 Hello for the invitation here, and thank you for being here. So I'm an experimentalist working on ultra-cold atom physics. And in these two lectures, I'm going to present to you. I'd like to give you a brief introduction into the field from my experimental point of view. It's complementary to what John will talk about, so I think you'll get two quite complementary views on different topics we're looking at with these ultra-cold atoms in optical lattices or in the continuum in low-dimensional systems. So I mean, I made an outline of some topics I would like to discuss. But the pace of how fast we go through all of this is really determined by you. So I have the tendency to always speed up. You will notice that. And the way to keep me slow is to continuously ask questions. So if you have a question, you just raise your hand. You ask a question, and that slows me down and also makes me aware of whatever I should go a little bit more into depth. So today, in the first lecture, I'd like to talk a little bit about introduction into the field, what we're trying to achieve in this field, and give you a brief review of these optical lattice basics of these artificial crystals formed by laser light that we use in our systems, and tell you a little bit about the detection methods we use. And then I'll focus actually in these two lectures. I wanted to focus on the strong correlation physics in the Hubbard model, tell you a little bit how we image the systems, what kind of correlations we can measure in these systems. Talk today maybe as a final wrap up about a nice kind of stat mech effect of absolute negative temperatures. It's kind of a nice, I think you will enjoy the topic at the end of the lecture today if we get there. And then tomorrow, we'll just focus on quantum magnetism with ultra-cold quantum gases. So there are a lot of different topics I'd like to go through, some of them very modern, very new, very recent results, some of them already a bit older. But I think they all build up nicely in showing you what the new kind of techniques and experiments are that we can use to unravel this kind of interesting phenomena, intriguing phenomena of strong correlation physics. So that's what I want to go through in these two lectures. I'll make the slides available as PDF copies to you so you can download them. I think you have that on the web page of the course. Okay, so let's get started. And since I probably imagine all of you have a very different background, some of you might be experts in cold atom physics, some of you are hearing this for the first time, I just want to go a little bit through the motivation why we're doing this. So of course, I mean, probably I don't need to explain a lot why we want to understand many body quantum systems. They are the basis for a lot of technological effects that are very relevant. They're of course now in this age of quantum technologies highly relevant in building quantum computers, building new quantum metrology devices, quantum sensors. What we're going to address in this talk is basically trying to get a deeper understanding of these strongly correlated quantum materials using ultra-cold atoms and explore kind of the new techniques and possibilities that we can afford in those systems. This builds actually on a very, very decade-long old kind of history in quantum optics. I mean, many, many more people than those two here. Those are maybe just the latest two kind of very famous people, Dave Weinlein and Serge Arosh who are some of the masters in taming individual quantum systems, individual photons, acting with a single atom, individual ions or ion strings in these systems. And today what we want to do is we want to build on this knowledge and build kind of complex quantum systems that allow us to study complex quantum matter in a very controlled setting as you will see and with kind of highly controllable techniques. This is of course what an ultimate quantum computer wants to do. What we're aiming for is a little bit less control but maybe larger systems and then we enter the arena of these quantum simulators as initially proposed in Richard Feynman's vision in the 80s in his talk at MIT. So now of course we're not the only system of ultra-cold atoms where this is explored. So we have ion traps and a bloods group for example in Innsbruck or Chris Monroe at JQI. We have the superconducting devices like in John Martinez group at IBM where actually a lot of similar physics is trying to be explored. We're gonna focus on these ultra-cold atoms in these optical lattices. So the physics that we wanna study in these two lectures is the physics of strong correlations of initially electrons on a lattice. So this is the original kind of motivation for this Hubbard model as I've written it down up here. So you have let's say two spin states, the spin up and the spin down electron which can live on this lattice here. It's a square lattice. The particles can move. So it's not just a spin system though the particles are mobile. So electrons can hop from one lattice side to the next which is given by this first tunneling term here and they can interact when spin up meets a spin down particle on a single site. They interact through this kind of onsite interaction. And this is the basic kind of Hubbard Hamiltonian for fermionic particles for electrons on a lattice that many people believe plays a very important role in explaining most of the correlation, strong correlation physics we see in strongly correlated materials including high TC superconductivity. So what we're gonna talk about is basically how can we realize such systems? What I'm not gonna talk about at all, how does a real material system map onto this system? So I hope you might have other talks dealing with that question. That's of course a huge abstraction. If you go for example to a real high TC compound like this copper oxide you see here whether that is actually realistically