 Here, let me start by thanking the organizers for inviting me and giving me the opportunity to give a talk here about one of our latest pieces of research. So what I would like to talk about today are quantum fluctuation relations, but not in the case where you start from a thermal state of the system, but rather it's off again. My mic seems on. Can you hear me now? No? No? All right. One, two, one, two. It's OK? So should I start again? Can you hear me now? No? Higher. Doesn't seem to be better now. No? Nobody can hear me, it seems. Very sorry about that. One, two, three. One, two, three. No? If I speak a bit more loudly, is it better then? OK, I'll try to speak up. So let me start again by thanking the organizers for inviting me to come here and give a talk about one of our latest research topics on quantum fluctuation relations. And I would like to actually use these quantum fluctuation relations, not to relate to as usually a kind of thermalized states, but to different kinds of states which might be pre-thermalized states or more like generalized Gibbs ensembles. And I'd like to look at how can we do non-equilibrium dynamics to find out a bit more about those types of states. Now the reason why this is a topic for us of interest is that in co-reactant physics as well as in condensed metaphysics, we more and more have experiments that actually take systems far out of equilibrium and try to induce by dynamically induce useful properties into these types of systems. And I've shown here two different kinds of non-equilibrium dynamics that are particularly promising and interesting for that purpose in the left-hand column. I got on the top an optical lattice experiment where basically a mirror is shaking. And through the shaking of the mirror, one tries to reduce the hopping rates and induce a kind of mod-insulating state from a superconducting state by shaking the lattice. So this is what you'd get here through a shaking mirror. You would induce a mod-insulator from a superfluid. And similarly, in condensed metasystems, and here I show some alkali-adopted fullerites. An experiment has recently been carried out where through shaking these buckyballs, so basically by driving the buckyball oscillations, one could induce a superconducting state into the system. So turn it from a conductor, from a normal conductor into a superconductor at very high temperatures just through driving. Both of those types of experiments have one thing in common. They take the system far out of equilibrium, and they induce some kind of interesting properties that make a macroscopic difference to those types of systems out of thermal equilibrium. And the second type of experiment is shown here, again, a kind of coratum implementation, and here a condensed matter implementation, where rather than driving a system periodically with some kind of laser excitations, what we can also do is do a sudden quench of the system. So in this upper example here, that's an experiment from 2012 in a Manuel Bloch's group. What they did was suddenly quench an optical lattice to see a density wave melt, and this melting dynamics was then followed as shown here, and we could in real time see how a density wave melts in an optical lattice. Similarly, here in the thin NIO3 film, what was done was a fast laser pulse that would basically excite here the interface between these two types of materials to induce the melting of anti-ferromagnetic order. And again, both of those experiments have one thing in common. There was a sudden quench of the system, and the sudden quench led to, in one case, the melting of a density wave or the melting of anti-ferromagnetic order. So I would like to do today with not exactly those examples, but with some simpler examples, is I would like to discuss how one could use these types of non-equilibrium states and characterize them through quantum fluctuation relations and whether it's maybe possible to use these types of fluctuation relations to detect symmetry-broken states. So some kind of conserved quantities that might take on non-zero value in these systems when they're driven out of equilibrium, whether those can be detected through these types of fluctuation relations. Now, fluctuation relations are, of course, a very kind of intriguing concept in that they take thermodynamics a step further and make some equalities out of what we usually know from thermodynamics as inequalities. So in particular, what we have in thermodynamics is that if we do work on a system, then the only thing we know is that this amount of work that we need to do needs to be bigger or equal to the change in the free energy of the system and that the entropy needs to always increase as we do such kind of quenches or work on the system. And that, of course, those relations are fundamental to the operation of all kind of heat engines, fridges and so on and so forth. But what remains is that we always have to deal with inequalities and then optimize those. What we don't have in traditional thermodynamics and what was only discovered in the early 90s are that we can turn these types of relations into equalities. And if instead of just looking at the work, we look at the expectation value of this e to the minus beta times the work, so where beta is the temperature of the system and we average this over many runs, then what we get is an equation here that equates this expectation value to the change in the free energy. And this is termed the Yashinsky equality and turns basically something that up here is an inequality into an equality if we average over many runs. And similarly, one has the so-called Krug's relation which relates the amount of work that is needed in a forward process. So we drive the system out of one state into the other to the work that is needed to drive the