 OK. Good afternoon. Unfortunately Boris is still not here. I wanted to check whether he recognizes this place. This is part of Norway, Spitzbergen. Very, very high above the polar circle. And this is nighttime. I'm thankful to organizers. For me it's a great pleasure to be here and to congratulate Boris, who is absent at present time. But still, I would recall our contacts. Actually, it's very hard for me to believe Boris is 60. Because according to Shakespeare classification, I know him since he was a schoolboy. And our contacts started in 1976 when my friend Arkady Aronov asked me to write evaluation for his master diploma thesis. Probably he found out something that I was not too happy because I had to write something and he told, don't worry, this boy is unusually good. And he was completely right. And since then I think benefited a lot working together with Boris. And most of this time he taught me to do some things. So I'm very grateful to him. We produce something like about a dozen papers together and always he was the person who taught me how to do things. So mostly we collaborated on what is now called nano physics. And to illustrate it, I will show you some picture, how we did it. And I'm showing this picture because the name of this dog is nano. So this dog belongs to Boris and his name is nano. So we did nano physics and he was very helpful. And we did several things with Boris and what I will tell about today is somehow continuation of the line about the coherence. So we wrote some paper, review paper about that and all participants of this team are here. I will talk about relationship between thermodynamics with simple driven quantum system and its coherence. And the questions which I would like to address is the following. Is thermodynamics of a driven quantum system related to its coherence? And which protocol can be used to determine this relationship? So it will be not a review, it will be one simple story. So why I started being interested that I am newcomer to this field? And mainly a driving force of this work is Yuko Pecala from Helsinki. And a Japanese group doing experiments, trying to find non-equilibrium thermodynamic of qubits. And everything is initiated by support from European Union project in fairness, which is spelled out as information, fluctuation and energy control in small systems. So it means that people are interested to study works made on some quantum system, try to find statistics of this work and try to relate it to some general principles. It's a very tricky business because if people work with small systems and low temperatures, it's not easy to measure actual temperature to see whether it does exist and so on. So people are hunting for some exact relationship which allowed to check validity of experimental data, consistency. And this project is more or less about that. This is a brief plan of what we're going to do. First of all, I started total motivation and I will tell something about the relationship which people quite come in to check at present time is Jarzynski equalities. And try to see how you can check this equality in a simple quantum devices when they can keep track of what is going on. This is my program. I wanted to give some examples but I put it in gray because most probably will not have enough time to go for that. Well, before starting that, I was very confused because I was grown up, educated in Soviet system. Everything was based on one doubt textbooks and according to conventional thermodynamic, I didn't understand how we can even speak about thermodynamics of a driven system. Usually what you can, if you open textbook, you can read that you have two states say A and B, both are equilibrium. You can define free energy between, for these both states, calculate difference. And if you make some work, this work should be at least bigger than difference in free energies. And you get equality if you do it very slowly across aesthetically. So you have only inequality and equality only one situation and you cannot propose any relationship which you can check. So auto-equilibrium according to conventional thermodynamics one has to treat each system individually, do some non-equilibrium science about that. And I can tell that that was my attitude at all that like happy families are all alike, every unhappy family is unhappy in its own way. So you have to discuss non-equilibrium systems just separately for each system. Then appeared this kind of formula which is called Jarzynski equality. And this is a question which relates free energy difference between some equilibrium systems and average special averages of the works needed to transfer a system from one state to another. So it's like that that you have to beat this inverse temperature different. So you have the system A which exists at some given temperature. Then you change external conditions and you construct another equilibrium system at the same temperature but the system corresponds to external conditions. For that system you can define free energy. So difference in free energies should be average of all ways of changing external conditions. So it doesn't matter how fast it takes place, still this should work. So you relate equilibrium states with non-equilibrium processes. And more or less there are several derivations which are more or less strict for classical system and somehow controversial for quantum system. I will not go through this derivation but they are mostly based on the reversibility of dynamical equations both in classical and quantum mechanics. So this is one of the examples. Suppose you have gas and a piston and this is state A. Then you push the piston and make, change the volume, arrive at system B and then you construct artificial system, equilibrium system at the same temperature which corresponds to this position of the piston. It's a lot of philosophy