described by this simple model system of just electrons moving interacting and that's of course another big question one has to ask and how much of that essential physics of this complex system is captured by the simplest Hamiltonian we can write down for this system. Okay, so the three central goals I think in the cold atom world to basically explore these phenomena of this strongly correlated physics on the one hand as you will see using new probes using new detection methods to analyze physics that might be understood already but to see it in a new light and using new analysis techniques. So this is what I would call new light unknown phenomena. The second I think big goal I think where ultra cold atoms have been extremely successful is really to test quantitative predictions. So I hear many of you are theorists so you might have your model out there that describes a certain system and these are really high precision test arenas where you can test your model. Okay, so we can really go in and say we take this model, this model and this model and we compare it to the experiment and there's no doubt that the mapping of the experiment onto the model is not correct but it's more whether the model itself actually the solution for that model is correct and then we can quantitatively test different predictions and I think a very good example of this is the so famous VCBCS crossover which I'm not gonna talk about in this lecture here where you've really been able to separate between different theories and tell which one is actually the correct one or which is wrong actually. And of course the third maybe most exciting thing is of course when we can look at new phenomena when we can realize new physics or look at physics and parameter regimes look at phases of matter and parameter regimes which are simply not possible in standard correlated materials and I think in the talk you'll see all three aspects of these central goals that we're trying to cover in the research. So just to give a basic idea what we're trying to do why this is such I think a radically different approach than an experimentalist studying real materials. So here I've sketched a real material for you okay so you can barely see it it's small it is angstrom size separations between the atoms in this crystal and of course if you want to study this you have to build a very good microscope like for example an X-ray laser on X-ray microscope that for example people are building in Hamburg with the XFEL to look into the system to study for example with X-ray diffraction what's going on in this system. Now another approach and this is basically what we're going to use here is why don't we just scale this whole thing up by a factor of 10,000 so let's just increase all spacings that are just angstrom size here let's make them micrometer size spacings here let's build a material which is micrometer spacings between the atoms and then of course when we have micrometer spacings between the atoms then we can just use normal very good optics, classical optics to look into this material to study what's going on in there. Actually this scaling up has another advantage that I will discuss actually in a second with you. So it's kind of this blowing up of the material this very radically different approach that we use to start build these artificial quantum materials and study them these enlarged quantum materials. So how do we do that? So let's discuss a little bit how we actually do that in the experiment. So first of all we have to make a lattice and the way how we make a lattice is not by letting the atoms bind to each other to form the crystal structure itself but we actually impose the crystal structure by creating an optical potential for the atoms. So the idea is basically that you take a laser beam and shine it on the atoms. If you take two laser beams you get a standing wave as you see here, two retroreflected beams it gives a standing wave pattern of bright and dark fringes and this intensity pattern is actually experienced by the atom as a potential and actually there's a very simple classical explanation to understand it. If you've never really discussed this, learned this before if you just think about a dipole in an electric field so let's place a dipole in an electric field then the energy of that dipole in the electric field will be just D times E and now let's say this E is the electric field of our laser so it's an oscillating electric field that we're shining onto the atoms, okay? So now you might know that neutral atoms don't have a dipole moment by themselves but if we look at an atom more microscopically and we think of it as this electron bound to the core if we shine an electric field, an oscillating electric field onto this atom we can make this electron oscillate and make its electron cloud oscillate by driving transitions between the atoms so we basically create an oscillating dipole moment which is proportional to the polarizability of the atom so this is where all the atomic physics is in this is what we have to calculate from atomic physics how easily this electron is moved around when we apply an electric field proportional to the applied electric field E and then times the E of t that you see here so this is what we call the induced dipole, D of t that we can induce with the laser light onto the atom and then you can immediately see that this is proportional to E of t squared and this is proportional to the intensity of the applied laser field because the intensity is just one half C epsilon zero E squared so what you see is that through this formula that we can create potentials for the atoms that are proportional to the intensity of the applied light and now whether this potential is repulsive or attractive solely depends on whether the sign of this polarizability this atomic polarizability is positive or negative so if it's positive then we have a minimum of the potential for maximum intensity