system back from the final state to the initial state in this so-called Krug's relation. And those types of equalities were first invented and derived for classical systems. And for instance, we're used here in this beautiful paper in 2005 to do some sequencing of DNA. So basically what people did there was they measured the force it takes to tear a DNA apart and from the fluctuations in the forces that are needed to do so, they could, via these Krug's relations, find out what the sequence of the different pairs in this DNA would be. Now, if one takes these types of equalities to the quantum regime, one can do so. But basically, considering the dynamics of a system that is described by a certain Hamiltonian is coupled to a Bartha temperature of beta and then is taken out of this thermal equilibrium by some kind of process that here for simplicity, I assume, is a process that is unitary. And if we take such a unitary process and work out how much work is needed to take the system out of this thermal equilibrium to some other state, then again, what we find is that this expected amount of work here in the exponent is equal to the change in the free energy in the system. So exactly the same thing as we had for the classical case. And similarly, for these systems, also the so-called the Saki Krug's relation holds that again relates the work needed to take a system out of equilibrium in a forward process to the system in the backward process and those two things are again related in exactly the same way as before for the classical system. However, all of those types of processes assume that we start from a thermal state. So the assumption is that the system is fully thermalized and we have here only one parameter, the parameter, the beta, the temperature, the inverse temperature of the system that characterizes this type of system. And what I would like to look at now is, oh, sorry, and also this kind of quantum Yasinski equality has been tested experimentally in an experiment in 2015 in a single ion trap experiment and one can see here that the agreement between the kind of theoretical predictions here in these color bars and the actual measurements agreed extremely well. So both the classical and the quantum version of these equalities have been tested and used in experiments. Now, what I would like to do today is actually to look at what happens if instead of having a thermal state, we start out with a state that is characterized by some Hamiltonian again, but in addition, a set of observables that commute with this Hamiltonian and are conserved initially. So basically that means that our initial state is now not described just by a quantity beta, but also by a set of more Lagrange multipliers beta k that are kind of constraining the average values of these observables mk. And so rather than having a pure thermal state, what we now look at is a state that has the typically to the minus beta h term here, but then a set of more Lagrange multipliers constraining these mk's. And so basically this so-called generalized Gibbs ensemble here is a state that will actually be the one with the most entropy under the constraints that we have a fixed average energy and we have a fixed average of those types of quantities mk. And what I would like to look at now is how do fluctuation relations relate the work that is done on this system to these types of generalized Gibbs ensembles and can this be used to detect say if such a conserved quantity exists and what its value might be. And in order to do this, what needs to do is extend the kind of meaning of work a little bit. So rather than just having the work defined as the difference between the initial energy and the final energy, we need to add to this all of these kind of initial values for the observable mk and for the observable ml with the corresponding Lagrange multipliers here. And so what we need to do is we need to replace the work that we initially had without these conserved quantities by the difference in this final value of this property A and the initial value of this property A. And if we do so, then indeed what we get is a generalized Yashinsky equality. This generalized Yashinsky equality will now relate this e to the minus w. So this generalized work that I just introduced to the difference in the generalized Gibbs ensemble free energy. And we'll also get generalized the Saki-Crux relations which relate the forward process to a backward process again in the same way as before. But now relating these generalized works to the change in the free energy where this change in the free energies or the free energy is now defined as the logarithm of this partition function of the generalized Gibbs ensemble state. And so basically what we now have is a set of equations that relates kind of work type quantity in any out of equilibrium process to properties of the generalized Gibbs ensemble in this system. And we'd like to look at what the meaning is of this in some simple examples that I would like to discuss now. And the first example is just take the Dicke model. So the Dicke model basically couples a bosonic field to an ensemble of spin one half systems. So what we have here is a Hamiltonian where we have the bosonic field. So this is typically a cavity field described by field operators a and a dagger and some energy omega a. Then here we've got the energy of the spin ensemble. So the J set basically is just the set component of this kind of sum over all the spins. And those two systems interact with each other. And the way they interact with each other is that they can basically exchange quantum between each other with a coupling G times here's a parameter one minus alpha where we basically create an excitation in the spin by destroying an excitation in the field or the other way around. And