 Michael thanks a lot also to the organizers for the invitations. I'm very happy to be here So let's hope this is going to turn into a fruitful exercise and the task that was given to me was to give them some sort of Overview over long-range interactions in cold atomic systems meaning okay settings in which quantum effects Do play a role and before I do so. I just want to briefly highlight Yeah, what we are doing in Nottingham Okay, and show you that even post-Brexit. Yeah, some interesting physics is being done there So Nottingham is well The Brexit jokes are all on me. Yeah, so it's really it's really sad. What can I say? Nottingham is here. Yeah in the center of the of these Midlands two hours north of London by train and our local heroes, of course Robin Hood and Together with him a number of people are trying to rule the realms of theoretical physics two postdocs are here Matteo Makuzzi and Yelgi Minar who are up there somewhere and Please pass by their posters and say I have interesting stories to tell And this evening and of course a number of collaborators from Mars and physics and Magnetic resonance imaging we are collaborating with and we are doing physics that is Basically centered around two domains. So the setting is basically atomic physics and quantum optics and We use these systems to do many body physics mainly out of equilibrium and just to sketch what kind of areas that encompasses this goes really from more fundamental stuff like Understanding the spectra of excited atoms and ions looking at dynamical phase transitions in open systems studies strongly correlated dynamics And then to more apply things like atom photon interactions for quantum information processing or atom interferometry or Methods out of equilibrium methods to enhance for instance magnetic resonance imaging approaches All right, so I just want to also say that there are open positions Is an open one open position? Yeah plural is maybe not the way in this area of research Okay, so let's get back to the talk Well Okay, it's not a too easy task to do to provide an overview over the field So I try to be as a simple. Okay, obviously I'm trapped in my own realm to be as simple as as possible and What I wanted to start with is I just want to briefly tell you what energy scales and length scales we are talking about and Now since one has cold atomic or molecular systems. What does one actually seek to achieve? And then I talk about a number of specific systems I want to start a trapped ions and tell you how one can Generate long-range interactions here I want to show you that one can describe them in terms of few level systems that are being coupled through phonons And then you can integrate out the phonons and get effective spin models and Christian rose at least to my understanding We will talk on Wednesday about subjects very much related to this from an experimental perspective So and the Next part of the talk will concern neutral atoms and polar molecules I'll tell you how they couple actually to the radiation field which then gives rise to long-range Interactions but also to interesting forms of dissipation That for instance result in the emergence of super and sub radiant states also the topic that will be touched upon in far more Detail, I'm sure the robot Kaiser's talk and a Roman Voschela's talk and then I tell you a bit about the basics of Vanda vals and dipole-dipole interactions and I want to conclude with something that relates to static dipole-dipole interaction and how you can actually Tune these dipole interactions such that they rise to funny potential shapes that go under the name dressed interactions Yeah, and so to say that you will see When I since this is a theoretical talk I try to provide you with some details on derivations Yeah, which are far from complete so all the derivations that I'm showing you are sketchy, but at least I hope okay I can outline the basics of and the underlying rationale of all these approaches Then the disclaimer which also pretty nicely links to the first talk we saw today what are long-range interactions? And yeah, and I think everything that extends beyond nearest neighbor in some sense Yeah, can be regarded as long range. So I don't think There is a very precise definition and you will see I'm switching between one over to the six and one over cubed and Anything that is not contact on nearest neighbor will be fine Good, so let's look at energy scales and length scales and clearly I mean we are not on an astronomical setting So I know mega-parsecs involved here So the energy at the distance scale is here usually micrometers And then this sketch summarizes somehow the typical interaction strengths that you can achieve in various Settings that relate to cold atomic or molecular physics. So here's the Here's the interaction scale that ranges from 10 to the minus 9 Hertz to 10 to the 11 Hertz But if I'm not completely mistaken corresponds to a temperature of 5 Kelvin. So it's also not utterly hot. Yeah And now if you look at Typical interactions and the first thing that can come into your mind are for instance Ground state atoms and we will see also later throughout the talk that these ground state atoms will interact with the Advanced potential that decays with the distance as a six power and you see these interactions are really on those scales tiny So the next best thing you can do is you can go to actually atoms with large magnetic moments Which is in the order of maybe 10 boars Candidates are chromium dysprosium erbium And if you look at these magnetic dipole dipole interactions, you see you do better Yeah, so it's like four orders of magnitude already and the and the power law this behavior is also different since it's Static dipole dipole interaction the power is one of our cubed So then on the similar a bit better than that are for instance polar molecules typically one is looking at situations where one has binary molecules and one looks at their Ground state or at physics close to the ground state. We will also discuss this later where they only Rotate and if you calculate the interaction potentials that can be achieved and these systems for instance in lithium cesium or Potassium or medium you see okay, you gain a few orders of magnitude here But still again you get this one over our cubed interaction for instance and then next thing you can do is Instead of looking at atoms in the ground state Two atoms in highly excited states so-called Rydberg states Robert love will talk about this tomorrow And I will also basically exclusively talk about those in in my second lecture tomorrow Where you can see you get even stronger interactions here, which either can be of dipole dipole With one of our cubed or one of our to the sticks and then last but not lead was it can also do is you can really Look at interacting charges for instance ions In pole trap or the panning trap and you see that you can get interaction strengths that are Significantly higher and of course much longer range is one of our but What does one seek to achieve? Yeah, so now you have those cold systems Yeah, and they're basically two things that jump immediately into your mind when you are in the cold atoms business So there is one thing you can do is you can try and and go and implement some kind of quantum Computation protocols that means I mean you encode quantum information in some states Yeah, we