 Okay, so good morning everyone. It's a pleasure for me to be here. Thank you very much to Joy and to Mauro the organizers For inviting me today. It's a pleasure for me to talk to you about the dynamics of open quantum systems I've heard that this is quite an informal set So please please interrupt me when there is something you want to ask or you don't understand and I'll be happy to to to try to To to put more details into my explanations This is a kind of an introductory talk or course Therefore, I'm gonna kind of touch a lot of different Realms where open quantum systems appear and and the main aspects of their study And I will not go into depth into any particular example So it will be about the theory of open quantum systems But I'm very happy of course again to to be interrupted. So Okay, so See if I can before starting I would like to thank all this bunch of People with whom I've had the pleasure to collaborate or to discuss extensively during during my research years and Also, I'd like to thank these people that are master students From last year and from this year in my group and to Carlos Parra Who is now a postdoc in my group coming from Colombia and When they are really I would say responsible of the best ideas I can teach to you today Thanks to them. I've been able to learn a lot Okay, so before starting what are the questions that come to our mind when we think about open quantum systems So open quantum systems are quantum mechanical systems that are coupled to some environment And therefore some of you may have heard of the word the coherence Defacing the coherent has to do with the loss of superpositions in the eigenstates of the quantum mechanical systems and dissipation has to do with the loss of energies From the system into the environment and then we have a set of questions like for instance If we couple an open system to a reservoir to an environment that is in a thermal equilibrium Does it always thermalize or not? And then for instance, we may ask where is irreversibility coming from because after all when we think about Quantum mechanical system is described with unitary evolution equations with Schrodinger equation or von Neumann equation Which are a reversible completely. So where is irreversibility coming from? And then we went to address the question us when we think about for instance a superconducting qubit and We say, oh, it's a noisy device. It's a noisy open It's an open system that is affected by a noise. So what is the Microscopic origin of such a noise? These are the questions. We are going to to address today This is the guide of the talk. So it's kind of Composed of eight blocks. The talk is all together around three hours We will make poses in the middle Well, I will start with very very general ideas What is an open quantum system? What is this? dynamical map so I will teach you Hopefully what a map is to those of you that are not familiar and then I will go through Microscopic modeling of both the quantum mechanical system and the reservoir and then I will discuss Statistical properties of the environment. I will also discuss the weak coupling and the Markov limit Which are the most important ones to deal with open quantum systems Then we will touch briefly what happens in the non Markovian limit I will explain why these fuzzy words are meaning for those of you that are not familiar and then we will see What happens when we want to actually tackle the full system plus environment dynamics? Which is kind of a huge problem as you will see and finally we will revisit the Microscopic modeling because as it turns out the simple models that we will be touching in the third point of the talk are not quite good to cover all the physical Instances where we find Open quantum systems. Okay, of course an open quantum system can be a molecule or an atom or Well, you name it or cavity mode or a set of them That Well in principle, you know in the origins of the theory of quantum mechanics people were saying, okay, let's Consider that this quantum system is isolated and is described entirely by this shreddinger equation Which depends on the Hamiltonian of the system? but as it turns out in in real setups what we find is that Quantum mechanical systems are coupled to environment like for instance when you think about atoms They are usually interacting with the electromagnetic field as a result of which they are emitting Photons or absorbing photons then you may think about this kind of quantum bronion motion setup in which you have a Massive particle that is moving in a solvent in a fluid for instance And it's been affected by the collisions of the surrounding smaller molecules And then you may think about also an electron that is in a lattice and is actually coupled to the phononic vibrations of the lattice and there you can Again think about the the electron as the open quantum system and the phononic field as The environment affecting the motion of the electron in the lattice another example of Of an open quantum system is when you think about for instance in a biological system in a in a quantum photosynthetic Complex you may think about an open system composed of antenna molecules that are receiving the light from the Sun and they are actually Transporting this energy from one antenna molecule to the other But as it turns out this whole transport process is occurring in the presence of vibrations of the surrounding proteins So it's kind of affected by an environment of phonons just as in the case of the phonons of a lattice and Then what it means is that we have to consider a Hamiltonian that not only includes the open system But also the environment and some interaction part And as a result of this what it turns out is that we have to consider a Schrodinger equation for the full wave function that includes both the open system and the environment and The bad news is that this guy here this wave function is living in a Hilbert space That is huge because it's kind of growing Exponentially with the number of degrees of freedom in both the system and the environment and remember that the environment can be very very huge and This is a problem because well, you know, we cannot compute the dynamics of such wave function In our laptops today or in big computers when we want to include the whole Hilbert space But we will see later on that there are tricks to do that by using what is called matrix product estates Some of you