 come pensi? ho ancora per sé mi stia succedendo odoscamente le fate anche la bella se non interfere il microfono questo dovrei far così si fanno il fede vabbè lahi che non trasc valuable ok se sai che avrei perfetto Sto qui in Trieste, ora posto l'Università di Tübingen. È andato a raccontare i sistemi quanti. Grazie per l'introduzione e per inviare me qui. È davvero pressione, è la prima volta che, dopo i miei posti, ho vinto e è sempre bello essere in Trieste. Questo è il mio ultimo spazio prima di lunch. Ho provato a essere lontano e non senza molte tecniche, quindi inizierò con un'introduzione, ma non è un'introduzione, perché è un buon punto di avere un pubblico specializzato, come tu, e in questo tema specializzato, che è stato grandemente discusso in previousi spaziosi, quindi non sarò in grado di riuscire a riuscire a resettare, almeno in un caso classico, e lo farò direttamente al punto principale del spazio, che è riuscire a resettare, ma in sistemi quanti. Ci sono due esempi. Il primo è il sistema di quanti quanti, con l'evoluzione unitaria, e il secondo è il sistema di quanti quanti con l'evoluzione dissipativa. Che è l'effetto di resettare sulle statistiche di migliori diversi, quando l'esettica è introdurata su questo. Per prima, discuteremo le dinamiche di resettia. Questo è stato specializzato in previousi spaziosi, ma mi darò solo una prima introduzione, con l'effetto particolar di quello che bisognerò dopo. Qui c'è un amiltonio, che è un amiltonio, ma è un esempio per questo spazio, perché molti dei risultati che scrivono sono generali. Io prenderò solo questo alla fine per l'evaluazione. Per cui abbiamo tirato questo, come primo esempio, è sempre la stessa risa. Il primo è un modello esattamente risolvabile, in cui puoi studiare molte cose analiticamente, e il secondo è un esempio prototipico per la transizione quanti quanti quanti, per l'evoluzione, che per la transizione quanta, più piccola di la valura critica, è una in questi uniti. Abbiamo un stato di grannone di 2 degenerati, molti per l'evoluzione, mentre per l'evoluzione più grande di 1 abbiamo solo un stato di grannone di 1. Questi sono per esempio per 0 transizione quanta queste stasi sono pure classiche e sono semplici per visualizzare, per le base di spin. E poi come il resettio di resettia funziona in questo caso è molto semplice. Stiamo da un stato in cui è un stato di grannone, in questo caso è l'unico blu, quindi è completamente polarizzato in avanti, ma, insieme, questo è un esempio. E poi hai un'evoluzione su di questo. Come ho detto, il primo esempio è in questo caso in cui questa evoluzione è unitaria, e che prendiamo l'evoluzione con questo amiltoniano. E vedete, questo è un stato di grannone per 0 transizione quanti quanti quanti quanti quanti, in questo semplice gioco. Questo amiltoniano ha non 0 transizione quanti quanti quanti, quindi, in Giargon, questo è quello che è chiamato l'evoluzione quanta quanta quanti quanti quanti. È un esempio prototipico, molto studente, questo è un paper, ma ci sono tanzi di più, e questo è un esempio prototipico di unitaria per far un'evoluzione non equilibrata. L'evoluzione si tratta di quando il prossimo è stato di grannone, in cui il sistema è rinitializzato all'iniziale. Questo è stato, per esempio, discutito in questo paper qui. E in volte, come sempre, queste tante grane sono toglie da alcuni tempi di distribuzione, tempi di risultati conseguarie. Poi, l'evoluzione unitaria inizia al prossimo risetto, e così on. Questo è abbastanza discutito, ma ho detto di solito, senza molte dettagli, avrò l'evoluzione iniziale, e avrò detto di quello che abbiamo imparato, che è una variazione di questo prototipico che combina i mesi e le risetze. Questi prototipico sono più importanti di più. Questo è quello che ho capito dal mio PhD. Quindi, ho schematato questo in termini di un prototipico, e questo è basicamente il prototipico. Quindi, si inizia ad un stato che potrebbe essere qui a un stato di grano, numero uno, nella fase ferro, per il magnetico, poi l'evoluzione, e poi, a un certo tempo, si tratta di un meso, un meso specialmente risolvuto di magnetizzazione. Perché magnetizzazione? Perché magnetizzazione è un parametro dell'evoluzione per la transizione. Quindi, è la più observabile di poter mescolare. In questo caso, quando si tratta di mescolare, hai due possible outcomi. Quindi, se hai un stato che ha una magnetizzazione positiva, quindi la maggiorità di reti in questo schettino, o in questo caso, non è una magnetizzazione negativa. E poi, depending on the outcome of the measurement, we choose the reset state. To be, in this case, either all up, in this case, either all down. So, it means that the reset reinforces the magnetization that has been measured. That can be listed. Yes, I mean, the measurement is probabilistic because it's quantum mechanics and then you choose. Yes. And this is the name, we use this conditional reset because the choice of the reset state is conditional on the outcome of a measurement. Taken right before. How can this be formalized? Be going beyond picture? Again, this can be formalized in terms of these two matrices. Here you have one first matrices with indices that run on the set of reset states. So, in this simple example, this is a two by two matrices. And