about that which I don't exactly understand. People tell that it's temporary violation of second law of thermodynamics because they speak about big fluctuations. There are no miracles here. This is the matter of correct, choice of correct set of representative fluctuations and taking them into account. There is a very beautiful paper by Luan Grossberg which they discussed, this generic classical case where you have a piston and molecules of the gas and they explicitly calculate distribution function and show which fluctuations are responsible for the Georgian's inequality in classical gas. It's not so simple because to keep track of that you have to take into account not typical fluctuations, rare fluctuations, you have to wait a long time and if the system volume is large and the piston is moving with great speed compared to some velocity for a very short time you have to wait exponentially long to pick up the fluctuations, the proper fluctuations. Any way you have exact relationship and people who work with a driven system are very much interested to make a cross check to see whether they get things correct or not and what I'll tell about one of the protocols how to make it with a quantum system. Regarding quantum system it's much more difficult because there are different definitions of working and people don't agree how to define work for rather complicated arrangements so I will do it for a simple system. So suppose you have two level system and then the total work contains a change of internal energy. Internal energy is just, if you measure it, you have initial state, a final state and the difference is the difference in internal energy sometimes it's called useful work in this business and also you can emit photon or phonon and then you have dissipation. So if you have a cyclic process, the Georgiansky inequality equality requires that if you take into account all the processes and all the works you have to get this average for one. That you can check and there are many papers today even in good journals where people really check this and claim that everything is consistent with this inequality. So how to do it with a quantum system having something simple? So we decided to go along the proposals, so-called two measurement protocol where the state of the system is measured so you know where you start, then you wait a bit and then apply drive. After some special drive you leave system alone and then it can decay here then you do one more pulse projected on initial state and then measure. Actually in the experimental setup which we discussed there is also some very precise colorimeter so they can also identify the quantum jump and they can discriminate between absorption and emission. So even from this what was initial state? It was emitted and then it was excited state it was sort of ground state. So for a closed system if you don't have any jumps and any inelastic processes then you have to get the average of U equal to 1 and this is not the case for realistic system because it's open and you can emit or absorb some quantum degrees of freedom, photons or photons. So what I'm discussing can we extract useful information from this average? This average should not be one it should be something different but is it reasonable, is it possible to extract something? My answer will be yes and I will show how people do it experimentally. So this is so-called quantum jump approach which was developed for quantum optics and that's what we use to calculate this average. So the idea is following suppose you study a system for a very short period and the probability to have a jump is very low either zero jumps or one jump then you can, what you have to do you have to analyze density matrix for the system but there is a simplified method which works on the level of wave functions and you construct some non-Hermitian motion and some non-Hermitian Hamiltonian which takes into account the possibility to have a jump quantum mechanics Hermitian so this is a model equation and you have to derive it if you study the dynamic of a system for a short period but you don't make a jump you have to take into account the possibility some final probability to have a jump and in this way you create this non-Hermitian Schrodinger equation what you have to do you can show that it's fully equivalent to a block rate equation for density matrix so this is Hamiltonian and now Hamiltonian we have this diagonal Hamiltonian this is a possibility to have noise or the coherence which you take into account which just moves levels in time and this is the part which takes into account final probability to have a jump and this is the drive of the system so we can do it and if you work with that we can represent the wave function as a linear combination of ground state and excited state and analyze motion of this amplitude this is technology and I will probably this is where the system has advantages compared to the density matrix because you can easily write diagrams for different process here and make it systematic was analytically and numerically okay, so let me tell answers suppose you are very long times in your protocol much longer than inverse rate of quantum jump then what you get all these occupation numbers are equilibrium and you get for the difference between exponentials of the work and one depending on temperature so if you have a long time if temperature is low then you have big differences if temperature is high you don't have differences and this is a way to check what actual temperature system has you can trace that you know what is the temperature of your device not of your thermal bars but of your device now let me answer the difference doesn't depend on the coherence fully thermodynamics okay, now let's discuss the protocol the protocol is following we do measurement of the system on the time t1 we apply