may be familiar with that The interesting thing is that actually we are only interested about the reduced dynamics of the open quantum system We are only interested about what is going on with the degrees of freedom of the quantum open system and The quantum imbalances of the open quantum system can be very very nicely described With this object that is called the reduced density matrix of the open quantum system Which is nothing but a trace over the bath or the environment degrees of freedom of this Projector of the total wave function So this is an object where we have already traced out the environment degrees of freedom But we are taking them into account It's not that we are forgetting about them But we are taking them into account in the sense that this object will correctly describe The effects of the environment namely the the coherence and the dissipation So the whole idea of the theory of open quantum systems is to be able to actually from the whole Hilbert space Describe the reduced dynamics of the open quantum system in its reduced Hilbert space So we want to obtain Evolution equations of the open quantum system within its small Hilbert space Which can be for instance the Hilbert space of a qubit or a bunch of qubits and this is of course a much smaller Mathematical space than that which is including also their environment and The key idea here is that we want to consider the environment But by but only by taking into account its statistical properties, and this is something we will also Be dealing with in in the next few minutes okay, so Let me go to the second part Which is to describe these dynamical maps, so let me Imagine that we have an initial state for the system and the environment which is The correlated states, so there are no correlations between the system and the environment initially This is a type of a state that we can certainly prepare and it is also mostly Well a very frequent situation that the environment is in an equilibrium state. What does it mean an equilibrium state? Mathematical is that it can be written as a sum of projectors over its eigen states This is an example of equilibrium state for instances give a state a thermal state in which case This lamb does depend on the temperature are there the thermal weights of of the environment state Okay, so then We can actually Mathematically without making any approximation Rewrite the reduced density matrix in terms of this quantity here, which is what I'm going to be calling a dynamical map This dynamical map is actually kind of a propagator because it's propagating The initial state of the system the initial reduced density matrix in time so it's actually a very very key object to describe the dynamics of the open system and As a matter of fact it comes in terms of these cross operators just to connect with the terminology that some some of you may know Which are matrix elements between? The environment eigen states of the evolution operator with respect to both the system and the environment and This object is kind of key in the theory of open quantum systems It has many different properties Which I'm going to discuss very briefly because they also connect with the physics behind these systems So in detail for instance, it is an object that will describe Very nicely the fact that open quantum systems are irreversible But on the other hand, I would like to also mention that this map in general is invertible So you may tell me okay if this is invertible Then mathematically I can always Reach back the initial state by simply applying the inverse of the map since it's invertible. That should be fine But the point is that This inverse of the map is not a universal dynamical map It's not a map itself and this is what irreversible means So but what does it physically means is that we cannot fabricate a weird? Environment that have as a universal dynamical map site T minus one Such that it can return me to the initial state So this is what irreversible mean and as it turns out I can only do that when Actually the open quantum system is a closed system. So actually when the map is just the unitary So only in this case not only you have an inverse of the map But also this inverse of the map is a map Namely the unitary evolution operator if you have an open quantum system and this map is not a unitary Evolution operator then the inverse of the map is not a map and this is what it means to have an irreversible evolution Okay, so another Property which I'm not going to be talking about is the fact that you can write them up in different Representations and you can write it in terms of what is called a choice matrix And you can write it in terms of many different Operator basis for the open system, but this is kind of more mathematical Mathematically oriented and today we don't have time for that, but what is important is that? Mathematically from this map we can show that a time-local master equation emerges and what does it mean is the following? So a master equation is an evolution equation for the reduced density matrix of the system Okay, and now if the inverse of the map exists we can always write such an equation simply by Deriving the the map and applying the inverse of the map. This would be size 0 Okay, and what is interesting is that one can by simply using the properties of hermiticity and trace preservation derived mathematically an equation which is having this form which is a time-local master equation for the reduced density matrix of the system and It turns out that this equation comes in terms of a first term that describes the the let's say The free evolution of the system without environment. This would be If you were to have only this term that would mean that your system is actually Independent from the environment simply evolving according to its Hamiltonian but now you have this other term that describes the dissipation under the coherence and This term is quite complex. It has as many components as system Transitions so it really depends on the dimensionality of the open system D And it depends on these guys here which are decaying matrices So these are kind of dissipative rates that are going to describe these these the coherence and dissipation process and By the way, these quantities here these G are operators of the system like for instance if the system is spin these guys would be For instance the Sigma the Pauli matrices Sigma X Sigma Y and Sigma