this is a projector over our magnetization sector with k, well, one can be the positive magnetization sector of the space with k, well, two can be the negative magnetization is the negative magnetization sector of the space. And what is the meaning of this? This is just quantum mechanics. So, this tells you which is the probability of measuring a k equal one positive magnetization having started since the last reset from a state, from a reset state j equal one or two from a positive or negative magnetization. Okay? Then we have to account the times are random between measurements and reset. And so we have to also build put the waiting time distribution ingredient in this story. And the waiting time distribution is here is this p of tau. And this builds up these matrices rjk that you say that as you see, depends on the initial state, the final, the initial reset state, the final one and the time elapsed between the two. And so as many of you discuss much more than me, this is a semi mark of structure. To my ignorance, I declare that this was the first time I learned these topics before I didn't know, but I learned much more from you in these days. But still, so you see here it appears this structure and this will be very useful. Then, again, pictures. This is basically the machine that we are building. So these q boxes, q stands for quench and m stands for measurement and r stands for reset. So fair enough. And you see that the evolution that we build basically at every time there is a measurement plus reset branches into direction and then goes on. And there are many possible ways this stuff, there are many trajectories according to which this process can branch. And in the long term limit we want to see whether this gives some stationary state of non equilibrium nature and how this stationary state looks like. And in particular, we would like to build some stationary state through this measurement in two reset states. Two, because for using model there is a z2 symmetry. But again, for other models we can choose a larger set of reset states, for instance. Here we add in mind the z2 symmetry. And why we do this? Because we would like to build some states which dynamically carries some signatures of the underlying quantum phase transition of the model. Something like this. So some crossover between a stationary ferromagnetic e a stationary paramagnetic phase. And... Ah, sorry. One ingredient I have not yet specified is the time between measurement resets which is the within time. As at all you see this block, this picture is irregular because accounts for the randomness of the times. And one example I will specifically consider is this kind of distributions which were suggested by discussion with experimental people that at least in Germany are working on the implementation of this kind of protocols. These are distribution with compact support. So you see after a certain time in this catch is 3, the distribution is 0. So it's like an exponential chopped at a maximum reset time. Here is 3, which we call tmax. Why this? Because indeed experiment is more difficult to implement distributions with long fails of course. And the exponential, despite being an exponential, are still a long tail. While here by construction you see the maximal time between consecutive reset is bounded by necessity to this maximal time. And why is also important because this tmax is a control parameter of this protocol machinery. So I lost something, sorry. So at least in principle one can tune it. One can tune it to be smaller than the typical coherence timescale of the protocol. Typically in every quantum system a unitary evolution after a certain time you have many effects like many body eating, particle loss is many more which destroy unitary time evolution. In principle here you can use this maximal time so you can use resetting to cut this long time effect and induce some stationary state which in principle is robust against dissipation. This was also the reason that was inspired by discussion with experimental people to use this kind of within time distributions. Of course there are no idealities but at least this is the motivation. Then just as belief, light on how to proceed, all this part was motivation just as light to give you how one proceeds operatively to compute things according to this protocol. So one bright density matrix has a sum over the density matrix Rho and conditioned five minutes. Okay, so sum of density matrix conditioned on the number of reset which can be from zero to infinity and then one makes the expectation over all the possible trajectories of this. The result I will tell you without calculation. Oh, better. It's given by this expression here for the evolution equation. You see this L is unitary evolution so Hamiltonian evolution in here you have two rates which are time dependent and they are time dependent in a non trivial way because they depend on the measurement protocol so on the measured value. So they couple the resetting dynamics with the measurement and plus the results are time integral on the previous time history of the process. And from this equation one can get the stationary state which is a statistical mixture which is not surprising because this evolution is effectively open. It's