if it's negative we have a maximum of the potential when the intensity is maximum so we can have two situations when we focus a laser beam so we take a laser beam we send it through a lens we focus it and then here's an intensity maximum at this focal point and if we are for a positive alpha then we can trap the atoms in the center here they will have a potential minimum there if alpha is negative then the atoms are repelled from this intensity maximum because the potential is repulsive and actually you can also easily tell by a very simple classical harmonic oscillator model what this alpha is whether it's kind of is this oscillator is the response of your atom in phase or out of phase with the applied electric field and this you can simply model by thinking of the atom as a simple harmonic oscillator where we apply kind of a drive field which is the oscillating electric field so this is our laser frequency and we have the dipole now oscillating because of the oscillating electric field the electron cloud now what I want to ask you is this dipole in phase or out of phase and to answer this question it's sufficient to basically think of a mechanical oscillator so just think of a spring with a mass attached and now you drive that oscillator okay that's your drive and now what I want to know is this oscillator going to how is this phase of the oscillator correspond going to respond to the phase of your drive field so you see what I mean so you have your oscillator it has a certain, it oscillates here with a certain phase and now this mass is going to oscillate right because you're driving it so what's the phase of that for a classical harmonic oscillator what's the phase of the oscillator relative to your drive field do you remember that? sorry let's go, yeah maybe you just raise your hand and then we can just go you want to try it, yes yes exactly then it's 180 degrees out of phase you can picture that right if you go very slowly you're just lifting the mass up and down just go to the extreme limit so it will be in phase with the drive if you go above the resonance frequency of the oscillator then you will actually see that this mass is oscillating 180 degrees out of phase and actually this simple picture actually also holds for the atom so if we look at the phase of our oscillating dipole which is kind of then the phase of the dipole moment here that we plug in it's in phase with the oscillator up to the resonance frequency omega 0 of our atom and then there's a phase jump to pi and of course you know how sharp this phase jump is depends now on the damping the atomic line width of the transition that you're considering so this is in general what you have and you can immediately see now when the frequency of your drive is below the resonance frequency so if omega is smaller than omega 0 this is what we call red detuning and the frequency of the drive is smaller than your resonance frequency or omega is larger than omega 0 and we also call blue detuning in this case the oscillator is in phase so D is parallel to E so we are in the case where we have an attractive potential okay because D is parallel to E so this scalar product is positive so it's negative here overall so we get an attractive potential whereas when we are blue detuned we get a repulsive potential because this alpha just changes it's 180 degrees out of phase D dot E is just negative and then you get a repulsive potential okay so with that you can actually very simply explain this what we call dipole force acting onto the atom a classical picture nice classical picture which you can explain this forces that are exerted onto the atoms and that's actually true not only for our atoms that works for all polarizable particles you might have seen this also in the context of biophysics where people have these laser tweezers with which they can use a laser focused laser beam to drag around cells and objects and that's simply because the cell is sucked into the intensity maximum of the laser beam okay and the same principle holds here that we use for this dipole trapping okay so if you know all this and everything's too slow you just tell me also we go faster over this so we now understand how we can make a optical potential for the atoms we can just take light fields and any intensity pattern will be mapped onto a potential pattern for the atoms so actually we can create a very simple 1D lattice by interfering these counter propagating beams creating this standing wave interference pattern and let's say this if this has micrometer wavelength then of course this interference pattern has this crystal of light has lambda over two spacing and now of course by Fourier synthesis you can in the synthesize almost any arbitrary lattice pattern you want you just overlap lattice beams in different directions and basically Fourier synthesize almost anything you want what's also nice what I'm not gonna talk about maybe only briefly later is that you can also realize spin dependent potential so you can make potentials where the two spin states that we want to use to mimic electrons in the lattice can see very very different potentials they can even be extreme cases where one of them sees a lattice and the other one does not see a lattice at all okay so they basically experience completely different potentials that's of course something you cannot do on a classical kind of electrons in a lattice okay I have to go a bit closer here so here's a picture that just really works so this is a very beautiful picture if you actually want to see how these pictures have been made this is from Ted Hench's labs which he took did this experiment so you can go to YouTube maybe tonight not now maybe later and you go to his channel superlaser123 it's his name on YouTube and there he has a series of wonderful optics experiments where you can basically see not only this how this pattern was made but a lot of other beautiful optics experiments