if we have a non-zero alpha here then also the opposite process can happen. And this is the two excitation are created one in the field and one in the spin ensemble or two can be destroyed at the same time. Now this type of Hamiltonian can be realized in iron traps. So basically if you have here an iron trap these irons would constitute the spin ensemble. And if we take two lasers one reticent one blue detuned then this type of model can be realized. And by changing the properties of this laser driving we can change these parameters G and alpha here. And now the interesting thing about this model is that if we choose alpha to be equal to zero then we have a conserved quantity here. So the total number of excitations in the spins and in the field is a conserved quantity and will not change as long as alpha is equal to zero. And so what we now assume is that we start from such a state with alpha equals zero then we turn alpha on for a while to some non-zero value and turn it back off again. And at the same time we increase this strength G to such a value then we would expect this M the number of total excitations to change with time. And if we now look at what that type of evolution would, how that type of evolution would look like in the standard the cyclic nucleation what we find is that actually there's a big difference in the work distributions between the forward process and the backward process. So here in blue is the probability distribution of the work needed to affect this type of change, this type of evolution here in the forward direction and in white is how this work distribution would look like in the backward direction. And so what we can see here is that these two probability distributions do not agree. And because they don't agree, we can actually conclude from this that some kind of conserved quantity we have missed out. So there's something that tells us this system is not yet thermalized and in particular this system was not really in a thermal state initially with no conserved charges. If however, we instead take the generalized the cyclic Rooks relation and relate the forward direction with this generalized work to the backward direction. What we find here is perfect agreement between those two probability distributions. So basically by identifying this quantity M, this conserved quantity as something that takes the initial system out of thermal equilibrium. We can again use this the cyclic Rooks relation in a generalized way to relate work in a non-experiment dynamics to properties of a generalized Gibbs assemble here. Okay, similarly if we want to take the Sinski equality we get a similarly interesting behavior. The first thing that you notice is that this expected sorry this exponent of the standard Gibbs free energy and the generalized Gibbs free energy they do not agree. So basically if you were to now measure the work to identify what is the change in free energy when we get a wrong result because we would always get the change in the generalized Gibbs assemble free energy and not in the Gibbs assemble free energy here. So those two differ and thus we will get a wrong result if one were just to assume that there is no conserved quantity here. If one then now measures the work for again a similar process as I've shown before. So we start off with alpha equals zero that is M is a conserved quantity turn this into something non-zero and then back into something zero and similarly increase this coupling G here. Then we'll be fine this that for very short time so if we turn this change it is quenches on for just a very short amount of time both types of your Sinski equality disagree with one another. So here we have got the triangles, the blue triangles are the standard your Sinski equality and they give exactly the same result as this generalized Sinski equality. So both of those terms agree and the reason for this is that if this quench is done very very short and quickly then there's not enough time for this conserved quantity to dissolve itself and to turn into some and to basically become thermalized. So that means there's hardly any change in this value of M and all these different types of works give us the same value for the free energy change. However, if we make this time longer then we see some market difference between the two types of work functions and in particular we see here that the generalized work remains exactly where it should be it's constant it is independent of the duration of this protocol so how long it takes to turn this on and off and the two agree as they should. However, the kind of standard work will actually deviate massively from what we've had before and the reason for this is that after about this kind of length of the protocol the kind of conserved quantity starts to get messed up it starts to thermalize and this leads to a change in the work that is needed to drive the system out of equilibrium. So basically in this case the Asinski equality can be used to kind of track the dynamics of non-equilibrium quantities and conserved quantities in the starting state over time and here for instance get that the lifetime of these types of M the lifetime of this M in a quench would be about 0.1 microseconds for this particular value that we've chosen if the protocol takes longer they will just scramble and thermalize. Okay, now a second example is now not one where we do a quench but I would like to do a second example where we look at what happens if we drive the system and whether again we can use these types of ideas to characterize driven states of quantum systems. Now in order to kind of do this in a concrete example I'd like to take the Hubbard Hamiltonian and the Hubbard Hamiltonian in this driven in the limit where the interaction energy U is much larger than the kinetic energy