 for me from South Africa. Okay. Thank you. Dear Ganesha, thank you for your invitation. And as my title of my talk suggests, I'll be talking about some unified approach, which as you can see, Lindblat is a part of approach which is familiar to you. And non-termission is somehow, for some reason I didn't see much of the non-termission dynamics during this conference. So I guess I would feel that they're going to learn quantum open systems. Let me switch off my camera. All right, so term open quantum system usually refers to some quantum system, which is surrounded and interacting with background. And obviously if we're talking about standard quantum, mechanic quantum field theories, of course they try to incorporate the background into the picture. So to make the whole system plus environment to be unitary evolving. Unfortunately in many cases it's not possible because environment can be very complex. Therefore we need to have different models of environment and here comes the quantum mechanics of open systems. Obviously there are different models of them. One could select out the most popular one, such as Boson-Barth models, less known quantum chaotic models, random matrix models, et cetera. They have all of these environment models that have a problem. So such that analytical complexity, obviously many body systems. If you do numerical computations, then in both cases your time computations would be proportional to exponent to the of N for N body systems. Number three, the problem is a lack of generality because you always, you have to postulate the environment from the beginning and your results. So there is no guarantee that it will be general or anyhow universal. And number four, sometimes actually, especially for practical applications, you don't need to full measure of environment and body environment systems, but we need to introduce some sort of decal dissipation into model and see how it behaves compared to experimental data. Then you tweak your terms and then you probably gave a better idea about your environment, which is pertinent to your particular system you're working about. So it's actually, it could be useful if you try to reproduce or restore from the model to reconstruct the original nature of your environment, as well as subsystem, of course. Therefore, the reduced models are very popular where effects of environment can be encoded in the correction terms in the Hamiltonian or evolution equations for the reduced density operator, such as example would be limblet master equations to be considered later. Common features of this reuse approach is that your total Hilbert space is a product of Hilbert space for your subsystems and your environment. And then you introduce the average of the density operator over environment degrees of freedom. And that's where you arrive to the reduced density operator for gravity will are gonna be dropped as subscript F. As for your total Hamiltonian that obviously leads into two parts and you are, since you are interested in your subsystem Hamiltonian, this is the one you would be most interested in to get analytical or by any other form of it. Okay, so first example is the windblot approach where actually it turns out that dissipative effects are primary encoded, not in the Hamiltonian, but in the some additional terms of evolution equation for the reduced density operator. And one could play with some mathematics, standard textbook exercise. So windblot approach, you separate your subsystem from the background, then you impose Markov and Bourne approximations, then you impose our rotating wave approximation, alpha, color, et cetera, plus do some auxiliary simplifications. And that way you arrive at the master equations, which looks like quantum phenomenon equation with the commutator part plus dissipator part, which is in a quantum dynamical semi-group form has this form where capital A's are called windblot operators. So this is believable when you have a background or environment then it will contribute, or it will definitely contribute partially into your Hamiltonian, but most of the contribution will come through the dissipator term, which is trace less. All right, so can the windblot term account for kind of this particular phenomenon and question number two, can this term account for all Markovian kinds of backgrounds? Unfortunately, it's answered to both questions, as you know, and if you can see from this picture, so you see that windblot approach is basically subset of secular approaches and subset of Markovian and Bourne approximations. So it's only upon assuming all this chain of approximations you eventually can say that your system described in windblot, it can be described by a windblot master equation. All right, so there is a, on the other hand, there is a totally different approach which was, which became probably popular since the classical work by Feshbach in 1959, who in nuclear physics, which related non-hermitional Hamiltonians to open quantum systems. Well, non-hermitional Hamiltonian is a Hamiltonian which has both self-adjoint and pure joint parts. And Feshbach found that in principle, if you take almost, well, quite general system and if you decompose into the bipartite time, then after some algebra, so you have your subsystem like a shell model plus interacting with all kinds of scatterings, and then scatter in stable environment and your system is described by some discrete states. So eventually your subsystem will acquire additional term, which is not only a non-local, but also complex. So it also energy dependent. So which means that actually you have to develop some very new formulas to deal with such systems. And let me explain you why. Let's try to start with density-operative formulas. So we walk a little bit by analogy with the Lindblat approach. So we say that our Hamiltonian is acquired on a pure joint part, which can be written as a some Hermitian-operative times imaginary unit. You can write down, of course, by now, like we're showing a picture evolution equation for the state vector, but the problem is if you try to apply a showing equation directly to the system, then you'll have a lot of contradiction because on the left part, you have a clearly Hermitian operator. On the right part, you have a non-Hermitian. Therefore, you have to be a little bit more smart about this. So instead of dealing with Schrodinger equations, you have to switch to the density operator. It's also good because it allows inclusion of mixed states, better connections, open quantum system, et cetera. To do that, you do a standard procedure. You formulate a joint equation, and then it's reduced operator, which is product, and then multiplying and adding these two equations together, you arrive at this equation for your operator omega. And then you generalize this omega to any arbitrary density operator. All right, so what we have in this equation? We have obviously the standard front-line commutator term, where H plus is the Hermitian part of your Hamiltonian. And then we have an additional anti-commutator term, which is basically contains information about your environment. And then you see, unlike this equation, it's self-consistent here. Your omega is still Hermitian, and all the structures, it has a well-defined mathematical structure. Well, why it's not the full story? See, if you take a trace of both parts of this equation, it turns out that your trace is not conserved in time. And so because it's derivative, it would be proportional to this trace of