set and well, this is just what I what I said and You can actually diagonalize this equation and put it in a Limbland form some of you might have be familiar with with Limbland equations and this pretty much looks like a Limbland equation where the Limbland operators are this L case But the key point here is that this is our mathematical representation of an exact evolution of a quantum open system The key point here is that both the decay rates and the Limbland operators are time dependent So it's not a Limbland equation properly where these two guys are time independent But it has a Limbland form and moreover the let's say the important question here in the theory of Is to come up with the actual form of these coefficients and these Limbland operators? This is a problem that is not solved in general So you may have some examples in which you can actually know What is the form of these coefficients and these Limbland operators? But in general you can only access them once you consider some approximations And as you will see later these approximations are related to having a large separation between the system the open system relaxation time scale and the environment time scale of recovering from the interaction with the system This separation of time scales is related to the weak coupling Approximation that we will see later But the important thing here is that we have this mathematical form, which is very beautiful Because this is telling you look No matter how Complex your environment is I don't know because I have not yet told you which environment I'm talking about What I can tell you is that if its map is Invertible which is a kind of a weak assumption because most of physical maps are invertible. I Can't tell you that the evolution of the reduced density matrix of the open system will be a time lock We'll have this particular form Okay, but what about non-marcovianity measures some of you Maybe experts on the topic. I don't know or some of you may only have heard about non-marcovianity and non-marcovianity measures and I'm gonna say a few words in this respect. I'm gonna focus on Nonmarcovianity measures that are based on the divisibility of the map Well, I'm gonna change a little bit the notation of the map now instead of putting a sub-index T I'm going also to refer to the initial time of the map So this map is going to propagate your state from a time t equal to zero to t and Well, it turns out that the map is set to be divisible And we will see how important this is for applications in the next couple of minutes If one can really Divide or split an evolution from zero to t2 into two pieces From zero to t1 and from t1 to t2 and both pieces are universal dynamical map This is what this acronym comes from These two pieces have to be maps have to have the nice properties of This initial map that we are splitting By the way, this is a property that is only fulfilled according to to the proposal of Rivas, Wellgas and Plenio. It's only fulfilled By Marcovian evolution. So it kind of implies in a sense a short of lack of lack of memory of the environment From the system dynamics the environment forgets about the fact that it has Interactive with the system roughly speaking But there is a stronger property than the visibility which is that the map is a Dynamical semi-group and this is the property that is linked to the Limbland Evolutions and these properties much stronger because it tells you that not only the map can be Splitted like this, but it can all actually be a split it into these two Pieces that well that actually have the form of an exponential over a Limbladian times t Or let's say for a map to fulfill this divisibility property it should have this exponential form and Well, as you can see this is an scheme of what happens along the time when the map is divisible and when it's a semi-group When the map is divisible you can really split it it split it into different pieces Each of each of these pieces can be different with each other's But for a semi-group each of the these pieces are gonna be the same Are gonna be just the exponential of the Limbladian times delta t So this is a much more convenient type of property when we want to describe An evolution in terms of the in the case of a semi-group in in terms of this Subsequent application of dynamical maps on the open system of the same map and This actually connects with How useful the maps are for for instance describing a quantum information protocol now imagine that you have a qubit superconducting qubit and You want to apply certain Certain unit areas on the qubit along the time at the time t1 you apply one and at the time t2 you apply another one and In between you will have the free evolution of the qubit when coupled to the environment because the qubit is going to be Affected by a certain environment and these unit areas are those ones in the middle These ones are going to be unit areas with the full Hamiltonian which also includes the environment And now what do you want to see is what is the reduced density? operator Conditioned to the application of all these unit areas and considering the environment and What we know is that if the map is divisible Namely the environment is Markovian in the sense of Rivas, Huelga and Plenio Then we know that each of these pieces will be a well-defined map So you can actually well I kind of skip a part of the explanation But you can rewrite this trace over the environment of these unit areas in terms of the application of Pieces of the map to the initial state Inter inter wind or or interrupted by the unit areas over the qubit So this is an scheme of the time evolution You will first apply the unitary then you apply the map then the unitary then another map and so on and so forth This is what happens when it's divisible but then You may have that the map is even better because it's actually a semi group and when it's a semi group You know that each of these? Map in maps in the middle are going to be defined through the same Limbladian And this is what people do in many quantum information protocols that where they want to Consider the action of of the noise or the action of of the environment is really to consider a particular model for this Limbladian According to some physical process like the facing or bit flip or or in general unitary maps And describe the protocol just like this But this is of course Implying some assumptions that we