will talk about this later that are actually rotational states or electron of a polar molecule of Electronic states of an atom and then you use this long range interaction to make those atoms or polar molecules interact And this then gives rise to so-called quantum gates Which which you can then build circuits and you can implement at least in theory all sorts of protocols here or what you can also do is you can use this As a quantum Simulator, let's say where you build yeah out of single ingredients for instance spin systems that interact with specific Interactions like power loss for instance And this is actually what I will be focusing on this talk in this talk exclusively spin systems that interact with long range Interactions and that just this picture that I was also my initial slide I took From the paper from the solar group and you see it was already in 2006 as it's ten years ago so this is not so recent and Still I hope I can convince you also in the talk tomorrow that there's yet a lot to be understood Yeah in this kind of domain all right, so Quantum simulation of spin systems so what does one go on about yeah, and let's Make it simple for a start. Let's look as an example at polar molecules Yeah, and we are talking about really low temperatures Yeah, where the only degrees of excitation that are relevant are the rotational degrees of freedom and then in the very simple approximation you can describe the Dynamics of such or the energetics of such a polar molecule via a rigid rotor and the Hamiltonian is very simple So it's just h times b b times Angular momentum operator squared and you know then the eigenvalues are h time b j times j plus 1 just Ordinary quantum angular momentum description and these the rotational constant which is in the order of tens of gigahertz for instance So then you look at the spectrum and you see well it looks like this Well, you have j equals zero at zero energy then j equals one j equals two j equals three and you see an important feature Here's that the spectrum is long nonlinear right so what you then can do is you can say well Let's just forget about this highlighting states j equals two j equals three. Let's just stick to those two down here and regard them as two states of a fictitious spin particle, so this is As simple as it is Essentially, yeah, so there you go with your spin one-half particle Of course, you can take into account more levels and then you get higher representations of spins But in the end, okay, this is what we are going to adhere to in the course of this talk So and then you can ask well now what type of Hamiltonians Can I engineer that describe the dynamics or that will govern the dynamics of this fictitious ensemble of spin particles? Yeah, and one thing you can do is you can apply local fields Yeah, for instance laser fields or microwave fields in this case that couple this down state this j equals zero state to the j equals one state Yeah, and You see what you do is the Hamiltonian that you can write down. I mean I promise you it was going to be extremely simple Looks like this. Yeah, so coupling from the downside to the upstate upstate to the downstate because it's coherent It's also reversible and then your Hamiltonian on a single-body level is just omega Which is kind of the strength typically called the Rabi frequency of this coupling times sigma x which is the Pauli x matrix So those are the local fields Yeah, this is not really what we are interested in because we want to do interactions Yeah, so now the classes of interactions that you can establish in these systems and we see examples Later on are for instance exchange. Yeah, where you see that one particle is in the excited state It will get de-excited and Then conversely another particle is promoted from the de-excited state to the excited state So and you can formulate this in this bra cat notation like this Yeah, or in the terms of Pauli raising and lowering operators like this Yeah, so this is exchange and how would how you would formulate it and On the other hand what you can also have is for instance density density interaction where you say this is actually just as a remark important in the context of Rydberg atoms where the Particles only interact when they are excited. Yeah, and the same game. Yeah, you just say The projectors on the excited states and then you have an interaction strength And this is how your Hamiltonian looks and now you're actually Prepared to do many body physics because you can now take Don't in the ensemble of particles put them into a lettuce for instance switch on the interaction What type you are preferring what not? Is your decision and then study for instance dynamics or phases? And I mean this can keep a whole field busy for the next five years without problems so Okay, let's be more concrete and let's now look at a particular manifestation yeah of this and kind of physics and I Think a very nice Example of how one can actually induce long-range interactions between fictitious spin particles is given by Irons that are trapped For instance in a in a portrait. Yeah, and you know what happens with irons Yeah, is when they are sufficiently cold they will crystallize in this trap Because you have the coolant repulsion and this is counteracted by the trapping potential and you see now you have this Harmonic trapping potential the irons repel each other. So and they assume an equilibrium position But what you can do now is you can of course push these irons a bit Yeah, and then they start to wiggle and they start to oscillate and those are phonons Yeah, and you can use those phonons there to if to mediate interactions. Yeah, and so how is this done in principle? So I told you now, okay the external dynamics of these Particles can be described by phonons you write down this Hamiltonian. So you see there's a number of modes Yeah, typically three times the number of irons if you are three dimensional and These a dagger and a k are just a creation annihilation operation these these phone and once So and now our goal is to use those phonons to create an effective spin-spin interaction between Internal degrees of the irons. So what are internal degrees? So and this is the ugly truth. Yeah, when you when you okay I'm theorist I can laugh at this but there are people who really have to understand this in the lab So this is for instance a Eterbian plus yeah that you can trap and you see there's a manifold of states here that you will also some of which You actually also use yeah to do manipulations But then you see the important bit now for our purposes are those two states down here Yeah, which form our fictitious spin one-half particle. Yeah, and now you put them in the trap and Still what you need to do is you want to make them interact with the phonons Yeah, so and this is not entirely trivial because what you want to do is you you want to couple the internal degree of freedom Yeah, those two levels here to the position degree of freedom. Yeah, which ultimately yeah whose dynamics ultimately is described by those Phonons and how can you do it you can do it for instance by Standing-wave laser beam. Yeah, so let's say you have an electric field Yeah, and you have let's say you have two lasers that interfere and you get a modulation in your field intensity And you see then you can now position the iron within this laser beam in a particular field So and