 I'm doing it with slides and this has the tendency to go a little fast so there's one trick you just ask a lot of questions the lectures are constructed in such a way that I mean I can easily stretch or compress in the second lecture specifically so let's just relax and we see how far we get it will just depend on you if you ask zero questions I guess I make it through if you ask many questions then I will compress and that's just fine on content I will just leave away stuff okay so we will discuss here driven open quantum systems which on the microscopic level violate thermodynamic equilibrium conditions and we will ask what happens if we morph from this microscopic scale to asymptotic many-body scale so this here is the outline in the first two lectures I will give you the theoretical background I will tell you how we can describe such systems which are microscopically driven and open and we want to understand what their many-body physics is so the main tool that I will introduce here is a Keldish functional integral where we start the starting point is a Lindblad master equation which will also go through but rather quickly anyways I again say if you have questions there we can pass the first lecture on that and then I will map this into a Keldish path integral so that's really a small point looks a small point here but it will probably take two days lecture and tomorrow I will then do a few more formal developments connecting for example this Keldish path integral to what is known as launch of my equation if you've heard about that I will specifically pay attention to the question how we can actually quantify the presence the breaking of equilibrium conditions including in stationary states so it will not be about time evolution but stationary non-equilibrium steady states and as applications we'll discuss phases and phase transitions then in the system specifically in two-dimensional systems from semiconductors called exciton polaritons systems that will do that that's mainly a lecture to then these four bigger points here and then in the last lecture I want to use this Keldish technology to discuss a topic that you've just heard in the lectures by Sun Wong on a measurement induced phase transitions we'll focus there on a specific a fermionic problem I will have to explain you a bit a weak continuous measurement concept as opposed to the strong projective that you've already discussed this is a limit of measurements which lends itself ideally to a Keldish functional integral techniques and I will introduce all the tools that we need in particular replica construction to extract the many physics in this measurement induced phase transitions from analytical path integral approach okay so let's start here Linblat quantum master equation with the idea that I first explained the microscopic physics and then kind of gradually we zoom out and on the way we get to this Keldish very basic what is a driven open quantum system well it's depicted here so the quantum system is in here and the specific there's two important notions that we have to appreciate here first of all it's an open system so this is immersed and coupled to an environment which conceptually is much much bigger than the system itself that infinitely many more degrees of freedom as the system and this is what makes the system open here and then the second crucial feature is that the system is in addition driven by external fields yeah so many systems are open phonon bars and condense matter but these systems we have in mind here they're in dish in addition driven by an external forces like a laser and this is actually the combination that breaks the conditions of thermodynamic equilibrium here is a simple example for that we have here the paradigmatic situation from quantum optics that's really where these driven open systems originally come from we have a ground state we have an excited state we forget about all the other possible levels of this atom and then there is a distance an energy distance between these levels between these two black lines here and we drive the system with a laser of intensity omega rabia frequency interpolating between these two transitions and the driving frequency almost bridges us up to the upper level so and this is the remaining energy distance is this detuning delta okay and so this is the aspect that makes the system driven and now it is also an open system because this atom when we have it here in free space it's just coupled to the environment of the radiation field photon field and this allows for the effect of spontaneous emission at a rate kappa so this is super simple but one thing you should notice and that's really a crucial point so drive in this problem here is absolutely essential to get a non-trivial dynamics even going imagine I put off this omega so and that's why I don't bridge energetically to the upper level then this system will remain in its ground state forever there's of course a little temperature from the environment but that will never be enough to non-exponentially small occupy this upper level and so we need the drive to get an interesting dynamics in this small system here and also there is important implications of the presence of this right physical implications and I would ask you to store these implications and we'll encounter them including in the many body contexts in these lectures over and over again for example there is no concept of minimizing the energy in the system so minimizing the energy would really mean I stay down there this is gone in the moment you start driving the system there's also no guarantee for this idea of detailed balance and I understand that usually this does not resonate well with students what is again detailed balance well that is just a statement that two states in the system say this upper and the lower level are connected by probabilities of going between these levels governed by nothing else but the energetic distance between these levels and without going in more into detail you can see that this cannot be the case here yeah so that the energy distance would be just the distance between these two black lines and there's many more scales in this problem yeah so there's this in particular there's this new the driving frequencies there's the driving intensity and there's his kappa scale many more scales that that just doesn't work to think that there would be only these two scales are the distance between the energy levels in the problem so detailed balance is violated in the moment that you switch on some drive and you operate a system like that we can we can even tune the detuning if we like yeah if we have the capability to tune the laser frequency we can also change the detuning it's just a parameter to show you what what the parameters of the problem can be positive can be negative it can be on resonance when it's zero so I'm just saying one point is crucial yeah the distance between these levels is huge and the detuning as a quantity that's typically on the order of the energy scale omega on the order of kappa and so these scales are in the same range but the distance with this is the biggest scale in the problem and if without this bridging we'd never reach it that's a point so it's energy scale separation another point that's maybe not that prominent and visible in this problem here is that in such a system in such a driven open system there is no guarantee or there's no need to obey the second law of turn it and thermodynamics for the small subsystem of course the total system and environment that will always have show entropy growth but I mean think of a fridge at home yeah so this is a classical driven open system where we can transfer entropy from the small subsystem in the inside of this fridge to the outside of the world and that also works in quantum systems in principle okay now how do we describe the systems here quite in general that's this beast here and and let's go all the terms one by one so the interest now as I was saying is the time evolution of the density matrix but now we have to qualify it's a density matrix of the system and what generates the dynamics what generates a temporal change of this system density matrix the first piece should be familiar should be really familiar this is the Heisenberg for Neumann equation if we forget about this other bulky stuff this is really a way of writing the evolution of quantum mechanical system for density matrices instead of state vectors so this is the coherent evolution and then come this bulky piece here you get to know no no worries we will spend some time on that but here I'm just describing I'm introducing what what what is going on here or what what are the the terminology these guys here these else they describe roughly how the system couples to the outer world they are called Lindblad