will see later on So are there any questions so far? This can be bad or good. I don't know Okay. Yes, please Yes, please a Bit slower. Oh, okay Okay, I'll go a little bit slower. Okay, so so far we have seen properties of the map which are as you can see some mathematical object that evolves the initial state of the open quantum system and So far we have not made any assumption of how does the environment looks like so the environment so far is kind of a Terra incognita is something that it's kind of well. It's a very abstract It's a very abstract object But of course to have more idea of how does this Dynamical map look like we need to model the environment We need to have more knowledge of how does the environment look like and this is the third part of the talk We are going to make some microscopic modeling of our environment So we are going to use some Legos Actually, we are going to Well, this is again what we are going to do is simply to model the Hamiltonian that describes the interaction between the system and the environment Well, that describes the system the environment and the interaction pieces And as you know the basic building blocks in quantum mechanics are of two types mainly things can get more complicated than this of course, but In quantum mechanics, you can either have Systems that are composed by harmonic oscillators that are mathematical objects that describe or that have Equispaced energy levels Or you may consider So for instance molecules or atoms that have non equispaced energy spectrum Which are described with the spin operators and these are let's say our Legos our building models for for the environment In the case of harmonic oscillators. These are described by creation and annihilation operators whereas spin operators are described with ladder operators and as it turns out back in the 60s of the last century When people were starting to to think about Well, not a starting mainly that came back from before but they were really trying to understand dissipation the coherence So of course Feynman and Vernon I say of course because this guy was I think Responsible of many of the brilliant ideas that we are today living from So Feynman was Vernon. We're realizing that most of the environments that are affecting That are affecting the evolution of quantum mechanical systems are actually harmonic. They can be described with harmonic oscillators An example of such type of environment, of course is the electromagnetic field There we don't have to make any any assumption So the electromagnetic field is simply described as a collection of Harmonic oscillators representing the different modes of the electromagnetic field but also vibrations can be described by by phonons and So this idea come back from the from the 60s and was later developed by Caldeira legged in the 80s and Then people realize that there is also another universal type of environment or Canonical model of environment in which actually the environment is composed by spins And we are not going to talk too much today about them But there is a very nice review by Prokofiev and Stam and they they really have different properties than than harmonic oscillators environments and it also Was concluded that in most cases one can really assume that the interaction Hamiltonian between the system and the environment Is a product between some operator of the environment that is describing transitions between the environment eigen states and some some Operator from the system that is also describing some some operation in the system some transition in the system and This is very nice because this means that no matter how complicated your environment is you can always or not always You will see later But in many many many cases Describe it in particular in terms of harmonic oscillators and this is what I'm going to talk about in the next few slides So but let's let me go to Back to to the these examples that I was mentioning at the beginning Just to mention you that these examples are examples in which we have open systems coupled Precisely to harmonic environments to environments that are composed by a set of independent harmonic oscillators So in this case we have the electromagnetic field as I said before Which is composed by modes corresponding to Well to two different frequencies and and polarizations of the photons in this case you also can Simulate a solvent as a set of harmonic oscillators The vibrations in this case and this case are phononic vibrations, which are also Harmonic oscillators of or can be described as harmonic oscillators of bosonic type Okay, so just to give you a flavor of how are these Hamiltonians looking like? There you see the familiar light matter interaction Hamiltonian Where you can see in particular the idea that I'm talking about so you have The light matter interaction Hamiltonian is composed of a piece that describes The the the atoms or the molecules that are interacting with the light The light or the electromagnetic field which a particular dispersion relation omega k BK dagger and BK are the creation and annihilation operators That are going to create an annihilate or annihilate a photon with momentum k And then you have some interaction piece So you can see that this Hamiltonian obeys the structure I was talking to you from the beginning it has some interaction piece where you can have a linear interaction between ladder operators that describe transitions Between the ground state and excited the state of each of the J atoms because you may have a collection of J atoms and The creation of a photon so you have an interchange of Quanta between the electromagnetic field and the atoms which are in our case the open quantum system and Here in this other model where you have the particle in a solvent You now instead of having an open system that is described as a set of two-level systems In the case of having atoms You now have an open you now have an open system that is described as a particle With a certain momentum P and a certain position q So this means that your open quantum system is described by a continuous degrees of freedom which are position and momentum and This Well, this is the Hamiltonian of the open quantum system of a particle or this heavy particle with a mass m Which is very large that is moving in a particular potential and now is gonna be coupled to some to some Harmonic oscillator environment where which is described by