now once you do this You find that you can describe the physics of this two-level system like this Yeah, so it's just a field. Yeah, remember two slides ago that couples that down in the upstate you can write down the Hamiltonia But this field let's say is detuned with respect to this transition. It's not really on resonance Yeah, and what you then can show I don't want to put much math into here is that these levels Acquire an energy shift in this light field Yeah, and this energy shift is proportional to the intensity of this electric field. Yeah, and now What you do is since these light field field especially in homogeneous you actually in induce a level shift so shift in the energy of these two Ionic states that is dependent on their position within the trap. Yeah, which you can formulate like this So this is our internal degree of freedom here Sigma Z. Yeah relating those two levels Times the intensity of the light field, which is a function of position. So Okay, so now if you are clever about this what you can do is you can position the irons such that they are actually Located on the slope. Yeah of the standing wave field and then you see already once this is achieved You can actually do a linear approximation. Yeah, so you assume it Okay, it's just a linear coupling here as a function of the position and then you can approximately write Okay, you see all these dirty details like coupling constants. I put under the carpet You find and in the linear approximation a coupling Hamiltonian that looks like this Yeah, where you see that Sigma Z the internal degrees of freedom are coupled to the photon degrees of freedom linear A phonon degrees of freedom in a linear fashion. So now you can formulate the full Hamiltonian which and compasses the phonon dynamics and the coupling between the internal and phonon degrees of freedom. Okay, so but now Our ultimate goal is of course to obtain an effective spin Hamiltonian So what we want to do is we want to get rid of those phonons and this we can actually do a relatively elegantly in this a simple situation we can apply unitary transformation which Factuates a shift in the in the phonon modes. Yeah, because okay, if you think back to your quantum optics lecture, you can identify this Unitary transformation as a product of displacement operators. Yeah that This plays your phonon modes In a way that depends on the state of the irons. Yeah, and once you do this You find you first can transform the Free term of the phonons. Yeah, a digger a so this will remain the same plus a correction So you see plus something that very much looks like what we want to achieve namely an interaction only between the spins Minus the interaction Hamiltonian So this is just what comes out and then if you transform the interaction Hamiltonian you get the interaction Hamiltonian minus two times This term. Yeah, and you see already if you put everything together The interaction Hamiltonians cancel. Yeah, and you end up With a situation where the sorry where the photon if at the phonons Decouple completely from the spins because you get something that is just a digger a plus This term added to that term so phonons and spins don't see each other anymore So you can disregard the phonons. You just look at the spins. Yeah, and this is a Hamiltonian you can get Yeah, this is how it works. So essentially the phonons. I mean like everything in nature somehow Yeah, so the phonons mediate interactions between the spins and then and The nice thing is that okay this Coefficient matrix JKN can be tuned. Yeah, so for instance, I will show you examples in a second. Yeah, it can Be tuned such that for instance at obey the power law behavior. Yeah, whose power can be changed. Yeah, it can Be done such that you for instance Get nearest neighbor interaction that could be ferromagnetic or could be anti ferromagnetic. Yeah, so you can control the sign Yeah, and for instance achieve a frustrated situation which we are going to talk about in a second and what I totally put under the rug Now, but you have to believe me. You can also generate terms like this. Yeah, where you create an effective Magnetic field for instance. Yeah, you can shine in additional laser field And then you see you can for instance study phase transitions in this problem when you tune the strength of the magnetic field with respect To the coupling so this has all been done experimentally and I just want to show you two examples, which I find nicer as a relatively old paper already, I think 2010 that's not that old, but it's also not that recent where in Chris Monroe's group they prepared so to say the paradigm as you wish System of frustrated magnetism. We'll just say, okay We have three irons in the trap and and they each represent a spin Interestingly, okay, the geometry in the trap is nothing like this triangle that you see here So they're just linear irons But as a matter of fact, you can now tune so to say the couplings between those irons one and two and two and three and Pretend that they sort of say were arranged on the triangle in space and you can now change the Interaction coefficient such that of course, I mean, you know that those two spins one to be in a fair anti-firm magnetic state, but then of course this Frustrates this spin who doesn't know whether it should point up or down So and another thing which is maybe more impressive. Yeah, which really brings this Onto it would say completely different level. Yeah, you go from three spins to hundreds of spins I think there are roughly 200 spins here and this Ex experiment conducted in John Bollinger's group and there they're shown nicely what really the power of these these Ionic systems is namely that you can tune the inter ion interaction or inter spin interaction To a great deal to a great degree. Yeah, you can for instance Get situations are each dot here corresponds to an iron and each Line so to say corresponds to an interaction between the irons and the color of the line Corresponds to the interaction strength. So essentially what you get here is some kind of all-to-all interaction. Yeah, and here you can Change your physical parameters with which you drive the system in a way that you yeah Change also the interaction behavior in a way that you can for instance go from a power law interaction between the irons with the power of approximately 72 to a power of zero which would correspond to this all-to-all connectivity So I hope I could convince you that this is sort of interesting and I put my faith into Christian I don't know. Yes. Yeah, that he will elaborate on this further and Tells us a story from the perspective of in Spock All right. I am here Christian. Here's your and but all right, so This is now Showing you the basic mechanism. Yeah, so you have these particles and something is exchanged between the particles you Integrate out so to say this this field that mediates interactions and you end up with some kind of long-range interactions between the spins and of course now we can do this also in free space Yeah, and this is what the mission is for the next part of the talk and So as a example that I want to pick and I will simplify it even further our atoms or molecules as I showed you before that Are described by such a level scheme? Yeah, for instance the low-lying states of a polar molecule that I just discussed are described by this system So we're j equals