operators and this index I that it carries here that's just a placeholder for degrees of freedom it could be for example a lattice side if you have a lattice and it's coupled somehow to the environment every lattice side so then it will carry an index I the index I can also be a spin degree of freedom so it can label all kinds of degrees of freedom that we might think of and then this entire thing so this combination of coherent and dissipative dissipative evolution this is what is called this Lindbladian or another word is also Leo willian Leo willian that comes from probability conservation and an analogy from classical phase-based dynamics which is also Leo will dynamics in the sense of probability conserving dynamics okay so how can we actually derive this equation yeah there's two structural ways how you can do that one is very good if you like to know how good are my approximations of course this is as anything in physics this is kind of an approximation and if you want to go that way you do a second-order time-dependent perturbation theory approach for an a microscopic model yeah which has here this age the system Hamiltonian it shows also shows up here and then explicitly models the bath yeah as a system the bath is here has the feature that it has infinitely many more degrees of freedom as the system yeah so that gives you a notion the bath won't won't change but the system can stage can change being coupled to the bath and then the bath system bath coupling how are they coupled that's really where these lint blood operators L occur and so and they are coupled linearly to the bath operator so this L can be any polynomial if you wish of system operators of the system operators but it is linear in this bath the degrees of freedom and the bath Hamiltonian itself we take it also in a simple way of modeling this as a really as a continuous collection of harmonic oscillators so this mu here you can see I wrote it as a sum but you may think of it really as a continuous as an integration over a continuum of these degrees of freedom this is this condition that we have here infinitely many more modes in the bath then in the system okay and then there's three approximations that would be a lecture in its own right to go through them in detail but just that you know the main ingredients so these are what is called born approximation that is the statement that's this coupling here this coupling G is smaller than the other energy scales in the problem say this epsilon here or the scales that are involved in this age when this is true yeah when we have a small coupling then we are always in the position of doing a kind of perturbative second-order perturbation theory approach and that is what what actually leads to the structure here and that one is using that the bath is unaffected by the system when the coupling is weak the next approximation that is used is the reason why this is a equation which is local in time so this is row of T here and on that side here we only see row of T so there's no memory back into the past this is this Markov property and what is used physically here is that is that the system evolution takes place on scales much slower than the temporal changes in the bath so that that is what makes this system Markovian and you can see it in this equation by the fact that only row of T occurs but no memory back into the past beyond this linear beyond this time derivative okay and then there's an approximation called rotating wave that's essentially selecting the energies in the bath that you're talking to I think we leave this out and just to be here again pretty concrete let's look at the example of the two-level system and make this connection yeah what is what in this bulky equation here in that case the Hamiltonian is really just this one here here you see the excited state is detuned from shows this detuning scale omega a delta here the transition between excited and ground state go via these scales omega here and the Lin blood operator that describes the decay from the excited to the ground state is just this operator here excited connected that maps any excited input to the ground state here and that or sigma minus operator that's the Lin blood operator for that case okay then there is an alternative way of thinking about this equation yeah yeah then you you would run into a highly non Markovian situation and you could not you would not be able to to justify this equation which is really Markovian which is local in time well then you have to then you have to approach the problem completely differently you have to approach the problem from scratch in the path integral approach this is called this Feynman-Wernon influence functional approach where you don't make any assumption on the rapidity or the fastest scale bath versus system you can still when this bath is still a quadratic variable you can still integrate it out but in this resulting functional integral you will then see that there's highly non local correlations in time already on the level of the microscopic action which emerges after integrating out the bath but that's the answer so so it's not we could not then we should not start then in this operatorial language in the Keldish path integral it's actually possible from scratch I'm just taking here this route so this is an excellent description for many very many systems it's still much more general than just looking at a Hamiltonian system alone so I want to start from this operator language that maybe people are a bit more familiar with and then map it to the Keldish but you can also start from Keldish from scratch with this yeah and then you could even describe baths which are much more general in the sense that you mentioned further questions if we have strong system bath coupling then I mean this is a bit more I mean so the bath remains somehow I think that there's no specifically interesting effect so it is it is mainly a technical so when this again in the Keldish path integral you could integrate it out immediately as a mainly you're using it to I mean to to to say that the bath is much bigger than the system and that is technically you can use this to to to claim that the this second-order perturbative approach in the operator formalism that the bath is unchanged yeah and therefore you can use it as I mean as you do second-order perturbation theory you expand around the problem which is only perturbed no there's no real it's it's I mean there's not really virtual states I would I would claim because it is a quadratic theory integrated out I mean the credit quadratic theory for the bath so it's all what is called tree level so there's no virtual processes involved okay good yeah and the the smallness that we that we use I mean here is the small G the weak system bath coupling so this is really the coupling and that measures how strongly is the system coupled to the bath and when this is small we can say this the bath remains unperturbed by by interacting with the system but it's really a picture that I mean the system gets a lot of change by infinitely the many degrees of freedom of the bath but the system does not feed back onto the bath right I mean it's like if you have a huge ball that's rolling and you throw a little ball to it yeah this huge ball will not feel it much but the small one gets an order one change and that's that's the spirit yeah so this big ball has many infinitely many more degrees of freedom and that's kind of the thing that we're using HSB is the perturbation in which we do the second order perturbation theory so in other words you can say that the gamma i is quadratic in g mu so in a sum over all the g mus that's that's that's what so that's the structure of the second order perturbation theory yeah you can also see the structure of the that it is second order effect so that here you see it occurring linearly the L's and up there L is coming in all possibilities of distributing it to the left and right of the then so that's again the structure of a second order perturbation theory and the gamma i will be quadratic in a sum over the g mus okay good I see we we are on a very good track we're going very slowly so I encourage really questions that's really the point they're better understanding something really than having a lot of stuff that I mean that's a kind of assumption yeah so that's a form that one claims for the bath yeah sometimes if for example this L is Hermitian it can be happening that this L is Hermitian when you can leave away these daggers here then we would couple directly to say b dagger plus b or b dagger minus b times I so x or p components that's just I mean it's a generic form it's nothing totally fundamental conversely what I can tell you of course if we would assume there's a good it's quadratic in these b variables then you could