 from Newcastle and he would talk about irreversibility in the driven dissipative optomechanical system. So now we make the change to quantum dynamics, I guess. Hello, Eric. Hi, yeah, we can hear you. Okay, and the slide is okay? Yes. Yeah, okay. Yeah, thank you. And then the organizers are for inviting me to give this talk at this workshop. And then I just to put also context as Edgar requested. I am a former, ICTP diploma students and originally from Nigeria and I did diploma program over 10 years ago now. And afterwards I proceeds doing some master's in nanoscience which took me to universities, Thomas University in Sweden, and then also Devth University in Netherlands. And I did my PhD in Germany with Eric Lutz, where I started in Osborg, but completed it in University of Elangen. So that's me and then this is just basically also to motivate the students in the audience and then also the diploma students that you can actually do it. So if I have been able to do some of the stuff up till now. So today I will try to talk about driven dissipative quantum system, in which you have some reservoir attached to rates. And this idea of this is important as we have been hearing over this workshop classical system and then also biological system. There's always this irreversible protocol and the understanding them is basically a fundamental problem in physics. So today's outline of the talk will proceed as follow first. I will give just a brief, in the production, why do we have to care about irreversibility and how to quantify it. And then the second part will be, try to give a concrete model and just go a bit through how one can actually quantify this parameter for an optomechanical system which contain in nonlinear. So, and the third part will be basically when you'll send this to more condensed matter like kind of system, the magnomechanical cavity which is becoming quite a real facade moment due to promises it's offer in quantum information and all that and I will conclude by giving some outlook. So first, whenever you think about irreversibility this is usually compute by the entropy production in thermodynamics and we don't need to go into this as I think we have had a lot talk on associated with this but if we think just a thermodynamic process and the second law put into our shoes is that you are changing entropy is given by this inequality in which you have the changing heat upon the temperature and then entropy production is basically defined as the difference between these two and is equal to zero for reversible process and is always non-negative. But sometimes these are easy to calculate they are using the rates so in time. So if you just look at rate of change of the entropy in time, this gives you this expression where this first quantity is the entropy production rates and then the second one is what is known as entropy floss so under steady state both of them are the same. So then the next thing that one can ask is why do we care to study this kind irreversibility? As I already said is important in biological systems, power plants, but if we just think of basic thermodynamic system you can actually do just a back of envelope calculation to show that the efficiency in steady state just combining the first law and then the second law that efficiency is given by this expression here and where what this tells us is basically that the entropy production is one of 10 that makes us not to be able to achieve cannot efficiency or perhaps that it puts some constraints on performance of heat engines. And on the experimental side also in 2018 it has been realized or shown that this entropy production here can be measured in emissoscopic quantum device and then in that case they did this in two different kind of set of one is optomechanical system and then the other one is a driven dissipative system in which you have a BC in a cavity and then actuated. So the group in Zurich did that and then the group of Nikolai also here did that of the optomechanical collaboration of experimental groups of modern astro. So then just to give now more concrete model then first we look at just say an optical cavity in which is mediated you have a spring attached which can shift the mirror citing some form of radiation pressure interaction and to model this kind of system first is to look what is your Hamiltonian of your cavity and then you have to also know Hamiltonian associated to the mechanical mode which is coming from the spring and then the Hamiltonian of the interaction between the two. So and then this is also related to the zero point mode and then the coupling here is the radiation pressure like coupling. So but if we just move because this with adding some just a standard driving here that you will be able to describe the experiment that was performed by the group of Nikolai in Vienna. So, but if we now move maybe consider a situation in which we have not just a driven optical cavity but this cavity also have a non-linear crystal. So which induces some sort of non-linearity in the system. That means that in addition to just having the standard driving Hamiltonia which is related to the sequence of the laser pump and then this is the strength of the laser laser and then this parametric amplification which a side crystal is having a squeeze form and then this results in a squeeze cavity. So sort of to say that this is parametric amplification induces some squeezing in the system and it has been shown that this actually enhances cooling and then entanglement and some other properties. So, but here what we're interested to look at is like how does this modify the entropy production in an optomechanical system. And to proceed them first is to write the Hamiltonian in a rotating frame is given by the first equation here where the delta naught is just the cavity detuning which is and then the omega L is the laser, the driving laser and then the omega C is the cavity frequency. And in presence of the environment then you can basically model this equation of motion and the resulting Langevin equation is given for A and B mode. The A mode is the optical cavity mode and then the B is the mechanical mode and then the copper here is just the induced the environmental decay rates and the gamma is now associated to the mechanical resonator. So, and if you linearize the system and then just take into account or you are interested in the small fluctuation that's on top of the mean field value then you can write the linearized dynamics in this form where the effective coupling capital G now becomes of this form two times the radiation pressure coupling and then the S which is the mean field here as there is the value of the system and then you now have the non-linear interaction which encompass the pumping of the nano crystal and then also the phase angle and then you can compute your amplitude as well as the phase angle which will be quite important for the analysis. So, if we will now introduce some quadrators and then also write the, you can write the the dynamics in compact form is where the capital R, the R of T is just the quadrature vectors of the two modes you are the cavity mode and then the mechanical mode and then the noise vector which you are described for different input noise and then the A is just the drift matrix which will some encompass all the components elements the system and then analyzing this equation basically tells you all the phases and then assuming that you have a two these two reservoirs or environments that are different temperature which results in breaking of the detail balance equation you can also just look at the non-equilibrium steady-state solution using the Luponov equation and by solving this in the steady-state then you can compute your covariance matrices and then this covariance matrix you can quantify different parameter bits entanglement and then the entropy production and others. So, if we now see to look at the entropy production then just following the framework put forward by Mauro and Mauro Brunelli in 2018 so this is just a diagonal matrix of the environment mode and then if you do the calculation then for two modes which is straight forward so what you get is the expression for this optical parametric optomechanical system is given by the expression here and then