zero state and a j equals one state Yeah, and there's degeneracy which I didn't mention before but let's forget for the moment that there are three degenerate states Let's just pretend that there's only one and then you can describe an ensemble of atoms here With this kind of Hamiltonian where we introduced this notation Yeah, but basically B is an operator that Annihilates an atom from the j equals one M equals zero state and creates an atom in the j equals zero M equals zero state It's just the lowering operator and be dagger is Conversely the raising the raising operator So now we have those two atoms and the atoms are of course embedded in the electromagnetic field here, which is kind of everywhere and the Physics of the field of the free field is described again by a quadratic Hamiltonian just like the phonons that I Introduced in the iron trap. It's a little bit more complicated because okay here and energy that is Proportional to the to the modulus of the momentum It's a normal dispersion relation for for light and also what you have to consider is that light has two polarizations Yeah, but in the end of the day. Yeah, it's similar It's just a quadratic Hamiltonian that describes the energy of the light field So but now the important bit is of course that there's a coupling Yeah, between the atoms and the light and the coupling is always in the lowest order approximation that we will be following here Described by dipole coupling. Yeah, where the magnetic dipole of the Sorry, but the electric dipole of the of the Atom or a molecule is coupling to the electric field at the position of the atom or molecule. So Just because I wanted to show you at least some to some extent, okay Well, how complex the physics is it's not really complex. That's the message here So in the end if you describe so to say the dynamics of the system, okay You have to commit at some point to a basis Let's say, yeah, this is usually basis one takes and then you find that you have now this operator here Yeah, which is the dipole operator you represent this in this kind of Matrix form when you see okay, you can now this operator effect rates transitions between the ground state Which is here and high-lying states and equals one zero and minus one and those coupling strengths Determined by these matrix elements. Yeah, and as I said before we just pretend We only have one transition and then very easily you can show that for the zero zero one zero transition The Hamiltonian looks like this. Yeah, you just sum over all the atoms and you see now each atom Couples to the mug couples to the electric field at its position All right, so let's put everything together now. We have an ensemble. Yeah of atoms So and now we have of course the free dynamics Yeah, atoms and light and what we have on top of that is now this interaction Hamiltonian Yeah, so which is nothing else, but the Hamiltonian we Had before the interaction Hamilton we had before but writing the electric field. Yeah in the eigen mode decomposition Yeah, so it looks like that and then you have an ugly coupling constant So, okay, this is what it is and now you have to again do the same procedure that we that we did for the Ions, yeah, and we will find that the interaction with light mediates a double-double interaction between the atoms and Funnily enough this interaction is not only coherent so it has also different has also a Disciptive component which leads for instance to collective decay which we will briefly glance over and the interesting thing is also that So to say the figure of merit that quantifies the strength of dissipation Strength of interaction and also the degree of collectiveness of dissipation is the transition wavelength So this transition the wavelength corresponding to this transition Divided by the interatomic distance and some factors of 2 pi which sometimes Will disappoint you as a theorist because the 2 pi makes a difference between strong and weak coupling All right So let's now follow the the same procedure that we that we followed for the case of the Ions and let's obtain an effective equation of motion for the atoms only So on the strategy is outline is followed Follow so now we formally solved the Heisenberg equation of motion for the radiation field. So which is governed by the evolution of these operators a Qp so Q again is the is the wave vector and P the polarization And we just plug this into the Heisenberg equation of motion. They are this commutator and out comes this yeah, so you see now the As well this is relatively easy to understand I mean there's a free evolution of course of the of the a operators of the inhalation operators of the field But then there's always this coupling To the to the atoms which is the interesting bit. Yeah, so we can formally solve this So again free evolution coupling to the atoms and of course now You have to make some simplifications and okay. I don't want to go through everything but okay There is a number of steps involved here But it boils down to to this thing that you say well You just approximately consider the evolution of the atomic operators is free So you just say that those objects here, yeah, which are time-dependent undergo a free evolution Yeah, so with the rotate with the atomic frequency. This also assumes that there's no retardation Yeah, okay, so what means that you put essentially the light speed to infinity Which is a nice complement to h by equals zero in this case and What you then have to use is in order to simplify these expressions further are some technical tricks Yeah, with which you simplify these integrals for instance. I mean you make use of this height lasita function which turns yeah, when you put this in you see you get integrals over over Oscillating exponentials. Yeah, you just we replace so to say expressions by those those bits here Which turn these integrations into principal value integrations that you can still handle but the important bit is that okay this expression here And this is ubiquitous to calculations of that kind be it for instance The consideration of decaying to level at him in the radiation field, which is like a Wigner-Weisskopf model for instance those Funny generalized functions here effectuate dissipation. Yeah, so on the one hand you have coherent coupling but on the other hand they Open your system. Yeah, and you will see in a second how this manifests itself and and generate decay All right, and this is the contour in contrast to the trapped ions. Yeah, but we just had very discrete set of vibrational modes Yeah, this is really a manifestation of a coupling to a continuum Yeah, where this difference comes in and you find this in all sorts of physical context like nuclear decay for instance I'm sure you have all seen this since in your respective field so and what you can now do is you can now Use this trick so to say and you will write down the evolution of your of your light operators, and then you see a nice thing namely that this suddenly becomes local in time Yeah, so and this is good because you can now there should be a T here. You can now really re-express yeah the light operators Yeah at time T through atomic operators at time T and this is nice Yeah, so what you now need to do is you need to calculate the evolution equation of an atomic operator This will be of course coupled by virtue of the Heisenberg equations to light operators