some position and some momentum and Position operators which can of course be described in terms of creation and annihilation operators, right? So this is just another way to write a collection of free or independent harmonic oscillators with Label lambda describing each of them which is coupled through this particular term to the heavy particle that is moving and Yet we have another example which is that in which we consider a set of antenna molecules which have each of them a particular internal level or is described by Some basis state m which is described Describing the fact that the molecule or the antenna molecule labeled m is excited So this is a Hamiltonian that is describing the hoping this is a hoping term of an excitation within these antenna molecules and some some onsite energy term and These these antenna molecules are going to be coupled again to a field of in this case phonons That are described with creation and annihilation operators and by a dispersion relation and the coupling is again described with this linear combination or this linear product between the antenna molecule ladder operators and some position operator of the phonons So this is again another example of of Hamiltonian or situation that Lies in the class of Fröhlich model both the polar on and quantum transport models Okay, so I hope by now. I have convinced you that in general We can write the many different examples of open quantum systems in terms of this general Hamiltonian where you have the open quantum system piece the environment piece composed of or which depends on some creation and annihilation operators and Some interaction term that describes the coupling between the system coupling operators and the environment creation and annihilation operators and By the way, the coupling between the system and the environment is tuned or mediated by these coupling strengths That are actually going to describe the coupling between each of the K harmonic oscillators and the open system Of course, we can have more than one Open system and we can have an index J in the interaction Hamiltonian That runs over for instance many different atoms in the case of light matter interaction or many different molecules in the case of antenna molecules coupled to the vibrations of the proteins around and In the end what you will have is an interaction term, which has the very same structure as Before but just including a sum over the different particles Okay so, I hope by now I have convinced you that the the let's say the ubiquity of this Feynman-Vernon or cadera legged model for describing open quantum systems and Why is this interesting is because this actually helps us to Characterize quite easily the environment in order to tackle the open system dynamics And this is what I'm going to talk about in the next few slides This is kind of a little bit of a mathematical slide, but we will go through it and emphasize The physical concepts. So remember this reduced density matrix that was Actually, well that we could express in terms of a dynamical map Well, it turns out that of course this corresponds to a trace over the environment degrees of freedom of The time evolved initial state, which we I remind you We're assuming to be at the correlated state between the system and the environment And now of course this guy here this unitary evolution operator is an evolution operator with respect to the full Hamiltonian of the system and the environment It's a kind of an ugly object Which we can nonetheless just to let you know the following concept not It's not an operationally Reasonable thing to do, but just for you to fix the following concept. We expressed in terms of a Dyson expansion and As it turns out when we plug this Dyson expansion here and here and we trace over the environment And we consider this particular form for the interaction Hamiltonian You will see that the reduced density matrix depends on different moments of or different Fluctuations of the coupling operator of the environment of B with respect to the initial state of the environment. This is kind of a funny thing I remind you that these bees are time evolving with respect to H0, which is just the free part of the Hamiltonian and The idea that I want you to take into account here is the following is that mathematically we can see that the reduced density matrix of the system will entirely depend on environment fluctuations or different moments of the environment fluctuations with respect to the environment initial state or equilibrium state this is kind of funny because Somehow what we care about to describe the open quantum system when it comes about describing the action of the environment into this quantum open system is How is the environment been altered with respect to its equilibrium state? How does it fluctuate with respect to its equilibrium state? And now you tell me okay, this is very abstract. What is the use of this and Well, this connects with this concept of statistics that I'm going to to discuss In the next slide So as it turns out There are two different families of environment And this connects a little bit with what you know from the central limit theorem as you will see in the following In particular, it's very easy. There are environments that are statistically Gaussian An environment which are statistically non-Gaussian, but what does it mean is the following in? particular Gaussian environments are Those guys so our environments composed by harmonic oscillators as all these examples as I was showing you before And they are therefore composed of a collection of harmonic oscillators Again back from the ideas of Feynman and Vernon moreover these harmonic oscillators can be of Bosonic type like for instance phonons or photons But by the way, they can also be a fermionic type You may also think about an environment that is harmonic is composed by a for instance a gas of free electrons like when you are thinking about a transistor Where you have some some? Well, some some switch in the middle of two electronic leads And it turns out that these electronic leads are very well described as a collection of free fermions, so this is also a type of harmonic oscillator environment Which which is well which corresponds to two fermions and Why are they Gaussian is? well because besides having a quadratic type of Hamiltonian that describes these collection of free non-interacting Oscillators The coupling