your anti-Hermitian part. Therefore, you have some problems with probabilistic implications. So as the trace goes to zero infinity, how would you interpret it? Also you have certain problems, how to define purity. And then you have some energy, you lost your energy gauge invariance. So all these three problems say somehow make you not happy make before this reason of this non-Hermitian systems were not popular, they're not become popular immediately, although they appeared as early as the 60s. Well, apparently there is a solution to these problems. What you have to do is you introduce a personal organization. You have to introduce your operator, which is measurable, we call it measurable. And because all the averages are computed as traces for this new operator. And this is your omega or omega divided by each trace. And that way you eliminate problem number one, you can problem number two, because you can define purity in some way. The only thing that's interesting is that your equation for omega turns out gives a rise to the equation for rho. And rho has a very similar structure to the older equation. Plus this new term, which is something new, it is non-linear because it contains average of your anti-Hermitian part of your Hamiltonian over your Hilbert space of your subsystem. And for example, if it's a full X space and it's a integration over the whole configuration space, and so it's non-linear because its gamma contains also rho inside and it's non-local because it's in general contains all those integrals over the configuration space. But it's always manageable because you can always use this equation and that and go back and solve it and then go forward and check what are the physical consequences. Okay, so this is actually was something you wish somehow people haven't really think about before 2012. And since we have a Lindblot approach, the old good one, and we have also this new approach that it's also tempting to create a hybrid approach when you have both terms, both the Lindblot term, which comes from basically Markovian dynamics plus all those approximations I mentioned before. And this term, these two terms, which actually come from top three, they don't use assumptions like Markovian or anything because they introduced a little opportunity to go beyond the Markovian, maybe this is part as well. We don't know actually that because you probably know that and from previous talks actually, the criterion for non-Markovianity are quite vast and sometimes they're not even clear which one is which to be applied. But so we in the meantime, we actually quite happy to apply to various systems and let me represent you, give you example of some application. Two level system, such as two level atoms sitting in environments. So we have a two level open quantum system. Obviously, this is one application is two more single atomizer, but it can kill second-level robot approximation in case of those two well Hamiltonian or two well potentials or systems where the distance between a bunch of levels is much larger than the inter-level distances. And let's introduce as an example, maybe pedagogical, let's introduce some technician Hamiltonian. So let's start with the original Hamiltonian which describes bare two level atom and a dipole interaction with electromagnetic field. So it's well known textbook Hamiltonian and let's do a trick. Let's move for this parameters omega note and omega into the complex plane because well for starters, there are probably some physical values and nobody told us from the beginning they should be real valued absolutely. So let's do a little assumption and then relax those real value requirements into the complex domain. And then that way we actually obtain the anti-firmation part which we can safely add to Hamiltonian. And in here, we immediately see the part which has a physical meaning of spontaneous emission which is quite popular in this models which study the curve at and also dissipation of other constant alpha which is imaginary part of omega would be dissipation of that over interaction. So we have created quickly this simply this extension of our original Hamiltonian. And then we, so total Hamiltonian would be our self-adjoint part plus this one, this part which is the spontaneous emission and dissipation and evolution equation would obviously contain this part. And the dissipator can come standard way if you remember in real-world approaching in quantum optics, dissipator has usually this from this plus this one. So all together and remember we also need to switch to remember that our observable so to say dense separator must be actually rho divided by trace rho of this one. Or alternatively we have instead of these two equations we have this just one equation. All right, so with this picture we have a, we create a system evolution equation when you write down as a differential equation. This is our commutator part, anti-commutator part. These two terms is your limb blood part and other remain three terms is also non-hermitian part. So we play, it's from optics textbook are rather plain to a two level is theoretically impossible because we know the population version it requires positive- Konstantin, if you can go towards the end because you have two minutes more. Sure, I'm almost done. So you know that the two mode laser is impossible for describe two level system but interesting that the semiconductor lasers they actually they are approximately two level system. So because they have instead of two levels they have a two bands. So the electrons are happily jumping between all of them and producing a coherent radiation. So the question is how to, how to make a, how to make a model which would be relatively simple yet accurately describe this kind of approximately two level system. And then yes, you can turns out that limb blood type of model would be would probably have problem with it because they as always produce negative contribution into the population difference. And but if you add non-hermitian terms with your Hamiltonian that actually your population difference can be made positive. And therefore you, those models can be used to model those approximate two level lasers at a very low, at very low computational cost. Okay, it's also well, there's also not additional effects which can be seen in the lasers like such absorption emission time and estimatory I'm gonna keep it and go to the reference. So the, basically the talk is basically on this article of me and Alessandro Sergi from University of Messina. We had some applications, we had some community. Thank you very much for your attention. Thank you very much for the talk. Are there questions here? Okay, a line seems nobody's a question. Are there questions here? So I have a question myself. In, so in the, in the mixed formulation can you engineer your limb bloodian and your non-hermitian part of the Hamiltonian in such a way you ensure convergence towards Gibbs measure or things like this? I mean, you, in such a way that you can study the equilibration of your system towards a thermal state. Konstantin, is this the answer or? No, I think the connection is lost. Hello? Konstantin, what should we do? Yeah, yes, he left. No, I think let me check. He left. So it was really hard question. That's the advantage of online conferences. You know, you disappear and that's it. I don't know if you will join it again. Well, if not, I think we can go to, we can thank him again, even at least morally. And then essentially we reach the end of.