not easily integrate it out anymore so it's a technical convenience but on the other hand this is a very prominent situation yeah so that that occurs and now I want to come to this other way of looking at the problem which is actually not based on on this idea that might help you to to see why this was a good model namely I claim we can argue for that equation here based on a few very fundamental principles of quantum mechanics and that is that was the approach that Linblad took yeah he looked at the Linbladian as a as emerging in the temporal continuum limit of a dynamical map and so what is a dynamical map it is just a prescription that takes your density matrix at time point t to a later time point t plus delta t and it gives us the action that there's some transformation of the state exerted by an operator had L here and now we can look in the quantum mechanics book what do we require from such a dynamical map to obey of course for one thing is the density matrix is Hermitian and that must be preserved under the dynamical map and that you can easily verify for our specific Linbladian here by saying okay so by just checking this condition that guarantees that the Linblad that row after the map is again Hermitian the second requirement is what is referred to as complete positivity maybe we keep it at the level of positivity here so that means that if row has positive or non negative eigenvalues as a emission operator at time point t we require that at time point t plus delta t it remains and non negative operator meaning non negative eigenvalues of it and then we have as a third requirement a very fundamental symmetry of nature namely the preservation of probability and how do we express the preservation of probability in a density matrix formulation we require that the norm of the state row of t does not change in the course of time and that is ensured and keep this in mind here that is ensured here by the cyclic invariance of the trace operation here you can see if I take the trace of this whole object here I can permute these operators through in a way that they just cancel out and that that in maybe I do this little simple calculation here it's articulate this structure here and let's look specifically at this Linblad operator for a single Linbladian so this minus I'm leaving away the hats now for being a bit of breath yeah so now you can use the cyclic invariance minus and again to see this is zero and this is how you can remember these signs and factors here so that you should remember these signs here they really ensure the preservation of trace here of probability physically speaking and the statement that Linblad could show is that up to a unitary transformation that acts in the space of these indices I here this is really this L of row is the mode this L of row that is written up here is the most general operator that you can find which is compatible with these three requirements so here you see we didn't know use here a specific form of the bath we just said okay we want a time local yeah so this is I mean an infinitesimal limit delta t to zero a time local operator that acts with these constraints here and that already spits out this form of the Linblad operator you mean why you can pull out the delta t right I mean so one is expanding here in the I mean so now I'm using already this this time continuum limit yeah so I should have noted that that I'm using that time continuum limit and so then I looked at right I look at this here and I apply the trace operation then to that side and then follows the little calculation we did here for the Hamiltonian it's really this commutator structure is also traceless right this is this thing you we use in the time continuum limit yeah that's why they should have should have noted that I mean on the other hand I mean of course I mean if we use this form here it's actually not I mean so you see then that also the trace of this guy vanishes so we have that the trace of row of t is the same as the trace of row plus t of plus delta t do you see that so and then I mean so for an any arbitrary L row sorry delta t yeah where I mean we use that L has this structure here but there is not that there is no strong assumption on this in the continuum of the tracelessness of this L operator okay further questions okay good so let's now I mean it's a bulky object let's now interpret this a little more and the interpretation it can go as follows and so we have here these two terms with the minus signs I can pull them together with a Hamiltonian with an imaginary contribution yeah so I just I square is minus one and I can pull together I can construct here a kind of non-Hermitian Hamiltonian yeah which means energy minus decay so this is what often is also referred to as dissipation yeah and that is not enough yeah nature doesn't allow non- Hermitian Hamiltonians standing around without further ado namely we have to preserve trace of the problem so we need this additional term here and that is also often referred to as fluctuation so this is a statement you may have encountered also if you discuss a long of our equations or something like that yeah so that a dissipation in a physical system must always be accompanied with a fluctuation and that is exactly what this limb blood equation tells you here so we can have this non-Hermitian Hamiltonian here as a piece but then it must be accompanied with fluctuation to preserve energy yet I mean it gives you a nice picture if you think of just friction so there's energy and there's decay you can ask the question why do we sometimes if we think about a pendulum a damp harmonic oscillator why don't we need the fluctuation there it's everyday observation right that a pendulum damps any suggestion my claim is there's even fluctuations in that in that case but there's a huge separation of scales the strength of the fluctuation is k Boltzmann times temperature k Boltzmann is for statistical physics what h bar is for quantum physics k Boltzmann is 10 to the minus 23 kb times temperature is also ordered 10 to the minus 23 and if I look at the macroscopic pendulum the fluctuations are just totally negligible that's where you can have a non-Hermitian but then you're on a form a fundamental point of view we are then violating probability conservation it just doesn't show up because kb t is such a small quantity yeah well I mean the point is these guys they are not very much constrained and so they can be the li's are free to choose I'm not even requiring them to be Hermitian operators or so you can think the other way around so that is the most general thing I can write I allow all L is a function of the microscopic a and a dagger operators that span your hill but space in which the dynamics is going on but I allow arbitrary functions L of these operators so there's actually no constraint H has a constraint H has to be Hermitian otherwise yeah so then we would not fulfill trace preservation here but the L's are arbitrary so it's really the statement one has to appreciate that so that this form is the most general that can happen and it's a non-trivial statement that you have to prove but it's actually not except for the complete positivity that I'm sweeping a bit under the cut that it's a pretty straightforward so you can look it up in the Nielsen and Zhuang book for example that statement one starts from this map for discrete time steps and then one one approximates the temporal continuum limit one can approximate this and then this is this is what is coming out when one starts from what is known as completely positive now was that called I mean the sum of a dagger a has to be unity so and then you approximate it we can discuss this later on if you like but there's absolutely no constraints on these else other questions very good point very good point so it's actually not so obviously visible here so you can say okay if I if these L's for example are realized by just a creation operators a dagger so then there must be something that pumps particles into the system yeah so that is an obvious way of seeing that it's driven we'll get to know to understand why this is driven from a symmetry perspective in this Keldish path integral okay good so we're at this point here we have the argued for this linblad equation and I mean the typical way how this is when this is written down yeah is in quantum optical systems which dispose of a few degrees of freedom yeah and now we want to look at a situation where these where these few degrees of freedom typical of quantum optics are really replaced by a huge number of degrees of freedom say imagine this here is really the index that is labeling the sides in a lattice in a huge lattice optical lattice or whatever so and then we are brought to this interface of quantum optics and many body physics yeah where the ingredient