power over 2 pulse then keep it for time t2 and then put another power over 2 pulse and then after some time do one more measurements this is the protocol which we use let me depict it, so if there is no coherence I do the following I start with some eigenstate for example spin up rotate it then nothing happens and then in the rotating frame and then you finish a measurement system in some other state if you have the coherence what you get you get the following you start with some state, rotate it when it's parallel there is no coherence due to pure defacing so we can split the time when you have defacing and then you have a trace of this defacing when you do measurements this is the measurement protocol and we can do calculations which we don't have time to explain but let me give you the answer the answer is the following so if we have at most one jump this average we can measure we can measure, we can measure since we measure system we know the difference in internal energy it's about a million repetitions of this experiment so statistics is made and according to the statistics this average depends on the times if these times go to zero you get one, system is closed and an elastic process doesn't take place for finite times you have this expression which depends on the temperature and it depends on the amount and mechanism the coherence because delta phi2 is the only phase shift which is accumulated during middle part of the protocol so this is our central result, very simple but it allows to make experiments and it allows to study this distribution function why it's possible, why don't have to wait so long because only few states is two level system so you can have either there is jump or no jump and probability for more than one jump is low so we did some numerics and this numeric shows that at small times it gives exactly the same analytic result but we can do it for longer period and study more jumps and what's good that we can also have we can calculate distribution function and distribution functions of the energies is measurable so this is cartoon of the experiment is made in Japan and this is Yasunakamura and his student and his student they have a transmount qubit and they do exactly the same protocol which I explained so these are characteristics of the qubit it's quite a good qubit so they made full spectroscopy of this qubit so they know left-hand side of this equation and right-hand side of this equation there are no fitting parameters so this is a check of the consistency and the quality of how this device works it's a lot of experiments because they have made many repetitions they do many repetitions this is why I'm still not allowed to demonstrate the experimental data because they are still in progress and they do double check but the first results are extremely encouraging so we get proper numbers of parameters and if you properly define temperature so the temperature turns out to be few percent more than what they estimate from the direct measurement of a qubit so this is consistent system and actually people now think that the system is fully characterized and they want to modify the system in order to make feedback related experiments so to make something like quantum Maxwell-Giman based on fully characterized device you have to characterize it otherwise it's very difficult to speak about distribution of the values entropy and so on so this is how it stands for today what is important and I will not spend a long time on that you have average of the cosine and delta-phi to it the same as you get for free induction signal if you work with magnetic resonance arrangements so you can study the mechanism of the coherence you can study how this average depends all the time and this is just in progress and I did it for a very simple model which is not physical I assume that distribution of phase shift is Gaussian and then you get the result which contains exponential of tau 2 squared we work a lot on the case of 1 over f noise distribution function is different it's not Gaussian I don't want to remove all this skip all this business because I don't have time for that otherwise I will miss dimmer but I will show you the distribution function this is a distribution function for 1 over f noise it's far from being Gaussian so then depending on different situation different function of time delay between two pulses they are working at right now and try to check how does it work I think I am approaching the end of my talk but I want to emphasize the following so the central result summary first two measurement protocol turns out to be an independent tool for studying some dynamic relations in driven systems and the central result is the average between this quantity there is a relationship between this quantity and the coherence temperature and it depends on the protocol on the protocol which people use if the right hand sign is known then jersinski equality can be checked only by measuring statistics of the internal energy you don't need to measure photons or phonons right now you are preparing system precise colorimeter which will allow measuring some quantum jumps for energy which is transferred to the heat and then it will be another functionality to check this relationship and another point if you can measure these statistics do many measurements and make average and then after this average know it as a function of all times you can find out the same information about the coherence and qubit from direct spectroscopic measurements or measurements in real time so this if you know these statistics you can use this relation to find out the coherence and the qubit well I think I can stop on that and thank you for your attention and of course this Boris happy birthday you haven't seen the pictures I showed in the very beginning but I will show you thank you thank you very much