operating in some coherent regime So it's the context Not the formalism Nice talk in which kind of reservoirs are you thinking on because for example in circuit quidi you have your You know your resonator which would be at the same temperature of the of the qubit But the environment could be a room temperature. So you unify and you choose a unique reservoir So I'm thinking I mean That you're talking about the Delta T is so it's not a single Resonator so this is something which is a resonator with the many modes and is the typical Device where people analyze the spin boson model for the so the qubit operating as a spin boson model or for instance the this this type of experiments in which so This the upper where you can sound how if these resonators which are in touch with the qubit are strongly coupled to the To the to the rest of the of some long transmission lines and so on you can define a reservoir out of that of course, I'm not thinking about the And these temperatures to be room temperature. So these are small temperatures because otherwise you don't have the structure of the So Maybe it's a little out of context. I'm not an expert on this sort of thermodynamics but it looks like this is closely related to the long land our principle because It's a matter of the qubit you're you're sort of using as a bus And the question of storing information and what is the efficiency of entropy transfer from one to another? In terms of efficiency, do you really saturate the land our limit exactly? Yes in some limit We recover that the limit. So this limit is recovered for instance if you make very long loops a very long cycles Which are very close to to quasi-static cycles and you recover exactly a land our Result We don't we are in general we are out of the We are away from a land our result so land our result is a limit where you have an almost quasi static quasi static cycle otherwise you are out of equilibrium and So it's it's worse than having transience because you are having the heat leak all the time So you are all the time operating out of equilibrium Yeah, I was wondering you have this experiment here that Where you kind of couple and the couple from the reservoirs because you have those filters at different frequencies So it's like a frequency selective coupling In your case it was like the coupling was turned on and off because of the matrix elements because you had the different operators So I'm just wondering that which one is more efficient or can you comment on that like your scheme or that scheme or? Yeah, not yet. We are actually now playing with other type of a couplings and even with the with two qubits I think that the I mean the The way to optimize efficiency is very complicated because of that I presented this geometric formalism because To optimize efficiency is not only to optimize this heat war conversion term but to optimize to minimize a dissipation and It's very counter intuitive to To figure out which protocols give you less dissipation. So the It's tricky. I cannot answer for for the moment If you have to go into the details, there is no easy answer for that Okay, thanks again. Thank you