T so the hopping between different sides is small and we have a the system is dominated by the interaction energy and in this particular case what we can do is can get rid of double occupancies in the system and what we are left with is basically a Hamiltonian is mostly described by this hopping between or fermions between different sides and so we have fermions moving along a lattice if they sit on top of each other a large energy that repels the two. Okay, and now we turn on some driving and this driving that we would like to turn on is a driving that basically shakes the individual lattice side up and down so we basically want the system to wobble around in time and then we look at what happens to this system if we do this so if we drive this system and make the driving amplitude stronger and stronger how does this system change in time? In fact if we remodel this by just having an onsite energy that oscillates in time so we add to the Hamiltonian sinusoidal modulation here of the onsite energies and we do this in an alternating way so if one goes up the other one goes down basically all oscillating phase nothing much happens. Now in this particular situation where we find this and this is well known from a number of experiments is that as we increase the amplitude of the driving what happens is that the bandwidth and hence the hopping amplitude between different lattice sides goes down with the driving strength like a vessel function and so if we start out here with a band that is brought that has approximately 40 width in a 1D system then this bandwidth will get smaller and smaller and here at this point the driving is so strong that any kind of hopping between different lattice sides is suppressed and we basically get all the band collapse into one point. This happens if our driving's frequency is large compared to all other quantities in the system so we have now a driving that is much faster than any kind of interaction strength than any kind of hopping and then basically we just suppress the tunneling of particles between the different sides. If instead we take this driving frequency below you then the behavior is slightly different so what we get is we do not get really compression of the band width to one point and thus kind of stop any type of dynamics but instead what we are left with is a certain bandwidth here that is determined by four times the exchange interaction so the T squared over U here and that never goes away so there's always some residual dynamics going on in the system and what we're interested in is in this limit how does the system behave close to this point here and in order to study this we first did some numerics and just turn on the driving strength in a slope as you see here and look at what are the properties of this system if we take a hubbub model and do this oscillation with ever increasing modulation strength and then we look at different types we look at different types of correlations particularly the density-density correlations spin-spin correlations and pair correlations and I'd like to focus here on the pair correlations so the pair correlations would tell us how two fermions pair together and then move together through the system and this is captured by this correlation function here that basically correlates a pair that is located at sides i and i plus one to sides j and j plus one and if we turn on this driving what we find here is that indeed this system turns from one that has a hubbub like ground state into one that looks completely differently and instead of having a peak here in the structure factor at four times kf has two peaks at two times kf and basically turns from repulsive Latinx liquid into an attractive Latinx liquid if we just look at the slope here so here is the slope at q equals zero and what you see is that for weak driving this Latinx parameter always stays below one so we have a repulsive Latinx liquid but if we drive the system very strongly we get an attractive Latinx liquid and it seems that the system moves from metallic state into a superconducting state we looked at that also through the spin structure factor and again conclude the same thing because what we get is in the structure factor two peaks at two kf turn into one peak at pi and this indicates a doubling of the wavelength of the spin wave so rather than having here a spin wave with two kf here seems to indicate that we get pairs and those pairs have a different types of spin wave and we conclude that this might be due to undeterminatically bound pairs in the system if we then look at the pair correlations this picture is corroborated even more so we turn on the systems driving and find that the pair correlation peak increases massively at q equals zero and seems to indicate that again pairs are formed and these pairs in the system would then bind together and move together through the system and that is what we find from the numerics but now I would like to use this type of fluctuation relations to find out whether one can say a bit more about those types of pairs and in order to do this what we do is we write down a low energy TGA model so basically we now assume that this driving can be effectively described as adiabatic motion or as dynamics in a low energy TGA model where we have as the first term the hopping again where single particles hop from one lattice side to the next with the hopping amplitude that is suppressed by the driving according to this zero sort of Bessel function the other terms it turns out are totally unaffected by this driving so if we now have here a bare energy here so we are basically a pair of binds together and that gives you the reduced energy J then that single bare binding is unaffected by the driving and remains the same as what it was without driving similarly we