and really fun to watch so this is a very simple experiment you take a laser beam you expand it make it very big and you send it through to apertures here two holes and then you focus it down with two lenses again and of course what you all know is when you have two openings and you send a laser beam through you get a double slit interference pattern and this is what you see here this nice kind of double slit interference pattern that you get this beautifully regular optical potential which our atoms see as an optical potential now of course this can now easily extend it if you take three laser beams now you see already we get this beautiful triangular hexagonal lattice okay we just have to make a third opening and we get this thing here if we take five then we get something like a quasi-crystal like pattern in the system okay so this is just optical interference this is no mathematical plot or something this is just light interfering and this perfect crystal structure that you get from this light interfering really is seen by the atoms as a potential so what we do in our experiments we make the atoms ultra cold so they can be held by this very feeble optical potential and we can trap the particles and observe them in these crystals of light so this is the first lesson to learn we create the optical potential we create the lattice potential externally by interference of light and we load the atoms into this crystals of light so in our case in all our experiments these atoms are loaded in a vacuum chamber and held in free space just by these crystals of light and have no contact to the outside world so typically this is done with a few thousand particles for the experiments I'll show you in larger systems up to ten thousands of hundred thousands of particles that can be trapped in such optical lattice structures and we'll see actually that we can study very different systems with these we can look at quantum spin systems we can look at particle systems bosons, fermions or even boson fermi mixtures in these systems and we'll see actually we can do that in interesting regimes where calculations become really difficult okay let me just recap why this whole idea of doing condensed matter physics in this artificial crystal works and why it's really the same thing as a strongly correlated material of electrons in a real material and you have one thing to remember of course that the only thing in stat mech that determines whether you are in a many-body quantum degenerate regime for example where collective quantum behavior shows up is whether the de Broglie wavelength of the particles to the inter particle spacing is large compared to one okay so this is what you have to achieve and now in our case of course I told you the D is now ten thousand times bigger than in a real material that's micrometer size which means of course that the lambda has to be ten thousand times bigger also to be in the same quantum degenerate regime for the particles and if you can achieve that then indeed you have the same kind of quantum collective quantum effects although in vastly different density regimes and temperature regimes so here you see these ultra cold atoms they work typically in a regime of 10 to the 14 atoms per cubic centimeters this is extremely dilute this is five orders of magnitude more dilute than the air surrounding us whereas real materials they are working at you know ten orders of magnitude higher densities but nevertheless the physics is the same physics we talk about in these ultra cold atom systems as in the one for strongly correlated electrons okay in practice this now comes my experimental slide this is not so easy and this shows a little bit that it's you know we need a lot of optics a lot of lasers on your table to make the atoms cold to trap them in the lattices to manipulate them and this is one of our kind of maybe most complex experiments that's not even the experiment yet so that's just the table where the lasers sit and we use these optical fibers to guide the light from this table to the actual vacuum chamber to shine it onto the atoms so this vacuum chamber some look something like this maybe there's kind of a glass cell a vacuum cell and we hold the atoms through light fields or magnetic forces created by these coils inside this vacuum cell of course we want to have a you know very good vacuum why because the atoms are at nano-kelvin temperatures remember we want to have a large deploy wavelength lambda over D has to be one and we want to have a large deploy wavelength so the atoms have to be very cold and they're just sitting behind this glass cell wall in the ultra high vacuum with a perfect thermal insulation from the outside world okay and just keep in mind I think that's really impressive I was quite stunned by this when I think about it myself so this glass cell is all at room temperature so there's no liquid nitrogen low liquid helium whatsoever in these experiments all the apparatus is at room temperature and just two millimeters behind this glass cell wall there sits this atom cloud in this in this you know crystal of light and that's at nano-kelvin temperatures okay that's that's how it is and it also is an experiment and that's I think important maybe not for the so much for the things we discussed today before a lot of other exciting experiments in the field it's an isolated quantum system as far as we can tell okay so it does not connected to any reservoirs that's very different from a condensed matter system where you have a thermostat you contact your probe to the thermostat you cool it down that system is always connected to a thermal environment okay here you really truly have an isolated quantum system more or less and and and this isolated quantum system allows us to study a lot of the interesting physics you know about thermalization and things like that how do these isolated quantum systems actually achieve a temperature when they're