operator of the system of the environment with the system this B Is actually composed of a linear of a sum of linear terms That are linear combination of creation and annihilation operators This is happening in all the examples. I was showing you before and it's the most common situation This condition here having this particular coupling is very crucial to preserve Gaussian Statistics, which I will show you what it actually means in in the next few minutes but interestingly is to say that Well We will probably not have time to extend too much today on this concept but it's very important that the diagonal terms of the Of each of these V lambda operators between eigen states of the environment is zero and this is connected to the fact that the second-order moment of These fluctuations will actually decay in time But let me keep a little bit this idea, which is a little bit more advanced and let you know that Precisely these Gaussian environments Have the property that among all these fluctuations that Let me go back a minute these different order Fluctuations only the all of them can be actually expressed in terms of second-order Fluctuations So this is what it means to have a Gaussian environment is actually to say that higher order moments or higher L Fluctuations can actually be Rewritten in terms of L equal to 2 in terms of second-order moments of these Fluctuations, so it's kind of a very nice statistical property By which we know that each of these terms that are key to describe the reduced density operator Can be described in terms of second-order moments only and by the way odd L's so Fluctuations that contain an odd number of B operators are 0 This is This is also linked to having a Gaussian type of environment and because of this We can actually say that all these Gaussian environments can be Entirely described through this function. This means Forget about all the rest this means that if you experimentally have access To this function here in principle. That's all I need to be able to describe the open system dynamics This is kind of a very very powerful result that emerges from having this Microscopic description of the environment and of the coupling between the environment and the system and let me go a little bit more detail how does this Correlation function looks like Because as it turns out it looks for an environment, which is initially in an equilibrium thermal estate Namely, it has a fixed temperature. It's in the give a state corresponding to a fixed temperature Well, be dice actually the inverse of such fixed temperature So it can be written in this form. This is the most general form for the correlation function of a Gaussian bath or a harmonic bath that is in a thermal estate and It comes in terms of this function here this spectral density, which is actually what people measure Experimentally, so people experimentally have access to this spectral information of the environment or it's also sometimes called noise spectrum and Also in the 80s, but this has been a model that has been used again and again Caldeir and Leggett were saying, okay Even if we don't have a access experimentally to the exact form of this guy What we can do is we can classify environments in different Well, we can classify them with a particular Phenomenological form of this spectral density and each type of environment will correspond to a particular power law for for omega So this means that depending on whether we have phonons photons phonons living in a two-dimensional material of Photons living in three dimensions two dimensions, etc The only thing that we have to care about with this Caldeir and Leggett Phenomenological model for the spectral density is to fix the corresponding s and This is a kind of a very powerful Result because this tells you okay We are still thinking about universality classes, right? So we are saying first of all we are saying no matter how complex where our environment is We can assume that it has Gaussian statistics and therefore it can be entirely described its action on the system can be entirely described through this function and Secondly this function which depends on another one, which is called the spectral density can be modeled with a sort of a universal Phenomenological form and Then there is some microscopic derivation for j omega which I will explain in a minute But let me tell you a couple of things more about this Caldeir and Leggett model and Also introduce some concepts that are related to the correlation function. So any questions so far? I'm still going very fast Yes, no, okay fine with her. I'm still speaking fast because you know Spanish people speak fast. So it's something But I'm trying to emphasize more things Okay Good so let me finish a couple of things that then we make a small yes By could you please speak? Second-order correlation. Yes Fluctuation so this is exactly this function This is a fluctuation of Well, this is a function that describes a fluctuation of the environment around its initial state this is the well if you Let me explain it in a different way if you solve if you solve the dynamics of the open quantum system for instance the qubit and you Imagine that you extract the The master equation that describes the evolution of the reduced density matrix of the qubit What you will find out is that the coefficients of this or of this master equation will entirely depend on this function This is what I was meaning. So this function is The only quantity from the environment that is coming into play in the evolution equations of the open system And you can call it fluctuation. You can call it correlation function The important thing is that you don't need anything else from the environment. You don't need any other information from the environment Okay, so So did I answer the question or okay So and of course depending on the S index That you choose in the cal data legged model for the spectral density the resulting correlation function Will behave differently in time and in particular it will have a different so-called correlation time as We will see later this correlation time. It's going to be very important quantity because it will describe Roughly speaking because it's It's kind of still an approximated idea It will roughly speaking describe how long does the environment take to recover from the interaction with the system or How much them how long does the environment takes to bounce