not unitary so it doesn't preserve pure states. And it's a superposition with some coefficients which are non trivial and are given by the semi Markov structure. Basically a stationary state of a two state Markov chain. Then application twice in model I will just give you here we computed the squared magnetization in a stationary state and did we see that there is some crossover here maximal reset time here transverse magnetic field from a ferromagnetic to a paramagnetic phase. You see it's a crossover because it's movement. It's not a sharp but at least it can be sharpened by a bit by tuning the maximal reset time. And then last part very briefly about open quantum dynamics. Here we consider system whose evolution is Markovian I just have two slides. And what in this case this is called quantum master equation or Limba equation was very well introduced in previous talks. Importantly for Markovian systems this W is time independent. So it's a generator which is time independent and these one important characteristics are quantum trajectories or better quantum jump trajectories which are basically every time you have a jump enforced by these operators you have a signal in the environment which could be for instance the detection of a photon a click in your detector. And these trajectories can be characterized dynamically by using large division theory so by counting the number of jumps occurring up to a certain time this obeys a large division principle with a rate function and the rate function can be obtained with the Legendre transform of the skeleton cumulant generating function is large division theory in two lines in a natural. And how one proceeds typically in Markovian systems is that one consider bias the tilted generator this is the name usually used. So a generator where there is an insertion of this exponential in k where k is the same variable of the cumulant generating function or moment generating function as well. And one computes basically this theta of k and therefore the large division function from the largest real eigen value of this tilted generator which as I told you is time independent. So diagonalizing is well diagonalizing it is a well post problem. We asked for the same problem in the presence of resetting in this case just resetting without measurement. What helps in this case is that resetting as a renewal structure as well been we have seen in many talks and therefore we can basically describe trajectories by considering density matrix on the number of emissions and the number of resets. And then we can ask for the tilted density matrix by tilting only the number of emissions because we are interested in the statistics of the emission which are counted by n. And basically this renewal structure allows to write an equation for the generating function in the presence of reset as a function of the generating function with the superscript zero which means without resetting. This idea was coming from other papers about classical systems where this was done in the case of resetting with exponential distribution. And this is remarkable because despite in this case we can write the tilted generator which is time dependent for non exponential weight in time distribution. So one cannot simply ask for the largest eigenvalue. Still using the renewal structure one has a way to compute the last division statistics. Very fast example is a three level system and then I co-finished. You have a three level system where you have three states and one of the two is weakly coupled. So the Rabi frequency of oscillation here is smaller than the Rabi frequency here. Why this is interesting is interesting because it has this is a system where quantum trajectories display an intermittent o blinking nature. So there are periods of time where emissions are plentiful and there are sequence of time where emissions are scarce. And this can be if you want to draw an analogy with equilibrium statistical mechanics is a dynamical first order transition because you have coexistence between bubbles of the inactive phase so no emissions with the bubbles of the active phase like coexistence of vapor and liquid in a gas. And how this is seen in the last division statistics is seen by this smooth net because it's a finite size system not thermodynamic crossover around k equals zero where this intermittent takes place. But in the presence of reset you see here is the maximal time you see that the large deviation function can be tuned very much and in particular you see that you can even have a large deviation function here in red was mean activity mean number of emissions is even larger than the one of the reset free process the dashed line. This is a bit counterintuitive because here the reset state is taken as the zero state. Here so with resetting you are fostering the population of the dark state where you don't emit but still you can have an activity larger than the one without resetting. Why? Because you give away to the system to escape from this metal stable state and to restart this cycle and so you can still trigger the activity. Of course if you reset too frequently if you increase the maximal time you see the activity gets reduced with respect to reset free dynamic and you encounter a quantum zero effect in