not connected to a thermal reservoir actually one question that sometimes comes up or people sometimes ask is what about black body radiation the system is not shielded from black body radiation doesn't it thermalize with black body radiation and just assumes room temperature through interaction with black body radiation so what do you what do you think about that is that a problem or eventually or not yes no not really that could be a problem so but there will be a tail you know which is resonant with the atoms still in that thermal radiation but what could you think yes that could happen but let's not let's not discuss that let's imagine we're good experimentalists we took care of all the laser beam noise and the laser beams are stable they're not shaking that would be bad of course so let's forget about that what about black body radiation is that a problem or not are the atoms going to absorb photons from black body radiation and exchange energy with them or not and by that they would thermalize but what might save us what might save us but think of the resonance frequency an atom has very few resonance frequencies compared to a solid compared to a molecule think of a big C60 molecule with all its rotation and vibration degrees of freedom it had tons of resonance frequencies compared to that the atom has very few resonance frequencies and exchange is of course energy with the black body radiation but it does so on very long time scales it takes very long times to thermalize with black body radiation and that's what saves us really from we do experiments we can cool the system to these low temperatures before because these thermalization rates with the environment are actually extremely low that would not work anymore if you have a complex molecule if you have a very complex molecule which has many more resonance frequencies you would have to put it into a kind of thermally shielded cryostat to block out kind of black body radiation because that would kind of thermalize the system much more rapidly than these atoms so it's a kind of a very convenient thing for us as experimentalists that we can do all this in room temperature environment and not worry about it simply because our atoms very inefficiently exchange energy with the black body system and would take very long time to actually thermalize with that okay so let's go a little bit and discuss a few detection techniques how to probe matter waves in these optical potentials and use that in the end so before I do that I just want to briefly recap again how do we think of these optical crystals so let's think of a one dimensional structure so let's think of a one dimensional lattice that we have created by interfering these two laser beams we have this lambda over two spacing let's say we have a lattice spacing of D here and then you all remember from your condensed matter course what you get is a simple band structure which is shown here as a function of the depth of the lattice so the depth is how deep is this thing here this V and this V is typically we express in units of the recoil energy it's a natural energy scale for the atom interacting with the light field and this recoil is just h bar squared divided by two M atomic where K is the wave vector of the light we shine on the atom so h bar K is the momentum actually photon carries which is transferred to the atom when it's absorbed or emitted by the atom so this is just h bar K squared divided by two times the mass of the atom so this is a natural energy scale in which to measure these depths of the optical lattice and this is just how deep this thing is and then of course you see that the eigenstates of the system are of course block waves we'll come to them in a second remember these are the block waves in this periodic structure and the band structure that emerges is shown here and you see those different energy bands so you see that a band gap opens up when we turn on the lattice and we get this lowest band first excited band second excited band and so forth what you also see is that when the lattice gets deeper and deeper actually these bands get less dispersive meaning they get flatter and flatter as you can see you explain remember why that is why do they get less dispersive or think about it again in the limit of very very large lattice depths what do you get in the limit of very very large lattice depths what are these block waves let's say in the lowest band let's just focus on the lowest band on the eigenstates of the system in the lowest band how do they look like yeah they're just localized you can think of it like this I mean it's always good to keep your harmonic oscillator in mind so think of these local lattice sites as harmonic oscillators and now when we're in the lowest band you should also remember that the lowest band is just formed by these superposition states of ground state wave functions okay so this is the ground state of the harmonic oscillator so it's this Gaussian so the eigenstates that we form these psi q of let's say n equals 0 the 0th band are just going to be some superposition states of these localized wave functions let's call them wx minus xj and then there's going to be a phase factor here in front of that so that's the that's the extended block waves that you get no you're shaking the head and you're not sure not it's not okay or it's okay okay so then so this is kind of the extended states that you get are just superposition states of these localized states with these different phase factors that here here and of course what you see is that the energy of those states will not depend anymore what phases you put here if you're in the limit of completely decoupled harmonic oscillators okay if the barrier is so high it doesn't really matter whatever phase factor you put here it's just the sum of all the ground state energies of the system is the energy of that state of course now when you start to put tunnel coupling