And then you can just replace the light operators by virtue of this expression through atomic operators and then you end up with an equation of motion that describes the dynamics of the atomic ensemble so and a convenient way of formulating this is to write down the master equation that Describes the evolution of the density matrix of this atomic ensemble and you see this where it looks very familiar This is just a for Neumann equation here Which is a bit modified because you have to include now these Discipretive processes and allow so to say that non-unitary Dynamics that is caused by the coupling to the continuum of of the light field takes place all right, so let's talk first about the coherent dynamics and For the moment. We just only make we make our life simple. We just look at resident contributions. Yeah, what does it mean? So these are just contributions that Conserve the number of excitations you see I mean the semitonian Is here composed out of the free bit? Yeah, which you see measures just the energy of every single atom and then you see this is now an interaction term which Describes a flip-flop interaction between atoms. So this is really coming out from this treatment It's just an exchange interaction. Yeah between the atoms where one atom is getting De-excited while the other atom is excited. Of course, this is Connected to an exchange of photons which is just integrated out in this picture and you write this down as a direct interaction between the atoms okay, so now let's look at the interaction strength and This is also not completely trivial and this is also related to the fact that these funny integrals appear that I showed you on the previous slide so You see that actually the interaction strength is expressed in this kind of Through this set of functions here, yeah, which are the straggler-bessel functions of second kind. Yeah, sorry And you see now if you it's more probably more instructive here to plot the interaction strength here You see for small distances, you see the interaction strength the case is 1 over r cubed Yeah, and then for larger distances and large I mean distances that are large compared to the transition wavelength as I told you before You get these leveling off to zero in an oscillatory fashion, which is all to the specific shape of those functions Okay, so this is the interaction we we get among the atoms and then Let's now consider this perhaps a bit more interesting term which Effectuates the dissipative dynamics and this is written in this form. Yeah, so you see it's not it doesn't have a commutator structure It has this So-called lindblatt-kosarkovsky form. Yeah, I'm hope hopefully the mathematical physicists like that because there are several names for this and and you see what it describes are actually Disciptive processes meaning that those are processes that are Non-unitary but yet that preserves the norm of the density matrix the trace of the density matrix and the positivity Yeah, this is important You can show that this is so to say the most general form of Operator that actually has those obeys those satisfies those two conditions preservation of trace and Positivity of the density matrix. So the interesting bit are now those coefficients. Yeah in front of and don't worry There will be a physical picture within one minute The interesting bit are now those coefficients. Yeah, because you can calculate them again and well Again, they look funny like the interaction strengths do so but now they involve the spherical basal functions of the first kite okay, so be it yeah, and Just so to say want to preempt the next slide. So what we will see is again like in this interaction case where we saw that this reduced distance. Yeah, this inter particle distance divided by Wavelength Place a decisive role. Yeah, what we will find that when this Reduced distance kappa KL is much larger than one. So then atoms decay independently Yeah, so they will couple to the radiation field. They will enter undergoes spontaneous emission Yeah, it's spontaneous decay by emitting photons into the radiation field. But now if we Consider the the other limit. Yeah, where this reduced distance is much smaller than one We will find that the decay actually becomes collective. Yeah, and this is going to be interesting so let's look at this and In a specific example namely of two atoms. Yeah, so this is again the structure of the dissipator and now let's look a bit more closely at this Interaction at this coefficient matrix as you wish. So, okay This blue plot you have seen before right? So this red plot is now important. So what this shows? Yeah, is the off diagonal element? Yeah, gamma one two as a function of distance Yeah, so you see this off diagonal element Level goes to a constant when as soon as the atoms approach each other and the case to zero. Yeah, when When the atoms are far apart from each other that means when the atoms are far apart from each other as I said in the previous slide Yeah, there are no off diagonal entries here meaning okay that this some is only composed of diagonal entries so it's just independent some summons that correspond to Radiative decay of the atom. So nothing interesting is happening. Everything is decaying so to say individually But now if you when you approach this limit here where both atoms sit on the same position Then you can write the dense at the coefficient matrix in front of this expression here as this matrix Yeah, parametrized by some constant, which is not important at the moment. Yeah, which you can also Calculate from first principles so now Clearly this is a is a matrix that we can diagonalize and we can bring it into a diagonal form by now introducing Collective modes. Yeah, so this is the diagonal form. Just look at so this is what we want to achieve So you see the important bit here is just follow the point. Yeah is that there's just a gamma k Yeah, not a gamma k l. So this is just a gamma k Yeah, so and you see the two modes that you need to introduce in order to accomplish this form are the Symmetric superposition and the anti-symmetric superposition of the of the two atomic Operators, yeah, and you see that Decay rate so this gamma one the eigenvalue of this mode of the symmetric mode is two times gamma So this atomic collective state decays it twice the single atom decay rate Yeah, so this is a super radiant mode clear super radiant decays faster But then there's also the extreme opposite limit namely the one that corresponds to this anti-symmetric mode So with zero eigenvalue, which has zero decay rate at least in this level of approximation So this corresponds to a sub radiant mode So alright, so how does this manifest itself? Now, okay What can you do? Yeah in order to probe the dynamics of such a of such an atomic or some and to in order to see Whether there's physics like the one that I just explained so you can take a laser shine it on the cloud of atoms and Look at the scattered photons. Yeah, so so turn on the laser shine For some time then you can also turn it off and look so to say at the fluorescence of the atoms Yeah, and now Let's consider a very simple situation Let's just show you some experiment in the second in a simple situation where you have just a bunch of atoms sitting on a Line so this is the geometry and okay There's a specific angle with respect to the laser doesn't matter for the moment. Yeah, so