from quantum optics is that the coherent dynamics it will describe by this and the driven dissipative dynamics they really occur on an equal footing if you neglect one of these pieces okay it's it's gone the physics is not properly described but on the other hand we also have the ingredient of a typical many body situation that in that we have a continuum of spatial degrees of freedom yeah and this combination of on the micro physical level we describe it in terms of such an equation here but we have the spatial continuum of degrees of freedom allows us or requires us to do the interpolation from micro to macro physics but of course we have to so this is realm of statistical mechanics then I would say quantum statistical mechanics out of equilibrium which however I mean requires us to develop some new theoretical tools to do that before doing so a brief a very brief highlighting of the possible platforms where such physics actually occurs with this what we just discussed is a good description for the microscopic system and that is here a platform of atoms so its atoms light and solids you know all these type quantum matter platforms have examples for that in the context of atoms one beautiful example is this driven open dicker model yeah which emerges as an effective lindbladian description of a Bose-Einstein condensate that is put that entire Bose-Einstein condensate is confined to such an optical cavity and then one drives this cavity and the interaction of light and matter gives rise to the macroscopic occupation not only of a single mode as you have it in usual Bose-Einstein condensates but actually two macroscopic modes and it has been seen that these modes interact with the light exactly in this way as does as a collective spin degree of freedom in a dicker model without going into the details and this dicker model is of course intrinsically open because yeah so the light can get lost via the mirrors of the cavity which never in real life never are perfect similar situation is these micro cavity arrays here so here you can confine light into microwave resonators and this light is confined but it still can hop quantum mechanically tunnel between the resonators you can make them interact via what is known as Kerr non-linearity and then you realize a driven open variant of the both Hubbard model it's easy to imagine that again in this setup as well light easily gets lost through these maybe imperfect resonators here so you have to pump it to get a many body stationary state and also in solids this is a platform I won't touch now we will discuss this a little later semi conductor hetero structures exiton polariton condensate as they are called so that's a solid state platform and another beautiful example from from atoms is for example Rittberg geysers that that are used that are operated in a highly dissipative regime and also more recently I mean systems move into the focus which are even more their microscopic physics is even more engineered to the point as in these platforms here and so here you have a real solid state system here you don't have perfect control of the microscopic physics in all these platforms here like the superconducting circuits Rittberg tweezers or trapped ions you really control particles the microscopic constituents on the single particle level and you can do fancy stuff with these like cooling into the ground state of a Tory code or so for example in more complex spin models here and of course I mean also these systems are not perfect and the coherences dissipative processes are important they are going into the many body regime so we also classify them in these driven open many body systems okay and from a theory perspective yeah we want to understand yeah whether on that transition here from micro to macro physics there's any interesting new universal phenomena that maybe that cannot occur in thermodynamic equilibrium but to do so we really have to first forge a bit the tools in the absence of a concept like minimization of free energy so this is gone in the moment that we break the detailed balance we need to something more general yeah I mean there is there's two statements that you can make yeah so in partial in a very first place yeah so usually you want to engineer especially in quantum problems you want to engineer a perfect Hamiltonian dynamics and then it's just the inability of a real-world experiments yeah to perfectly isolate from the environment yeah so this is one source of this dissipation and that's called noise yeah so it can be electrical stray field say in these superconducting circuits so then it comes just as a nuisance more recently people have also understood yeah so that I mean one point that I was mentioning there is no second law of term of second law of thermodynamics so in principle you cannot only have decoherence yeah there's also the possibility for some coherence so you can there's nothing fundamental yeah so you can have fridges in the classical world so you can also have fridges in the quantum world where you extract energy entropy from the system so you your density matrix becomes more and more pure by suitably engineering your dissipation yeah so there's also beautiful experiments that are doing that yeah so then dissipation is not really a nuisance it's even a resource but all of this is described in this Lindbladian framework so these are the two answers that I think one can give more questions okay good right so let me let me put here a model that looks bulky and but it will occur few times in these lectures so now this is an important slide and I would refer to this as Lindblad 5-4 theory so if you do quantum field theory course also you learn 5 to the 4 theory as the paradigm and the workhorse model to do your first Feynman diagrams and so on and so forth so let's write a 5-4 Lindblad yeah and that is this problem here so we have here a Hamiltonian yeah which describes the dynamics in a many body system now and in this many body system we have a kinetic energy so this Laplace operator we have a rotating frame let me call this neutrally rotating frame and we have some elastic collisions yeah some two particle processes particles get annihilated and they get created again yeah so two annihilation to creation operators in this scattering elastic scattering process then we ramp up now our dissipator here that that enters the Lindblad operator and the first contribution that we want to describe is a single particle pump I was already referring to that so that's a system where you can see here a particle at position x gets created yeah we hammer this operator on the density matrix and the particle gets created and this can occur at every position x in the system so we have to sum over all these possibilities we have also a single particle loss here built in yeah I'm not saying this must always be like that but let's look at this 5-4 and then we also want a quartic term so something fourth order in the operators here in the dissipative sector yeah so we want we describe this by two particle loss processes so you can see here at this point x two particles are lost in the same instant of time what's the basic physics of this complicated looking model really looks complicated but I give you a very simple intuition yeah excellent so this is this is the following assumption I was saying it implicitly so I only came to if the question comes imagine I have lattice sites and then I have the idea of this limbo as I was saying at the idea if you want to derive this is at every lattice site I have an infinite amount of bath degrees of freedom important important point so how is this compatible well that's fine yeah if the wavelength that comes out here that is emitted into the bath is much much smaller than the distance between the sites then although I have coupled the entire lattice system to a single physical bath it looks like in core uncorrelated baths because the emissions here they see destructive interference so the mathematical description of independent infinitely large baths is for such a lattice system is fine next thing first statement statement number two now we are coarse-graining our resolution over many lattice sites that gives me in the continuum limit this spatial X but that is the I mean there's a few steps I mean that I that I bar out here okay good so more question yeah we can't I mean just writing something generic so like a 5 4 theory in quantum field theory if you start with that you can write a spinner quantum gravity whatever I'm but if you want to develop the first steps you start with kinetic energy potential energy by scattering so that it's not totally boring not a Gaussian problem yeah and that's that's a reason you know so it's just natural and we're thinking about non relativistic particles in a condensed matter environment then