 Starting to record the session right now. I believe it is being recorded. The record button just disappeared. I assume it's being recorded. Okay, so with that, let me introduce the first speaker Ariane Sauré, who will talk about thermodynamic consistency of driven quantum optical master and I will all let you know after 10 minutes that you've come to the 10 minute mark. Okay, thank you. All right. So should I share my screen. Yes, please. And I'll start. All right, let's do this. Okay. Here we go. So can you see my slides. Yes. Yeah. Okay, so hello everyone. Thank you for the introduction. So yes, there. So I will be talking to you about the consistency of optical master equations, a project I recently did at the University of Luxembourg with the Massimino Esposito. And it should be on the archive soon under the name thermodynamics of atom photons interactions near residence. So slightly different name. All right, so the main focus of the talk will be to discuss the coherent energy changes between photons and atoms near residence, but before diving into the core of the topic. Let me recap some basic interactions in electrodynamics. If you consider the interaction between photons depicted here by a dotted line and atoms represented by two level system with a ground state in an excited state, you have three basic processes. First, the photon can be absorbed by the atom and then the atom goes from the ground state to the excited state, or a photon of wave number K can be emitted by an atom. And then you have scattering processes in which an incident atom of wave number K is absorb and remitted with another wave number K prime. So the thermodynamics of these processes are well known at equilibrium. So light at equilibrium with matter is described by the black body radiation. Basically in this setup where you would have some matter so an atom here inside a cavity with with a radiation at equilibrium, the exchanges between the atom, the energy exchanges between the atom and the radiation would satisfy the first law where the type of the energy is just a heat. So this is one well described. Things become more complicated when we add a driving field a laser, which drives the system out of equilibrium. In that case, on top of the spontaneous absorption and emission processes which happened between the atom and the, and the photons at thermal equilibrium, you have now stimulated absorption and emission from the laser source. And the question is, what is the thermodynamics in that setup far from equilibrium. Usually, later drives are described by an external time dependent Hamiltonian, which I wrote here this term VAL for atom laser interaction, and it's time dependent, and quantum thermodynamics predicts that the work performed by that laser on the is given by this expression so the rate of the work is given is related to the time derivative of the Hamiltonian. Now the problem is that when we go near resonance meaning when the frequency of the laser becomes really close to that of the atom, and in the strong complaint limit that first law can break down. There are two reasons for this. First, when, usually when doing the semi classical descriptions of the laser, the number of photons in the drive are assumed to be constant, whereas they do this number does change because of stimulated absorption and emission processes. Second, the strong coupling between the laser and the atom can create coherences, which need to be accounted for to have a proper thermodynamics description. So the topic of the talk is to address that question. Before diving into the thermodynamics, I will give you more thorough description of this of the physics of atoms coupled to coherent light. So, the first assumption we're going to make is called the long wavelength approximation, meaning that the wavelength of the at of the of the laser photons is assumed to be much larger than the size of the atoms and molecules, or in this case just at the larger than the size of the second, we will make the dipole we will use a dipole representation for the atom. And in this representation the interaction between the field and the atom takes this simple form here. We will furthermore assume that all the radiation here is quantized, meaning that the radiation field is written in this form, where the a and a daggers are creation and annihilation operators. Finally, we will only focus on the case where single atom is interacting with the photons meaning that we can choose that atom to be at located at r equals zero and remove any space dependence in our interaction Hamiltonian. So the basic setup under study is this one represented here so we have on the one hand, an atom which is in a cavity where we have some thermal radiation, and all of that is also driven out of equilibrium by a laser beam. So I highlight at this point that maybe that picture can be a bit misleading, because here, the laser is described by Hamilton, Hamiltonian which is independent of time so what I actually have in mind here is simply described that omega L by a single which is in a coherent state. The total Hamiltonian of that system has is written here has three parts hx describes the atom plus laser field. So hx itself is decomposed in h a which is the Hamiltonian of the atom, hl which is the Hamiltonian of the laser so written here as a single mode of frequency omega L. And the L is the interaction between the atom and the laser. All of that is also coupled to a thermal bath, HB and coupled via Hamiltonian Vx B. The first assumption we will make is to assume that the laser is nearly resonant with the atom. This allows us to perform the rotating with approximation, and to simplify the interaction between the atom and the laser, and to keep on either non of resonant terms. With this approximation, the Hamiltonian hx becomes black block diagonal in the basis, and G and plus one so what, what does this notation mean this is the joint basis for the product tensor take a space of the atom and laser. It describes the excited and grand state of the atom, and is the fuck basis of the lasers. In that basis and under the rotating approximation, the Hamiltonian for the atom and laser can be written in that form. I regroups in the first matrix here the atom and interaction terms for reasons that will become clear in a minute. And the second term is describing the lasers, the, the photons of the laser. The first assumption is to say that the statistics of the, of the drive of the mode L satisfies this condition. This is the case for instance, if the mode L is in a coherent state. And what this implies is that we can replace in this interaction term here that we can replace the square root of N by the square root of the average, and replace that term G zero times growth of N by an average G. In that case, each block of the, of the Hamiltonian hx can be diagonalized using the same unitary operator. So, it will be easier to understand this procedure, looking at what happens in the on the right here on the basis. So, when I diagonalize the Hamiltonian hx, I can rewrite it in the form here on the