you allow tunnel coupling between them then of course the energy of those states will change what kind of phase factors you put here and the band becomes dispersive okay so this explains at least this qualitative fact why this is in the case here deep lattice is flat and dispersive in this lower lattice regime okay here are just some of the functions you should remember this is the block state with q equals zero with quasi momentum which is basically this e to the i q xi factor that we put here and so now if we go to a lattice site where we have quasi momentum zero that means this q is zero so all these wave functions are in phase superimposed in phase that's actually the lowest energy state of the system so this is the state with q equals zero and the state at the band edge is the one where you see you have pi flips in sign between those alternating sites and this is also something you probably should remember again from your simple double well so it's again good to go back to simple model system so let's just remind ourselves of the double well if we put that there is two Gaussian ground states we can form two eigenstates of the superposition by either let's call this the left ground state and the right ground state we can either form a state one of us grow to W left plus W right which is all we can form a state minus W right so this is the state where we superimpose them in phase and this is the state where we basically superimpose them out of phase and you should remember that the energy difference between those two states is just two times the tunnel coupling in the system so this is the ground state this is the actually I should have drawn it the other way around that's the ground state that's the first excited state in the system and in the same way you see now for many sites these kind of lowest state q equals 0 remains in the band edge but then there are of course many states in between which simply are characterized by the phase difference from side to side so this phase factor actually has a very simple interpretation that's just the phase difference you get in the block wave when you go from one side to the next side so if it's 0 they're all in phase you made an extended state I can come back to your question in a second you make an extended state with all of them in phase when you had q equal pi that means every site when you go from side to side the phase flips by pi so you have 0, pi, 2 pi, 3 pi, 4 pi and so forth and if you're at a different state in between the quasimentum just tells you the phase shift is between 0 and pi okay alright so what's the question yeah so we you have completely right if I would be at very high energy states here which would correspond to higher bands in our picture then this harmonic approximation will become less and less valid so what I am talking about we first of all want to mainly focus on the ground state physics which is the lowest band physics so I think actually in the talk we will forget about all the higher bands everything I will say even the strong correlation physics will be in regimes where the particles only occupy this lowest energy state okay and what's the separation to the high next band well that's approximately this vibrational splitting between the harmonic oscillator states so if you go back let's go back to the you see actually here this energy difference between the lowest band and the first band in the limit of very deep lattices is basically just a vibrational onsite vibrational splitting that you get okay and everything we want to discuss in this lecture will be done in a regime where all the interaction energies and energies in the system are smaller than the splitting so we can for that energy that state will not be touched so we can forget about it we can eliminate it from the problem if that's not true if for example the interactions between the electrons become so large that they approach this vibrational splitting then of course we have to talk about a multi-band model which takes into account lowest band and first excited bands as well okay more questions or you're good so far okay so then let's test your knowledge if you're good we make a small test and so here is now let's look what happens if we would suddenly and of course this is something we can't do in a real solid I can't turn off the crystal potential suddenly but here I can do it because I can just turn off the light field very suddenly I have my matter waves in there I turn off my light fields and we ask what happens so we have the following situation you have your block waves for example in there which are these different superposition of the Gaussian ground state wave functions and then we suddenly turn off the lattice okay so now what's going to happen what's going to happen to each of those Gaussian ground state wave functions what happens to a Gaussian wave packet in free space it spreads absolutely it spreads so it's just going to spread this one's going to spread this one's going to spread so you see what's going to happen they're all going to interfere right so all those wave packets spread they will interfere and they will give rise to an interference pattern just like in optics if you do a diffraction of a laser light from a material grating here we do with the opposite here we diffract matter waves from a light grating but when we look at the resulting pattern it's actually just the same okay it's just the roles of light and matter have been reversed and now we can ask so with this knowledge that it's just the same experiment as interfering light beams from a material grating I ask you what do you get well of course you get a multiple slit interference pattern which is the one I show here which is the one you also see in the experiment which we'll come to in a second so now the question is you see different length scales here you see an envelope you see peaks here with a separation you see a width of these peaks