the leading term you can write is a kinetic energy which is a Laplacian assuming there's a spatial inversion symmetry no direction preferred that we would have to add a linear gradient H is a many body system which you can read off from the fact that we sum over all positions X yeah so this is many degrees of freedom on living on a continuum in space right okay and so is the Linbladian yeah so this is a huge system it's a exponentially living in an exponentially large fox base that's what this described here okay oh I think I wanted to say something about the rough physics of this problem here yes so to extract the rough physics let's do something very simple let's do what is known as mean field theory so we study the evolution of the field annihilation operator we could also take the creation operator so this is described by the trace of this operator with row of T yeah and then we take as a mean field ansatz we factorize the density matrix over space so that's a strong assumption yeah but let's do something simple to see qualitatively the physics and moreover we say okay and on every of these sites in space we assume that the system is in a coherent state so these are massive assumptions but it's not so bad sometimes for these for these problems in the right regime and then we even make our life simpler by saying okay we forget about the spatial dependence and so it can't get much simpler but what you can out get out then yeah is an evolution equation for justice field expectation value psi which then under this assumption only depends on T and it's governed by this nonlinear equation here which is a oops nonlinear equation which which which has a picture in terms of a potential landscape and this equation describes damping in a potential landscape in such a quartic potential here so what this problem it's an overdamped motion in a quartic potential landscape so in the regime where the pumping rate exceeds the loss rate of the problem so we pump more particles then we lose yeah so then we destabilize the minimum where the field expectation value would vanish and instead yeah it is stabilized and by the two-body loss yeah we get stabilized by the two-body loss we get a finite expectation value which settles spontaneously in one of the possible degenerate directions in this sombrero landscape here and that is what we call spontaneous symmetry breaking so upshot of this yeah this nonlinear complicated quantum master equation here operated in the right regime and under certain strong assumptions gives us a spontaneous symmetry breaking and describes the condensation of this field amplitude or the field expectation value phi of t yeah also a very good point yeah so you're not forgiving me anything here here you see I wrote imaginary part of V yeah where V is actually this equation I want to write it like that and this V is a complex function as you point out yeah and the state is fixed by the imaginary part the physics of this is that the damping yeah will fix where what's the size of this condensation amplitude that's the imaginary part and then the real part is slaved on that I insert the solution for the value of psi absolute value into the real part and then I will solve this equation by adjusting this rotating frame parameter in view this is still a free parameter very good point so that's why I don't refer to it as a chemical potential because it's actually in this dynamical systems really much more a rotating frame that adjusts to find the solution for the real part okay right now we will translate this into a Keldish path integral and then go through a few of the structural properties I was already announcing so this Keldish path integral first point will be a construction of this so maybe one I keep it very short but why why would you even do this why would you change gears we have a description of these systems nice microscopic description in terms of this quantum master equation I can only argue here with the words of Feynman the inventor of the original path integral idea the formulation that we are going to develop here is mathematically equivalent to the more usual formulation there are however therefore no fundamental in your results however there is a pleasure in recognizing all things from a new point of view that's certainly true and also there are problems for which this new point of view offers the distinct advantages so what could these advantages really be yeah so first of all I think I mean this is a prophetic paper here that I mean has proven that it has just proven that path integrals are the language for describing systems with many degrees of freedom if you want to do actual calculations there have been these techniques diagrammatic techniques collective variables are easy to introduce renormalization group concepts very important if you want to zoom out from the microscopic to the macroscopic scale all this is not possible from my understanding in the second quantized operator formulation and specifically did this childish path integral I mean what I would advocate is that it's much much closer to real-time formulation of quantum mechanics so then for example also then compared to equilibrium imaginary time path integrals it can give us really a view of thinking about these systems by thinking in this language and in particular it opens up also the toolbox that I want to demonstrate later on of quantum field theory for this driven open systems okay so here is now the core developments I do this in the form of an onion yeah we'll go here very quickly through the concept steps then the next layer of the onion I will describe these steps in a little more detail and finally I will do one calculation of the simple most system in some real detail so here please just follow the logic first logic goes in three steps so we start here or think of remember Schrödinger equation we start from evolving a state vector so this is the Schrödinger equation here and we can write it either in a differential form or in an integral form where we use the exponentiated Hamiltonian operator so the same logic or can be applied to what is known as the Heisenberg for Neumann equation recall this is the part when we leave out the dissipation then this is actually the density matrix evolution here and it is equivalent to the Schrödinger equation if rho is in a pure state so because then I can these two branches here described by this commutator they are just separable so the state rho is separable when it is pure and then we can map it back to the Schrödinger equation and its conjugate equation of course but for general density matrix which are not of this form mixed states then this is the right way of describing a quantum mechanical evolution what is shared with the Schrödinger equation of course that it's a linear equation linear in psi this is linear in rho and so we can also pass from the differential form to the integral form over here yeah so and it's just really I mean applying the same unitary on both sides of the density matrix the key point that I want to make here that will explain you why we need the Keldish the structure of the Keldish path integral is that the difference here is we are evolving an object with a single index psi a vector and here we are evolving a matrix so we need two branches the same structure is actually obeyed as we as we already said in some sense yeah by the Lindblad equation it's again a structure where we have a left and right action of some operator on the object rho it is still a linear equation in rho and because it is linear we can at least formally we can find a solution by exponentiating something so now we have to declare and they'll do it on the next slides in some more detail we have to declare how this object acts but it's definitely a structure that's still linear in rho and we can find an exponentiated version of this what is called here super operator super just means action from both sides on to rho so we'll see what what what this how this can be given meaning but this so far for the very basic structure okay let's use these three points to learn bit by bit the meat of this construction and the first point is I want to refresh your memory on the functional integral idea itself yeah so how do we construct and path of functional integral for vector evolution of the Schrödinger equation well we have one time string that's running out here and this time string describes the action of the unitary operator here and what we then do to derive a path integral is to chop the evolution into small pieces with a reason that we can then linearize this exponential yeah after in each little of these little time steps with this delta t in the limit n to infinity goes to zero and then we can act with the ingredients of the Hamiltonian operator onto the insertions that we put in here and