and now I Also told you that the figure of merit. Yeah, that determines whether the system is super whether the system is collective or not is the distance divided by The way transition wavelength so and what we now do is we Decrease the distance between the atoms if as we go from bottom to top Yeah, so it's not very easy to see but okay what you see now So this is sort of say the the frequency axis and now you turn on your this laser Irradiate the atoms and look at the light that is coming out whenever there's light coming out So to say the signal then this gets darker and you see that is it there's a big blob here around zero Fine, so this is just a light scattered of your atoms nearly at resonance But then you see this big peak here actually gets broader. So it gets you think it's it's vanishing. It's not vanishing. It's really Fading away. It gets much much broader as you increase the distance Yeah, decrease the distance between the atoms. So you make the system more and more collected So this this peak becomes gigantically broad. Yeah, so this corresponds to the super radiant mode But then at the same time You see that There these these finer lines appearing those are very narrow and the width of these lines is determined by their decay rate, right? so but you see You do the fact that they are very narrow. So I mean I tell you this is sort of say here the single atom decay rate You see there's a very broad peak This is much much smaller than a single atom decay rate You see those are sub radiant modes and they bend away because also these modes in your system They are interacting. Yeah, so this is what I told you before It's not only the patient's also the interaction that shifts then these lines away from resonance and you can now See them as sub radiant modes and now if you just take two cross sections you see here This is the big peak. Yeah, this is the super radiant peak which dominates everything So these and and here these small peaks that are the sub radiant modes which become more and more separated The more closely those atoms become and you see these super radiant peak. I mean it was more like this Yeah, so it's Entically broad Okay, and the red dots are just a probability density distributions not so important. Okay Let's look at the experiment and I Made the slide before I was Aware of the fact that Robon was going to be here So him hopefully forgives me and he's I guess gonna talk about this in far more detail. So What what he observed in the experiment was exactly this behavior Yeah, of the emergence of these long time scales of these very Small decay rates when Irradiating a dense atomic ensemble. So and I just show you the settings. So you have your probe laser you radiate the system Yeah, and then you detect what's coming out So those are the scattered photons and you detect the signal as a function of time and then you can see this is a lock Lin lock plot you see this becomes a straight line you can think of this as kind of an exponential curve and you can fit a Decay rate here. Yeah, and what you find now is If you make the sample denser and denser that this decay becomes more slower and slower slower and slower and slower So and you see this must be at least this is the idea due to the presence of these sub-radiant modes Yeah, because those are the modes collective modes of your in your atomic ensemble that take a long time To decay and just for sake of comparison to see how dramatic this actually is Look here at the single atom decay signal. So this is basically a vertical line Yeah, so so the timescales are much much prolonged over what you would see in the non-interacting case and Then a second Slide which Is actually interesting because it shows you that not everything is understood in this system is It's this one. Yeah, so it's exactly about it's similar. Let's say so again You you have your ensemble of two level systems You shine in a light at a specific frequency And you measure this frequency with respect to it's detuning with respect to the atomic transition frequency and you see okay, you get a peak so you get a peak you get a peak you get a peak so and now These peaks correspond to different runs of the experiment that are conducted With different with different samples with different densities. Yeah, so the density is increasing. So you would expect The system to become more and more collective Yeah, as the density increases and in fact you see a hint Yeah, you see that the peak gets broader somehow. Okay, which shows you I mean there might be some density related effect But what you don't see is any shift Yeah, of this line. So if you if you just this is experimental data, you see pretty much this line is centered. Yeah, at Zero at the resonance frequency, which is today not really understood. Yeah, and contradicts actually The our understanding of light scattering in dielectric media because they're what one would assume by using standard electrodynamics is a shift of the Of the resonance line that is proportional to the density of the atomic ensemble Yeah, and this really has to do with the fact that the response of the system is collective Yeah, that they are these delocalized modes that you cannot do so to say standard mean field that you usually do in electrodynamics Yeah to describe the response of the system with respect to the light field. No, you have to really consider the collective nature of The response of the system. Okay Yeah, this is what I just said Good So I'm actually a bit too fast. I think but okay, you'll be fine with that. I guess and right so Now I talked about these Interactions that are mediated by the light field and the phonon field and the dissipation that came about so Now let's just really consider the limit where we have static dipole-dipole interaction. So standard Electrostatics where you describe your atom or your polar molecule just as a displacement of two charges You do the multi-pole expansion. Yeah, this is what you would do and you get this kind of Dependence of the interaction potential between them. So this is the static limit and where DK is again the dipole operator so now Okay, this is a very simple exercise you will see but I just want to make the point. Yeah that These although there's dipole-dipole interaction that is sort of say the fundamental interaction between those objects There is not necessarily a one over R cubed Spatial dependence of the interaction energy. Yeah, that's really very much depends on the Situation that you're looking at and I just want to walk you through this. It will be very quick And I'm sure you already understand this but Surprisingly It's not clear to everyone. Yeah, that when you talk to people that also have been working in the field that this seems to be Maybe it's not anymore there, but I witnessed misconceptions concerning the what actually the interaction strength between polar molecules is Yeah, okay right, so What we do is again we take our beloved Polar molecules here and now what we want to ask is What's the interaction between two polar molecules in the ground state? Yeah, so that means okay. They are here each on those two states so for this we need to Commit and and write down explicit form of the dipole operator, which I choose to be this form Yeah, it's just again transition dipole Because each of those states doesn't have a permanent moment despite the fact that those