back to its equilibrium state? This is why well this connects to the concept of fluctuation. So the environment when coupled to the system will It was initially in an equilibrium state But now it's coupled to the system and it will be moved out of its equilibrium for for a certain time which is Related to this decay time of the correlation function and then bounce back Hopefully to its equilibrium state and I say hopefully because this is not always the case in particular when you are at the strong couplings But we are now yes Right. Yes. Yes exactly more or less this time will tell you the time is scaling which the system and the environment will become correlated and And and somehow when I'm talking about correlation time Well, the first of all the fact that this function decays to zero is telling you that somehow there will be some Well, the environment will will be Recovering from the interaction with the system. So we'll go back to some Kind of equilibrium after it after it it is not always the case that the correlation function decays to zero as you will see later Yes Yes, oh, sorry. Yes. No, I you didn't miss anything. I didn't say it So this corresponds to the real part and the and the green line corresponds to the imaginary part Which yes, please I'm gonna approach to you because Yes Yes, indeed, it's it's connected very well is connected to so during this time There will be chances, but not always it will be the case But there will be some chance for having a backflow of information from the environment to the system Yes. Yes, exactly. Just during this time. This is the time in which These guys may become correlated either classically or quantum mechanically And and moreover not only they will be correlated, but also It might be the case that some of the information that the open system have lost Into the environment will come back to it Exactly very well Good. So and by the way, the fact that there is an imaginary part in the correlation function is telling you that the I mean it is telling you that the environment is quantum mechanical. We had a classical environment This the let's say the imaginary part will be negligible one way to reach the classical limit Is by increasing the temperature there should be one parenthesis here Sorry, there should be the hyperbolic cosine. It's growing with the temperature So this means that when the temperature is is higher and higher The real part of the correlation function is going to be much more relevant than the imaginary part And this means that we will have a real correlation function mostly and Well, we have therefore our whole theory Can be a theory in which we can assume that the environment is classical Because classical environments have a correlation function that is real Right. This is a sight remark. So please. Yes Sure Yes, I don't know if I know by heart. Okay, let's see Omnic environments And let's let me start with sub omnic environments are those that are for instance correspond to When you have the electromagnetic field, right? confined in a particular In a particular architecture in a particular material like for instance a photonic crystal. We will see later on A photonic crystal is a material which is periodic in the refractive index This means that there will be some frequencies of the electromagnetic field that will be scattered out of this material Will not appear in the material and as a result of this the spectral density of the electromagnetic field which in the vacuum without the material is scales like om super omnic omega To the cube in a photonic crystal. It can scale like omega over one half. So this is one example and then I the different so what I can tell you is that this Is that is in the power of omega for instance when you are talking about phonons Will correspond to so different values of s will correspond to the phonons in the material Now I'm jumping into the case of a phononic phononic environment Will correspond to phonons being encapsulated or existing in or corresponding to vibrations of a materials of different dimensionality So this s will grow with the dimensionality of Of the material that is vibrating I've also read that some values of s correspond to Fractal environments so environments that have some dimensionality that is fractal like so it's it's a it's a very nice question and well These are just a few examples To let you know but we can discuss later But indeed they really correspond to different physical situations And as I said in the case of phonons it corresponds to different dimensionalities of of the material Okay, so And but okay, this is a phenomenological approach to access these spectral density We can think about Microscopic derivations of such a correlation function directly when we are When we know a little bit more the Coefficient of the of the Hamiltonian. Let me explain this a little bit more Hopefully not jumping too much So this correlation function for zero temperature for instance But well for finite temperature as we saw before it can be written in both cases in terms of Remembering the Hamiltonian. There were some coupling strengths That were tuning the coupling between the open system and each of the k or lambda harmonic oscillators So these are coming into play in the correlation function and then this omega k corresponds to the dispersion relation of of the field so this remember this Hamiltonian that we had I Don't know if there is we have some chokes here. So we have Hb being a sum in k of omega k vk dagger vk and then the interaction part was a sum in k of gk of Bk plus bk dagger times some coupling operator of the system Okay, so as you can see these two quantities omega k and gk are those that coming into play in the correlation function and of course there are cases in which we have Well, we have knowledge of this dispersion and we have more or less knowledge of these coupling operators so not coupling operators, but coupling strengths and Well, for instance when we talk about the light matter interaction We know with the dipolar approximation what the form of these gk's is and What we know is that for instance when the light is not confined so the electromagnetic field is in the vacuum Well, you know that actually the Well this function is going to be in particular this Well the density of states which is one of the components of the spectral density is going to be smoothly growing with frequency So this means all