we choose them to be coherent states so coherent states are a crucial concept here yeah to derive the path integral the coherent states have this proposons have the property that the annihilation operator acts as a number on these states as a complex number they have this normalization here and they have this completeness relation that's displayed down here so these are the three key properties of the coherent states in particular will insert this resolution of identity after every of these little time steps so now let's look at a single of these time steps and here I want you only to refresh your memory hope you've seen that somehow before yeah we'll do this for the lindblad in more detail here I just want to observe the structure the structure is this yeah so on this link or on this insertion I insert yeah this resolution labeled with an index n yeah for time step number n yeah and I also write I don't write the integral but I write this e to the minus factor here then comes the action of the unitary constraint to this little time step and then comes already the next insertion of time step with time step n plus one and what happens yeah really just watch the structure here what happens is that this matrix element here yeah I can evaluate it using the smallness of the time step and I can see that the Hamiltonian operator reduces to a functional or to a complex valued Hamiltonian function this much more similar to a classical Hamiltonian here it's just no longer operators it's just complex numbers that are that is Hamiltonians a function of and by these overlaps of that we there's also a time derivative structure we'll do this in more detail yeah and then if you do many steps where we just have to take the product of all these elements and then we get this object called action yeah but no worries it's just a structure yeah but I hope it generates an expectation how this has to look like when we do the same idea for the Lindblum okay the operator age in this process is reduced to complex time-dependent functional age of these complex valued fields fine good the next step yeah now we have to we just dealt with this vector evolution yeah single time direction now we want to look at the density matrix evolution so we have to not work with not only a single time string that's going out for a vector we have to work with two time strings yeah that are sorting out of the two indices of this density matrix row and in the case of this Heisenberg for Norman equation it's really just the operator you and on the other side you dagger and we take again this decomposition and then have the exact same idea now we have to insert this resolution of identity on both of these time strings okay so we get here two sets of degrees of freedom for the matrix evolution and then the identical program can be done for this Leoville generate or Lindblum generator here when we take into account now we give meaning to this object here the meaning of this is okay let's define it via its little time steps yeah so let's define it as the action in on a small time step of L on row naught yeah so the first step will give us row one and then we have this row one and we apply the next dynamical map onto this one yeah so iterate out yeah and graphically this means we are going from the center here yeah we are mapping out all the little steps yeah so we can treat this problem exactly in this path integral language idea yeah so using the definition of this exponentiated Leoville operator or Lindblum operator in terms of this representation here which becomes exact when n goes to infinity we understand that we have exactly the same structure as we had it for the von Neumann equation yeah and now the last step that we still have to do is of this graphical structure so if we look at these two time strings that are sorting out now we want to construct something like a partition function so this partition function is the trace of row so if I have a matrix index i and the matrix index j here the trace means to some set equal i and j and sum over this graphically this means I not together these two branches and that gives me then the structure of this closed childish contour time pass here where and we are starting here at row initial and then we are taking an endpoint we take this in infinity and that's where we take the trace so we we weave these guys together here and that gives me a closed time path as it's called and often one refers to one of these branches as the plus contour and the other as the minus contour and of course yeah so that's a statement I like to emphasize and so we are representing the number one in one of the most complicated ways you might have seen but I guarantee that this will allow us to extract very useful information by asking by interrogating the system at any time on this time string we can insert now operators and in this way we can generate correlation functions and that's how we then extract information from this unity from this trivial factor okay so now let's do a little calculation for this precisely for the the limblad operator itself and to this end we take a very very simple example and the simple most that that I can imagine and that is really the single degree of freedom attempt quantum harmonic oscillator with a scale omega naught so that's the oscillation frequency and the kappa rate a damping rate of this single bosonic oscillator you can think of it as a cavity as we've seen that this micro cavity system yeah which a bosonic quantized photon mode and this can decay with rate kappa and we will use this formula that row of T can be chopped up into many of the small time steps and what I want to do now is to compute the single time step update on the density matrix at time n yeah which we have represented yeah with these coherent states so now let's write this down so we take time step T n which is an initial point times n times the small time step delta t and row n is just row of T n set everything and now we represent this matrix density matrix at point n in terms of coherent states so how do we do that we write here follow this picture here minus n yeah this minus just tracks on which side of the density matrix we are like this and the same on this tracking the other so then I don't like to write it like that I just write it a little differently to isolate so to say the indices of this matrix and I just pull out this is a number here so I can write it like that so these are now really the indices that this big matrix hex and I have represented this thing in terms of coherent states and now to be really accurate we have to still I mean use the resolution of identity here so and this is the product sigma is plus or minus where I integrate over d sigma n star d sigma n just this yeah so we are integrating so this is this formula of coherent state integration now with these two indices plus and minus and here I must not forget the normalization factor which I can write in this way sum over sigma is really a small sum plus and minus good and this is our matrix element yeah so that we want to now evaluate so no that's not quite right we want to morph this matrix element into the next time pin n plus one find now goal row n plus one which is the matrix element phi plus n plus one density matrix time step n plus one so this is the really the non-trivial update on the state that we have to do yeah in this representation of row n this coherent state representation and that's now the goal and let's now look yeah what how we can construct this matrix element yeah for the density matrix at step time step n plus one to this end we have to apply once one plus dTl so the matrix element that we are constructing here involves where I mean I forget about these three factors just not to just to save a bit writing phi plus n plus one and then comes the action of unity that's written up there fire plus n plus one times application of L which acts now on these guys here and then we close the matrix element here and I want to keep memory of the insertions okay so and then comes the calculation so we have here we just can evaluate these these first sandwiches it's very easy plus n plus one n plus one plus now comes the little time step delta t times and now plus plus one now this linblad operator has as we said it has a pure and right action component which is minus i omega naught a dagger a look at this Hamiltonian up there minus gamma a dagger a that's a piece coming from this linbladian or did I call it kappa kappa is gamma I take gamma and then this is folded together with plus n minus n like this yeah so this is the pure action of of the terms that come from the left then we have a term where there's no action on the plus contour which comes here remind the commutator i omega naught a dagger a leave away the minus gamma a dagger a minus n