the end for zero in max. And this is the case where you have a reset to the state number two, the metal stable one for this choice of rubby frequencies and you see as you increase the number of resets to state two you basically foster the metal stability of this system because this basically gets flattened around zero and this tail of the distribution gets basically killed. And so moral of the story about is that you can tune the activity of the system by using resetting. This is a three-level system but we are working with some more complex example and I conclude it. So we have seen how to construct stationary states with reset with unitary dynamics and measurement how this can carry some signal of phase transition in terms of crossover. It could be in principle robust defecivation and we have a possibility to control dynamical phases in open quantum states. And then there are many open questions which we can discuss which we are also working. Thank you very much for the attention. Thank you very much. Let me proudly say that it was my pleasure to see you. Okay, so are there questions here? I don't know if it is a proud. Yeah, exactly. I don't give responsibility. So online there are no questions. Are there questions here? Yes, Satya. So can you use this resetting protocol for this quantum steering problem? You know this Schrodinger in his very original paper, he was asking the following question that if you start from an arbitrary state, suppose that you have a desired state and final desired state where you want to go. Target state, basically. And so by a combination of measurement and kind of resetting, you didn't think about resetting at that time, but you can problematize it. But you can probably drive it to the final state. So I don't know if, since you have a combination of these two, perhaps you can actually steer a state to the final constant state. So basically you would like to optimize the mean time to reach a target state. Well, in principle we could do it. Una idea I have... No, ok, but I can use it. Sorry. In principle, the way I would implement this is by running this protocol and then you want to ask the first detection of a given state. So you have to count basically all the trajectories among all the trajectories that are possible, only the one that reaches that state. But you cannot put it as a reset state. Is it used as a... If it is a reset... I don't know, I'm just asking everything. Speculative question. No, no, ok, boy. The way I would implement it is basically you have this measurement and then conditionally on the measurement you have some state that you want to reach. And you can ask for the first time, you detect basically a magnetization with the sign that you want to reset to that state. And that would be easy to implement given this because you just have to count the trajectories which have, I don't know, positive, positive, positive, positive. First time you measure negative magnetization over the normalized but total number of trajectories. In this way, I can think, I can implement it quite naturally from the calculation, it was shortly presented. Are there other questions? Yes. Thank you very much for the very interesting talk. I'm interested in the large deviation function. Is it possible to calculate it like in the classical system with this Laplace approximation or you need this generating function? Laplace approximation, you mean large deviation principle? Saddle point. Saddle point, but I think it's more appropriate to use Laplace approximation because we don't have complex function there. So, but this is the saddle point method, yes. So what year? The gender transform, I don't know. So if I understood correctly, you would like to write the generating function as some, basically, integral over... Let me rephrase the question. How did you calculate this, the saddle point? Ah, the large deviation function is obtained from Legend-Fenkel transform of the generating function. So, seemingly he had no problems with that, yes. No, I have checked all the poles of this expression, it seems to be very fine. So, I have a question myself. So, you show that in general, because of this protocol, you are changing a phase transition into a crossover. A forizing model. My question is, if you, for example, if you change the distribution of your setting times, can you recover anyhow a phase transition? Because I imagine... Miscontinuous. I mean, you change the distribution, for example, you make, I don't know, with the certain characteristic times, which... I don't know. I mean, my understanding is that, here you have a Isim model, Isim model, and between resets you have a quenched evolution. So, quenched time evolution is effectively thermal. Isim model is a short range, effectively dampened, so you don't have phase transition. I don't think that if you change the weight in time distribution, at least for Isim, you can get some discontinuity. If you get more complex models, which is what one would like to do, maybe you get something more interesting. Okay, okay. This is my interpretation. Yeah, yeah, so maybe you can even think about not assigning una effective temperature. Okay, are there other questions? If not, let's thank the speakers of today, Gabriele, and also the speakers of the...