left. Okay, now I will explain where does the two, these two terms come from here on the left hand side of this diagram you have represented the initial joint basis for the atom and the photons. So those states GN plus one, et cetera, and represented in two types of dotted lines you have the two processes that play on the one hand stimulation emission absorption by the laser, which connects states with EG with a variation of one in the number of photons of the drive, and then the spontaneous emission to the cavity, which connects states where the atom is excited or in the ground state, but with the same number of photons in the drive. Here we diagonalize the Hamiltonian. Since the diagonalization is happening in each separate subspaces here, the egging basis here is only constructed by a linear combination of the initial EN plus one gene plus two for instance. But interestingly, now the spontaneous emission process couples egging states from those different neighboring Hilbert spaces. And instead of having a single frequency omega q you have now three different frequencies that you can observe in the emission spectrum, called the fluorescence triplets. Now, back to the picture on the left, how do we get there. So if you want to diagonalize your Hamiltonian HX you notice that you can do the the following mapping from the initial Hilbert space h a h l to one of DADL what so what I call DA is for the dressed atom. It's the atom dressed with the photons from the drive and DL would be the lasers but dressed with the atom in the sense it's the laser slightly modified with the presence of the atom. And then the initial Hamiltonian HX is a sum of two terms, one describing the dressed atom, the other one the laser, but you see that the interaction term is now completely removed. Okay. So after this long introduction to the physics of play, I will, with the time I have left, talk about the thermodynamics in this setup. So I will begin by deriving a fluctuation theorem for the. You have you have about five minutes and that should include questions. Yeah. Oh, okay, so it will be I will go as fast as possible but I think the main point was to try to understand the physics at play, which I have done now so we want to study now the energy transfers between this coherent state and the dressed atom to do this we can do we can use a double point measurement technique with counting fields. We don't have that here but what you just need to remember is that with this procedure measuring the fluctuations of some quantity a can be done by computing the moment generating function GL, which is the trace of obtain of this density matrix row which is the solution of a tilted dynamics where we dress the unitary evolution operator with the counting fields of the quantity we want to measure. For example, measuring the heat amounts to putting a counting field on the Hamiltonian HP. Now, if we're if we want to measure the all the energy changes between the dressed atom the dress laser and HP since they all commute we can measure them simultaneously and define a joint generating function. So when the laser is in a coherent state so coherent state is defined by this depends on a parameter alpha, which is related to the average number of photons and its variance. If we have a coherent state where our files very large, then we have a state which satisfies the assumption to that were required for the for the setup that we're interested in. In terms of functions you can notice that, in fact, the entropy of the coherent states divided by the energy of the field is in fact negligible, which means that we can consider the coherent state to be a work source. And using that condition here. So this is a simplification theorem in terms of symmetries of the generating function, which is expressed here so this is relating the reversed generating function to the forward one. And the condition is that the symmetry for the bus is that you should shift the counting fields with the inverse temperatures, but if you have a work source. The symmetry is that in fact, there is no no shift required. The interesting point is that if you also impose a strict energy conservation, which can be expressed in terms of symmetries of that object as well. Combined with the initial, the previous filtration theorem you can derive a crux relation for this specific coherent drive. All right. You have about a minute left. Okay, so I will maybe just give you the conclusions. And what was useful with this framework is that then we can describe which master equations are consistent or not. And the conclusion is that we reexamined the there are two main optical master equations that are used in the literature the floquet and the block one. What we find is that the floquet master equation is invite fully consistent so it satisfies this fluctuation theorem, whereas the block breaks this fluctuation symmetry, but is consistent on average. So that would be the maybe the technical message coherent light access a work source, we can derive a fluctuation theorem for such sources, the floquet master equation satisfies the symmetry but not the block one. So that would be the semi classical limits, semi classical limits in which the laser drive is described by a time dependent Hamiltonian do require to make the born approximation between the atom and the laser. And that approximation is valid for the block equation but not for the floquet where the floquet master equations in fact correspond to strong couplings. Just jumping here to the conclusion is that what is interesting is that the block equation although it is not consistent at the fluctuating level. You can, it gives you the correct thermodynamics on average in both pictures in the dress atom picture, or in the semi classical limits where the field is treated by an external drive. Leave the slide of conclusion for some questions I apologize for the rush towards the end. Thank you very much we have time maybe for one quick question. So, maybe I'll have one so you wrote down the these two different versions of the fluctuation theorems in terms of a generating function and then a crooks like one. Can the work actually be, I didn't quite catch it can the work actually be measured there in terms of some scheme or other otherwise. Exactly that was the point that the work is in fact obtained by measuring the Hamiltonian of the lasers, and you could either you could either measure it in the dress atom picture so measure this HDL term. You could either measure also in the in the original basis and measure HL and then if you just may measure the HL Hamiltonian you would recover the expression of the work that you would also get. If you were computing the time derivative of the of the semi classical Hamiltonian, but you get quite different expressions if you if you compute HDL instead, but in both cases it corresponds to yes, actual measurements of the Hamiltonian. Thanks. Thanks for clarifying that I think thank you for that very interesting talk, and we need to move on. We don't have time for more questions. We need to move on to the next speaker.