plus one and I have also we have this term this probably this fluctuation term yeah we have left and right action of the a operators up there this comes with this factor of two and it looks like that the annihilation this is this guy okay and now we can use the coherent state property to reduce these operators here to fields yeah so that's where we use the coherent state property and you can see this guy yeah I can act to the to the right in order to reduce the operator to a number complex number and the dagger operator using the bra cat and the joint relation I operated on that side so here we get phi plus n and here we get five plus n plus one now let's go through all these steps here so the same obviously here here the same with a plus is replaced to minus and in this term here we get five minus n I have to act it on that side while this here goes to five plus n this is a yeah like that okay and now we can collect these things together I bring a little bit of order into this problem I write it as I use one plus delta x something small yeah which is controlled by the smallness of delta t is e to the delta x to write this as e delta t and then I phi plus n star minus phi delta t yeah so the local one in n that comes from the normalization factors here and these guys here they overlap to this factor here when I completed from the property from the overlap of the coherent states that's this term and then I get the same plus and minus exchanged which reads in detail like that minus star minus n and that's one over delta t so you can see I pulled out a delta t and then minus I square I write minus one as minus I squared so yeah and and that okay let me that's everything all the terms that I collect together here and let me write the maybe the final result when we take now the limit delta t going to zero such that you can see that in this limit we produce here a derivative with respect to dt and then the final result is and so there's two points yeah so this becomes a derivative and also we can use yeah because there's a pre-multiflying factor delta t here in we have contributions that are not exactly local in time so they involve steps n and n plus one but since they are pre-multiplied by a smallness factor delta t we can approximate them as time local because we're only taking linear terms in delta t into count so that is why yeah so while this is here or not exactly local in time in the limit delta t to zero we approximate this yeah the sign is different yes can do the calculation the sign is different yeah so this is because these matrix elements here they come in in the non-local ones they come in conjugated ways yeah so the local term has the same sign but they did not exactly local the one connecting n and n plus one this comes in the opposite way so that's the reason why they come exactly with the opposite sign and it's important and so here yes we have e to the I put it there in the delta t goes to the dt to an infinitesimal increment minus i dt phi star on plus now you see something that looks more really like a field theory minus i phi minus star minus minus i times and now I fix all the eyes or I hope so omega not by plus phi plus star minus omega not here this minus sign reflects the commutator structure of the Hamiltonian term and this is the left action or the action well that acts to the right on on on the density matrix this is the right action term so this is this one is the second and this is the first one and now I'm doing the same for all the other guys so now I'm sorry to be a bit messy but I would like to fit it into this line namely if we collect everything together we get two times gamma phi plus phi minus star so this comes technically from this left and right action terms and it describes the fluctuation term in this path integral language minus with a factor one yeah so the gamma is can pull out phi plus star phi plus minus phi minus star minus this and that was a huge bracket over there for the exponential so that is now the result and well if we take many time steps but then I mean we just multiply now all these matrix elements together and that gives us really the structure of an action many time steps we get by multiplying these exponential factors together we get a sum in the exponent exponent and a sum over these DT's or delta T's no matter that morphs into an integral DT so this is a typical action is an integral time integral over a Lagrangian function and we take this from T initial to minus infinity yeah to sample the whole time interval and this here to plus infinity the final time and in particular if you are about science and to write the action in a neater form we can under such an integral from minus to plus infinity we can use partial integration to write this as minus plus so and then then you you produce the form of the action that I'll display on the on the next slide yeah I'll summarize this again so these are the details of the calculation and the next thing is I want to summarize this this discussion again so that's right yeah but I'm using the the I'm using this property only for the plus term so I want to you'll see it on the next slide there's now a little glitch yeah we are now here many steps in the temporal continuum limit I want to add one more detail before summarizing and that is the following thing yeah I've said we did this construction now for a single degree of freedom yeah which is even a free degree of freedom not no interactions and now you can say but this we want to do many body physics yeah so do we have are we in the position of fulfilling this promise my statement is yes so the key step is really how to operate how to proceed on the time domain yeah this is the quantum mechanical evolution and at this moment if you like we have this formulation we have a single degree for freedom phi where this index is plus or minus depending on the branch of this contour and we have a single temporal integration so this is what you would refer to as a zero plus one dimensional field theory quantum field theory description or a quantum mechanics problem now we can have many degrees of freedom sitting on a lattice yeah so then I have an additional index I which however really runs from one to infinity we have an extended system and the integration over time is supplemented by a summation over all these spatial indices and if we want to describe a many body problem in the continuum well this lattice index yeah morphs into a continuum spatial index the time integration goes into a time and space integration and we end up with a d plus one dimensional quantum field theory description of the problem so and this works analogously so the writing it down can be done for interacting system whether you can still solve it then is a different story and I just we leave out a few subtleties here yeah there's a subtlety when we have nonlinear operators L but I can't cover this so this can be fixed by a time splitting prescription it's a regularization issue and for fermions their mind that it's not the correct action if you want to write the fermions because there's additional signs due to the fermionic structure of the of the problem so but here is the summary discarding now these subtleties we have started now from the Lin blood equation yeah which is characterized by left and right action of operators but linearly on the density matrix we have transformed this problem into an equivalent you're no approximation made just a rewriting into this Keldish functional integral here where the structure of the action that emerges is written down here now you can check again the signs so the relative signs this you see that the how to remember that I mean the IDT is Hermitian IDT is a Hermitian operator yeah so and then if you evolve says it's you and you dagger so that comes with a relative minus so actually that's an anti-Hermitian operator because there is an I overall yeah so so an anti-Hermitian is what did I so that's that's how you can remember that and check that that this is the right way so good and and and then the key structure is yeah so the Lin blood yet so the Hamiltonian commutator that we have up there really the rule is extremely simple yeah when the dust has settled this goes into Hamiltonian functional of fields H which only depend on the plus or a minus field so that's the notation here H plus means it depends only on fields five plus and same for minus and now you can see also how the this structure up here in prints in the field theory language left and right action go into contour an index yeah so we need to remember somehow does it come from the left does it come from the right this is remembered by adding the index in the field theory so like you remember a spin degree of freedom and you recognize really this Lin blood structure here's now now unfortunately I have