here have terms where the bears hop from one side to the other and that hopping goes like alpha times J with alpha is one half and again this type of hopping of pairs is unaffected by the driving so it turns and finally what we do is we protect out any double occurrences because they are still high up in energy so it turns out that in this low energy model the only thing that really depends on our driving strength is the hopping of individual pairs and if we now basically change the driving strength all that will happen is that this hopping term here changes and in particular if we go to residence where the hopping term is zero what we'll have is the number of nearest neighbor pairs that are described here by those two terms in the Hamiltonian will be a conserved quantity and what we're now going to look at is what happens if we start from the situation where the system is described by this low interaction, sorry by this low energy Hamiltonian in the case that driving is such that the hopping term is switched off this number of nearest neighbor pairs should be a conserved quantity and if this number of nearest neighbor pairs is zero so this means if our driving has not induced any nearest neighbor pairs then the evolution of our system should be perfectly aligned with the standard, the Sarkich Rooks and the Ashinsky equations if our driving has actually induced non-zero number of pairs then our evolution should violate the Ashinsky equality and instead agree with the generalized Ashinsky equality and so what we did in order to study this is we looked at an initial state that is just like a GG state that I introduced before so we here have a temperature Hamiltonian with zero hopping and then some kind of chemical potential here for the pairs that fixes our average number of pairs we start dynamics at t tilde equals zero so no single particle hopping and turn this towards one and then we look at the standard work so basically if we do the standard work what we should get here is that the expected amount of work that is needed to drive the system from this from this value of zero to 0.1 is just zero on average and the fluctuations are shown here so the W squared is shown here so this means that our standard Ashinsky equality would predict zero work is needed to do this if instead of course we take the generalized work then what we find is a very different behavior so the generalized work goes like the number of pairs in the system and hence by just measuring this generalized work in this quench of t from zero to some finite value we will see a change in the amount of generalized work that is needed and now of course this measurement of the work needed to drive the system from t equals zero to t equals 0.1 could therefore be used to test whether or not we have produced some pairs in our driving process and similarly of course we can then work out all sorts of higher moments of the generalized work and of the magnetization and constrain the probability distribution of our pairs produced in the driving by measuring generalized work of that system now I should add that all of these results are still preliminary we haven't really published any of this yet but what we can see here is actually if I go back is a very good agreement between the generalized work and the number of pairs that we've produced here between the two cases so it seems that this is actually already working very well now with this I would like to come to my summary what I hope I've shown today is that it should be possible to use extended quantum fluctuation relations to track the formation and destruction of charges in quantum quenches so if one has some conserved quantities that have produced some non-zero value for conserved quantities then one might be able to track their evolution and their lifetime and their destruction and creation through measuring the work that is needed to drive the system out of this kind of pre-thermalized state one can hopefully use this also to detect pre-thermalized states because if there's a pre-thermalized state then the standard of the Szynski and the Sarkic Rooks relations will not hold anymore and that means it should be possible to identify them through measuring work needed to drive the system and also to identify conserved degrees of freedom that one might not know about so if one measures actually English faith and assumes that the system is thermalized and then finds that there is a discrepancy between the Iazinski quality and the Sarkic Rooks relation and the experimental results then one can try to start thinking about maybe there are some conserved quantities that one needs to include into the consideration to make this work and so hopefully be able to identify some of them and then of course by doing the same thing it should also then be possible to indirectly or at least observe dynamically induced symmetry breaking like I showed in the last part of the talk in the last example if we basically take this system and create pairs and they start to form a condensate of pairs then again this should constrain the system even further and violate even this generalized Iazinski qualities that I've shown but we haven't really shown that yet here are some references and I'd like to conclude by thanking the people who have been working on this in particular Jordi Moopetit who has been the leading senior poster from this Jonathan Coulthard who has been doing all the calculations all the numerical calculations and our collaborators Amanda Rilano and Rafael Molina in Spain and of course I'd also like to thank all the other members of my group postdocs and PhD students and of course most importantly all people who give us the money to do what we do every day alright then with this I'd like to thank you very much for your attention, thank you