objects are called polar molecules This is due to the fact that the ground state is actually a rotationally symmetric So there's no There's no Preferred direction which was right would give rise to a static dipole. So now we can write down the Hamiltonian. Yeah in this Basis describing those four atomic states that we construct out of these Two atoms and well, it's not no surprise. I mean you see zero both atoms are both Molecules on the ground state one is excited or the others excited or both are excited and then if you take the dipole operator and just plug It in here and make some simplifications Then you see okay the interaction is indeed proportional to one over R cubed, but of course it's off diagonal So now you calculate the ground state energy and then you see well, they're actually two limits which are This distinguished are separated by a characteristic length scale which is given by this figure here Which somehow compares The strength of the dipole interaction to the energetic separation Yeah, of the levels of the j equals zero and the j equals one level of the polar molecule So when you find that now, I mean, this is what you would expect Yeah, if and now in the van der Waals interaction so the interaction in one regime where the Atoms are far polar molecules are far apart the interaction is just not strong enough Yeah to overcome the single particle energy that means You you will find a regime in which you really dominated by this bit here If you now do the expansion of this of this energy you find you get a one over R to the six Dependents of the interaction strength. So for large distances, you have a van der Waals potential So not really surprising and then for small distances. Yeah, when your interaction strength allows to overcome this Single-body Excitation then you can hybridize your system and you obtain actually Interaction curves that have a one over R cube behavior. Okay, so this is just not not very deep, but I think it's worth Spending 30 seconds on thinking about this All right, so now how can we actually generate static dipoles? Yeah, because I mean for instance with these polar molecules This is one thing that they were advertised for we can one promise was that they can for instance be used to Engineer instances of matter in which particles interact with one of our cube potential So what you need to do this for this is you need to expose these polar molecules to strong electric fields Yeah, let's say for the sake of simplicity. We take a homogeneous electric field We point it in the z direction and then you see that the Hamiltonian of our polar molecule is just bj squared So this is just rotational degrees of freedom plus this coupling To the electric field as the dipole coupling again and D is the the dipole moment of the Polar molecule which is on the order of one to ten D by and So now what you find is when you look at the at this spectrum of a polar molecule as a function of the electric field strength then you find of course at Zero field you again find the field field states clearly it's fairly symmetric no permanent dipole moment Interact with the van der Waals interaction and then here and this is important Yeah, it's strong field where you actually hybridize These rotational levels you find a linear dependence of your entire of your single-body energy With the electric field strength and you know well that the energy of a dipole In an electric field is actually proportional to the electric field strength, and this is nice. You cannot directly see read off Essentially the dipolar that the strength of the electric dipole here by the slope Of this curve and now if you have two such polarized objects You can write down the interaction energy between two polar molecules in the presence of these electric field It hybridizes the states and you find the form that is very often used also in the context of magnetic atoms where you where the Electric field is just replaced by a magnetic field in order to align the magnetic dipoles And you get this familiar form of the dipole-dipole interaction where you see also this magic angle When this is just the cosine squared is just one over squared of three You see that the dipole-dipole interaction is vanishing. So this is just to clarify this and then I want to Conclude you on the next three slides or so With some let's say more recent developments. I mean also, I mean this started Probably if you dig deep enough in the literature 10-12 years ago that relates to the use of dressed interactions to engineer many-body systems with which Have funny behavior exhibit funny behavior and I will show you some examples All right. So What do we need? Yeah, so our starting point is now the situation where we select the state Yeah, which has no permanent dipole moment starting dipole moment and a state with that Has a permanent dipole moment. Okay. How how can we achieve this? Yeah, we can for instance achieve this by Looking now at those two states of our of our molecule And now we we see that if we choose the electric field in a specific way And namely we set the electric field to be at this point. We find a situation where this state here the blue one is Having a slope so it has a dependence its energy depends on the strength of the electric field. So there's a dipole moment Whereas here the slope is zero. That means to first order. There is no permanent dipole moment in this state so and You see I mean before I call this state down and this state up. So just for the sake of convenience, I I changed this notation round and and Elabel those states like that. So now what we do is we take those two states and we couple them With the laser beam of resonantly in this case That means we have here the term that excites the downstate to the upstate and vice versa because its quantum It's reversible, but still if we wanted to excite the the highlighting upstate. Yeah, we have to pay a penalty Which is given by this parameter delta in this so-called detuning So and now by virtue of the choice of those states we find also that only Particles in the upstate actually interact with an interaction that is proportional to 1 over r cubed And it's actually a static dipole-dipole interaction that I showed you on the previous slide. So now In order to now calculate the effective interaction between those particles. Yeah, so they interact In the excited state with this interaction. They are driven Off-resonantly by this radiation field and prepared initially in the downstate So then what we have to do is we just have to Do fourth-order perturbation theory and the small parameter omega over delta and we obtain an effective Interaction potential between those atoms and this is looks kind of funny. So this so-called dress state potential has a long range 1 over r cubed Tail, yeah, this one here in the dress regime. I will come to Explain this in a second and it has this short range flat top Yeah, and the characteristic length scale is given by this combination of the detuning and and the dipole moment So now let's consider the regime to the right Yeah, so where the stress potential is approximately going as 1 over r cubed. So what is happening here in the physics? Yeah, so you find in this regime you can just Conveniently consider both atoms separately. You can just say, okay. There's this laser Yeah, this laser couples weakly the downstate to the excited state