and all that when you calculate for this particular case the correlation function It will decay very very fast and this is how in quantum optics in most of the situations of quantum optics We will have that the electromagnetic field when interacting with an atom in the sense of for instance Changing a photon with the atom will recover instantaneously What physically the picture that you should have physically is that you have an atom that is now in the middle of the space and It's now emitting a photon and this photon will never ever go back to the atom, right? Because it will immediately Get lost so there are there is no back action from the environment to the system There is no backflow of information or to say it in different words the correlation fun The correlation time or the decay of these second-order moment or this fluctuation will be Very very very very very fast as compared to the evolution of the system and But then this is just one example of the many because now imagine As I said before That you confine the electromagnetic field into a photonic crystal remember the photonic crystal was this periodic structure in the refractive index which is kind of the light equivalent of a of a Semiconductor in the sense that you have a periodicity but not in the potential for the electrons But in the refractive index which is what the light sees is kind of the potential for the light if you want and because of this periodicity you will have some black scattering of The light outside of the crystal For those wavelengths of the light that are related to the periodicity of the material And because of this bracket scattering of this particular range of frequencies Related to the periodicity you will have a gap being formed in the density of states Before the density of states in the vacuum was grown with omega square But now it kind of look like look pretty ugly Because you will see that it presents a gap So this means that when these materials were discovered by by Sajev John I'm in Toronto in Canada Back in the I think it was in the 90s They were saying look these materials have the vacuum that is more empty than the vacuum because If you think about the vacuum there is always density of states around there are always modes that are available to receive photons Like empty vessels even if there are no photons in the field There is always the possibility of having them because you have the party the modes the density of states The vessels for for for receiving or creating photons but here you really have a gap of Frequencies or a range of frequencies where no photons are allowed and Now it turns out that when you couple a two-level atom with the frequency that is Either in this band of allowed frequencies or in the gap It will have it will suffer a strongly non Markovian effects And why because if I calculate the dynamical equations of the atom and I told you before that Such well we focus on just one band. We could also consider this other one But if I calculate the dynamical equations of the atom and remember that I told you before that these dynamical equations are going to be This is for zero temperature are going to be entirely determined by the second order moment or correlation function It turns out that for this particular Density of states This function will decay very very slowly will decay polynomially in time So this means that such type of environments will be strongly no Markovian will recover very very Poorly from interacting with the system and actually it's kind of funny because We will not have time today to discuss about these but if I locate the atomic frequency here in the band gap H Or very very close to it This will be so extreme that there will be a bound the state being created between the photon and the atom So there will be an entangled state So imagine how far we are from having an environment that bounces back to the vacuum equilibrium state That after interacting with the system you will the environment will actually Not only not relax back to its initial state, but create an entangled state between the The photons and the atoms so it will be Well, it's a very extreme case of having no Markovian dynamics Well, just to give you a few flavor of how you model this correlation function it's actually by thinking about the fact that You know that the dispersion for these particular materials by the way also when you have an Atom that is coupled to the one-dimensional field in a waveguide This is I think one of my last slides before the the post because I see people are already a bit tired So by the way, this is also the dispersion relation of the light when it's confined to a one-dimensional waveguide So this is kind of funny, right because such a different Structures and still the light will have the same dispersion relation, which is tight binding type and This tight binding type of dispersion relation will reflect precisely this band gap structure that we saw before When when plotting the density of states and this is simply because the density of states is in this particular case related to the derivative of the spectral density of the dispersion relation, so the density of states is Related to the derivative Minus one of the evaluated of course of K of omega if you like it's a little bit Just to let you know that both Quantities are connected to each other's right before we saw a density of a state that had a gap And this is emerging from the fact that the dispersion relation of the light has also a gap a Gap of frequencies that are not allowed and it also has a band and Okay, and well when you work out this Dispersion relation that is cosine like this is just the Hamiltonian of an atom Coupled to such electromagnetic field with this particular dispersion Relation and this coupling Hamiltonian, which is corresponding to light matter interaction It turns out that it really I mean you can really solve in the continuum limit And now this depends on the spectral density the density of states Well you can really solve this integral and it gives you as a solution a Bessel function That at long times, you know in the case like one over a square root of t So this is again what I was expressing before the fact that we have Very very slow recovering of the environment from the interaction with the system I think this is a very good moment to to a start for let's say 10 minutes or so I Think everyone will be happy about this Thank you