put here one half and here one yeah so while in the other formula sorry I had two and one but this is trace preserving that's the key point and you have this rule so that we need yeah and and what we have done on top of it yeah so this step I left it out we have taken the trace and identified the indices to close this this childish contour okay so and maybe I would like to little bit of this still before tomorrow so we have a few structural properties so that I would like to discuss and so and that will teach you a bit how fluctuations are treated in this functional integral and I hope you can if you've heard about functional integrals so then I think this this will make a nice connection otherwise you you have to learn on top now how this functional integral works so but but it's really the same way as a standard functional integral works with a few my view interesting special features yeah so in particular for these structural properties I want to discuss three points yeah first of all how is this fundamental property of probability conservation reflected in this functional integral language and we've seen I mean probability conservation in quantum mechanics in pure state quantum mechanics you would replace this probability conservation by unitarity of quantum mechanics so it's a very deep symmetry principle of nature so how is this seen in the childish functional integral and then we so this can be seen as the absolutely trivial most approximation that you can do for this childish action yeah and then comes a little bit better yeah it's a kind of first order approximation gives us what is what we can understand as a deterministic limit yeah so it's like the superclassic limit where the whole functional integral which sums yeah by virtue of this summation here which comes really from all these coherent state insertion the product over these we can still do an approximation at least formally which singles which approximates this whole sum by a single configuration the one that minimizes the classical action so how is that coming out of this childish path integral and finally of course I want to explain how we can integrate fluctuations in this in this problem okay so this probability conservation sometimes also referred to as causality for reasons I think people didn't know a better word but I would rather term this probability conservation yeah so this is the property of the Lindbladian that trace of row has no time evolution in the childish path integral this is the statement that trace of row is normalized for all times yeah so we always produce with this Keldish partition function the number one okay and what I want to now argue yeah is essentially that this Keldish action has a very special property that other field theories do not have and which is the property that if I well evaluate the fields at identical configurations on the left and on the right contour then this is precisely the property that ensures probability conservation and so and this is a very special property of Keldish actions if you take a an action an imaginary time action functional integral you look at the action you insert a specific configuration you get a sum number but for this Keldish action if you identify the configurations on left and right side then we get zero so how to does why is this reflecting probability configure a conservation this is due to a nice redundancy that this Keldish that that's actually we have due to cyclic invariance of the trace and this redundancy is written down here the operator expectation value for any operator it can write it like this operator on the left or by using cyclicity operator acting on the right right now when we translate this into a path integral this means that I cannot distinguish whether I insert the operator right before I take the trace at time t on the left or on the right this is precisely in the operator language cyclicity of the trace means in the Keldish language that I can insert operators either on the plus or on the minus contour and the result cannot change that is not physical information whether I insert on left or right something one should appreciate and that means in immediately that if I look at and the time evolution of z so that will be the result let's look at the time evolution of z for the fun of it when I look at the time evolution of z you can see in in the action the action now runs up to the time t and I write it as a kind of time integral over a Lagrangian small s now if the time differentiation at the endpoint in time t acts on this time up here so it spits out just this Lagrangian function down there and if I evaluate this this Keldish this Lagrangian for equal arguments then precisely by this formula you can you can see it here if I evaluate this formula I say now Phi plus equals Phi minus so then this here cancels with this one and this is also Phi plus Phi plus so all these terms here cancel out so indeed if I identify these two contours then I get a zero for this this a bit if part and same thing here yeah if I identify Phi plus Phi minus these guys will just cancel out and it's precisely this property of the Keldish action that I mean if if we insert here out this is now zero as we observe yeah so then we see indeed this is the probability conservation reflected in the Keldish path integral so this is I mean upshot of this discussion very simple if you look at the action at equal at equal arguments yeah if I we identify field configurations on left and right then you can see explicitly that the action vanishes and here we have the interpretation of this vanishing of the action as the probability conservation of the problem okay so the last thing and then tomorrow we go on with with fluctuations and the last one one thing that that I would like to motivate is that this property of the Keldish action motivates something that is known as Keldish rotation namely you can pick much more clever coordinates for these fields and if you read in books on Keldish you will see this immediately now this is how we rationalize this so we can make the probability conservation for which we argued now much more handy yeah so the starting point is that we have these fields on the plus and minus contours and we understand now yeah or we have seen that if we evaluate them at equal arguments yeah so then this implements probability conservation now we propose to do a rotation of these fields we switch the coordinates of this problem and we switch them into center of mass yeah so we look at the sum of these guys and relative coordinates yeah and with this trick yeah we then produce the property which is a bit more easy to handle yeah the action evaluated for any configuration of this of the center of mass coordinate this one for any configuration but for this relative coordinate evaluated at zero which means identifying the two configurations yeah must spit out this zero so this is how in this new basis for the field variables we represent the probability conservation and the interpretation of this is the following yeah so they are called C is called referred to as classical field Q as quantum field this terminology has nothing to do with entanglement or so it's just related to this following thing here the classical field can acquire an expectation value yeah so this is the field that can describe condensation and spontaneous symmetry breaking yeah so as you know in many body physics you've heard this yesterday in norm's lecture condensation yeah is a physical phenomenon that we will see in this classical field expectation value yeah while the quantum field yeah by this redundancy yeah so one of possible choices for an operator is the field operator itself and we've just seen there's this redundancy in description that tells us that these expectation values don't see the value of this plus or minus yeah so therefore the quantum field can never acquire a field expectation value so by by this redundancy of the kelvish path integral and that's why it's called quantum field yeah so if the classical field describes possibly condensation phenomena the quantum field expectation value is always tied to be zero so this looks maybe a bit like a technical detail in fact it is yeah however I'm emphasizing it because in any book when you look at kelvish you will most usually find this choice of basis here because it's much better for practical calculations and that I wanted to motivate that physically that it is a beautiful handy implementation of this probability preservation property that the kelvish path integral has okay so we end on this technical node and tomorrow we go on and flesh out the merits of this quite tedious calculation thank you and see you tomorrow and of course if you if you have questions just ask them now or you can come come to me so any more questions for today yeah you