And now you can calculate the eigenstate the actual dressed atomic state which looks like this So you see that it's just mainly the downstate With a weak at mixture of the upstate. So and this is a small coefficient. So Meaning we get mixture. So now you can calculate the interaction energy You just calculate the expectation value and then you'll find that you indeed get an interaction strength now that is again the dipole-dipole interaction Yeah Go as 1 over r to the 3 like here but With a pre-factor that goes as omega over delta to the fourth So it's weakened with respect to the original dipole-dipole interaction But the important bit is that now if you are distances below this critical radius It is not sufficient to anymore consider only the single-body picture. So we really have to consider the whole picture of pair states. Yeah, so you just just a Collective picture as you wish. Yeah, and this will come back also tomorrow. I guess in robots talk when you talk about blockade and and Collective Rabi oscillations and whatnot. So it's essentially the similar story then What you find is that in fact The picture looks like this. Yeah, so you can now Laser photon can bring you to here and to there or to here and to there Nowhere in both cases. Yeah, so this is the the simple but sad truth. So there's no coupling whatsoever. Yeah between Within some approximations of this laser to the system Which gives rise to any interaction effect. So here basically no interaction is taking place Yeah, and you see that in effect this results then in this flat dependence of the of the interaction potential So one of our tail Unflat top. So now Is this just fantasy? No, it's not. Yeah, it has been observed now in more and more labs But I think the first ones that really did this In a way that was conclusive Where those guys in the in the Biedermann group I think in in Sandhya if I'm not mistaken and what they do is they excite Rootberg states. Yeah, so I haven't really talked about this today. Yeah, we will talk about this tomorrow in detail. Yeah, but The details do not matter. So the setting is exactly the same Yeah, as I unfortunately one cannot see it really well due to the resolution the the general setting is exactly the same that I just That I just outlined doesn't matter and what they do is they they They do spectroscopy on the system and also the details don't matter. Yeah, what they can do is with the spectroscopy They can monitor the interaction shifts. Yeah between Atoms that are Broad closer and closer and closer and closer Yeah, and you see now just like in the case that I discussed before here at large distances You get some kind of power law Growth. Yeah, just flipped upside down due to the nature of the experiment. They are conducting, but it's the same physics And you see there's some power law behavior here and then it results in the end in a flat top And people this is was observed now on the level of two atoms and of of course the challenge is and and for instance the lab of Christian Groß and Emmanuel Bloch and Garching they they are doing experiments along those lines and Stuttgart people as well Yeah, well you have started with that and and They want to transfer this to the many-body domain and study interesting many-body physics. So what could this be? So and I just give you Two examples here. So one example is this is also Say from six years ago is you can now Look at a 2d system. Yeah in which bosons Interact with this funny flat top potential. So this was done by Guido Pobilo in Back then still in Peter Zoller's group. So and those are quantum Monte Carlo simulations and what you see are snapshots of the density of the Snapshots of the exactly density of the of the bosons now for different temperatures So and what you do is you start from high temperature and you decrease the temperature And then what you see is of course at high temperatures It's kind of a random gas but now when you lower the temperature What happens is you have this formation of the of bubbles Yeah, and this means that those are particles that are kind of accumulating on the flat top Yeah, so they the with around on the flat top of the interaction potential So they are confined in the region where there's effectively no potential visible for them And they build a they they form a blob and now you have a number of those blobs Yeah, and these blobs now crystallize. So what you have now is And within each of these blocks kind of a mini bc Yeah, and you still have a crystalline structure that is imposed By this van der Waals tail. Yeah, it's like a Wigner crystal. So to say of these blobs Yeah, and then you can also have exchange between those blobs and this is perhaps can be thought of a Well manifestation of a super solid. Yeah, we have this coherence over the entire system But at the same time Translational symmetry breaking All right, and then another thing, which is also theoretical and also it happens to be from the Zoller group I should have taken a different example is More complicated than that. I guess I don't know. I mean the experimentalist. It's probably everything complicated Robert is laughing. I don't know probably Yeah, all right. So what one can also do with them one can Generalize this concept of these stress interactions one Can look more more complicated settings. Yeah, and then one idea is was to use them The details are written in this paper here for instance exotic types of matter like spin ice Yeah, we have many body interactions and also many body exchange interactions in order to Make these system quantum. Yeah, and this is sort of say the perspective Yeah, there are lots of things that still can be done with that. I'm sure Uh, they are by no means exhausted those possibilities, but still I mean This is a real challenge for experimentalists to get to this level of control All right, so this brings me to the summary Basic message cold atoms molecules and ions allow to engineer spin systems Yeah, with a tunable interaction range You can go from exchange interaction, dipole-dipole interaction, van der Waals interaction up to this funny dress state interactions We have not even talked about angular dependence and what you can do with that. So there Still is still a plethora of possibilities so I showed you that these interactions are mediated through photon exchange and this also gives rise to collective dissipation, which you can observe in experiments through super and sub radiance and tomorrow We will Hopefully be Little less basic. Yeah, we really will do some many body physics with those systems at the moment There was just few body physics with some tasters on on on many body effects But tomorrow we will focus on root book atoms. Yeah, starting with Robert's talk and then I try to Hopefully add more things to that from a theory perspective where we will discuss interacting atoms in root book states many Moneybody dynamics with competing classical and quantum effects And I will show you that dynamics is more than static. So really looking at the evolution of systems where you can have a situation the Stationary state that means the equilibrium state is completely trivial and structural as but the evolution towards it is highly intricate and Interesting and also we will be talking about non-equilibrium phase transitions that you can observe in those systems Thanks a lot for your patience and attention