the first term is associated to the cavity mode while the second term is associated to the mechanical mode so the N of B here is just the thermal occupation number of the mechanical mode at a given temperature and with one look at a steady-state so if the system is a steady-state what you get is that basically this covariance matrix up here is just equals to your denominator and then you have that entropy production is zero. But if you write the full expression which we can do in this case I didn't want to go into that because yes I'll show you that because it's just you have to cumbersome and then you don't get much information maybe out of it but one thing that the parametric amplification does is that even it always gives you entropy production that is greater than what you will get without the nonlinear crystal in the setup so if we then look at some plots then the first plot here is when you write the entropy production rates against the effective detuning in the system so what you see is that as you increase the nonlinearity then the entropy production increases so and this also even if you figure D here is when you have many number of photon so if you have many number of photon in the system then you basically increase also the amount of the reversibility you will get in the system and this is controlled by the phase of the of the parametric pump that you use so and you can see that clearly from the figure B and the E here so if we we just scan through different phase you can see that you have some phase which correspond to maximum value of maximum value of entropy production rates and also if you go have more number of photon in the system you can even have some regime where you are actually below what you will get for a nonlinear crystal so in principle with this one can say entropy production is modified by the presence of this nonlinear crystal and then by tuning that you can actually maybe parametric range which will be good for operating either your heat engine or your rectifier so the next thing we it also was to look at what happens if we increase maybe the amount of the decay rates so and what you see here is that the more you increase this okay then you can have orange but there is always a value at which it deviates and this the decay rates of the cavity is sorry when the decay rates of the cavity is equal to the square of the amplitude and then multiply by the square of the phase so in principle you can have this modification over a given time but once you always be know that you once you get close to this value then these two parameters becomes comparable then the system becomes unstable so there we also look at the correlations in the system because this system is basically Gaussian states so it is easy to compute the mutual information as well as quantum which is good because it can actually get the correlations beyond what one can get understood from the entanglement and also there is some sort of relation between the quantum and then entropy production so from our analysis then the first plot on the left is when there is no crystal in the medium and what you see is that there is almost one to the blue line is the information and then the red one is the discord so I see there is almost one to one mark between mutual information and then the entropy production so but as we start to inject or have some crystals in the medium then you start to have some deviation and the remarkable part is like close you have very small detuning then you see that while the entropy production is going down then there is some increase going on in the information what mutual information as well as the discord and this is basically yeah later to at this point what we felt is where this you have yeah this breakdown of yeah the breakdown in your cavity strength being almost equal to the square modulus of the amplitude of the pump so we further extend this it's now past you know past the 20 minutes so just let me know so then we further look at your situation okay where we have maybe your three different kind of system so this Magno mechanical cavity model so just to give you a pictorial understanding of this is just having phone on your couple to your Magnon and then also the Magnon is coupled to your photon and there's some driving a citation of the Magnon mode which in principle you can be tuned it even to go just not being a splitter but some other kind of a coupling and then the third term here is the coupling between the phonon mode and the this is the coupling between the between the atomic mode and then the Magnon so and this is the driving where the last term basically just to take into account the noise components as well as the external driving force we're seeing here in this picture and if we just do the same framework again and then couple our yeah the quantum finding right the equation of motion and solve the quantum Landgeverne equation the only difference here now instead of having four by four matrix then we'll be having yeah six by six matrix and then solve for the entropy production you can compute every other system so then I just show you the results for the entropy production here so where we plot this against the effective detuning of the Magnon mode and then what you see here for different coupling so photon Magnon coupling so the black line is when there is no coupling between the photon and then the Magnon and what you see is that as you increases yeah this is for small value of phonon number and what you see is that as you increases the coupling then you have quite big modification in your system and the second peak we see here is from the hybridization of the two mode so and they have not only shift in this peak and then you have also like this appearance of the second peak so and if we now also look at for high value for non-occupation which is the figure B here you see there is large number of coming the system and then your entropy production rates is high but the system is almost same but a bit different because what we are looking at here this figure B is having is plotted with a pointer one of the gamma the environment decay rates of the cavity photon cavity so then if we look at the C then what we did is to change the coupling of the cavity mode and what you see is that there is still some different behavior but it is not the same as we have it when we have a small decay rates of the optical mode and then the D is basically just when we have high number of mode and then also for this same kind of coupling and other parameters are just in the experimental range so with that I come to the conclusion of the talk and what I have tried to show you is that non-linear cavity which basically transcribers with cavity modifies entropy production and secondly when you have magnum in your system you can also modify your entropy production rates and this will be useful from what we have seen since looking at this figure that we can go from region where you have maybe increase to decrease within the same side band so it will be useful in using this to study how this can be done and how this can be done and also modeling quantum heat engines and see how this can really can play in enhancing performance in a refrigerator or heat engine and in North Shell this study does show you the interplay between the quantum and the quantum and how this can be done and so on and so on and so on and so on and so on and so on and so on now what is the hematonic express in a rotating frame and does this validity persist when the ground that system is altered specifically does actor be production any changes I think there is no this is just basically for easy calculations so it doesn't change so but yeah you want to write your then you can be able to write this drift matrix or compact and having also good interpretation. But yeah, it doesn't change because the effective detuning that you have, when you check the range of feet with coupling, then you see that it has little effect to the bare detuning of the system. Okay, thanks a lot. I hope that clarification helped, I think. So, we are a little bit tight on time, so further questions, please keep them in mind for the discussion section and let's thank Obina again for the nice talk. Maybe Nicole, you know how one can share her screen over these.