so now I want to know what determines what so let's start maybe with the separation between those peaks what determines the separation between those peaks in an interference pattern that's good that I ask no? I'm sorry sorry? the number of splits no? no? well there are finite choices so you can just try another one or you'll get there no? not the yeah? that is constant absolutely so the spacing between these maxima that's just determined by the spacing of our particles okay what determines the envelope of the interference pattern where does that come from? a bit louder maybe at the dimension or what do you mean by dimension? you mean diameter or yeah diameter okay yes the diameter yes exactly because remember what you see here is the Fourier transform what you had in the lattice remember in an optical interference experiment what you see on the screen is the Fourier transform of what you have in real space on the slits so everything that has small spacing in real space becomes large spacing in time of flight or on the interference pattern so the envelope actually the largest scale here is determined by the smallest scale here in real space and the smallest scale here is the width of our Gaussian ground state wave functions okay so let's call this sigma w so if you make the slits very narrow if you make the wave packets very very localized this means you would get a very flat almost flat kind of envelope in this case it's a Gaussian envelope why Gaussian? well because our wave packets here are almost Gaussians and the Fourier transform of a Gaussian is a Gaussian okay if this would be a rectangular thing here what would you get in momentum space? you get a sink function exactly very good okay yes yes so it can be 10 typically it's a factor of 10 so we said it's like the spacing of the lattice sites is 500 nanometers half a micron typically Gaussian wave packets on each side can be 15 nanometers it all depends of course on how deep the lattice is the deeper we make the lattice the narrower the smaller the harmonic oscillator becomes but it goes like very weak scaling if you remember the scaling of lattice depth on harmonic oscillator with it doesn't scale so strongly so 50 nanometers pretty pretty standard value I mean you could push it any way you want but that depends on how much laser power you have available or you could go to making different lattices if you make like non sinusoidal lattices you can have very different things of course as well right you could have we're not going to talk about this in this lecture but there are ways to do that okay all right so the final thing then is of course the width what determines the width of these peaks system size the number of slits that are illuminated basically so how large is the system how many wave packets are we interfering okay very good and now you see basically what you what you what you get what you see in the experiment when you do this from a 2D lattice is precisely that so we took here actually Bose-Einstein condensate which is like large number of bosons all in the ground state of this lattice all in this wave function with quasi momentum zero and we release them and we detect where the particles are and indeed you see precisely this two dimensional interference pattern matter wave interference pattern emerging from them which tells us actually which block state they were in so we have a powerful way of telling which momentum states the particles were occupied by just turning off the lattice and looking what the interference pattern is that we get okay that's a way to analyze what block states have been populated in the system actually let me go back now you also know how these different ones look if you add the block wave with quasi momentum pi then you immediately see that now there's destructive interference for the central one and you get constructive interference here so you get a very different interference pattern if you add the quasi momentum of pi over 2 with the phase difference of pi over 2 between neighboring sites then you get this asymmetrically skewed interference pattern so every interference pattern corresponds to a distinct block wave in your lattice so that's a very nice way to analyze what block waves have been occupied okay so here's the dispersion relation again that you get for this ground state band this is cosinosoidal dispersion we didn't derive that but let's just write this down so the energy of the these block waves in the lowest band is minus 2 times j where j is just the tunnel coupling how strong are these sides coupled to each other cosine q and well I called it d or a the lattice constant in the problem and you see actually you have the lowest energy for the q equal 0 state and the highest energy for the q equal pi state which is like in the in the double well that you have okay now let's say we want to look at wave packet propagation let's ask something different again we want to ask how is a wave packet that we form in a lattice how fast does it move through the lattice for that I need to calculate the group velocity all right if you want to know a wave packet propagation we want to calculate we're talking about group velocity so how do you calculate the group velocity of that knowing the dispersion relation somebody else maybe you know it I'm sure if you give you any dispersion relation how energy is linked to momentum how do you calculate yeah it's just a derivative versus q gives us the group velocity so this is what I've shown here right so this is this green curve here and you see actually that in a lattice if we form a wave packet centered around momentum q here it will be at rest group velocity zero and then there will be also kind of the wave packet will also be at rest if it's at the edge of the a bryon zone so that's actually a very high energy state will also not move in the lattice because of destructive interference actually in that case and we have maximum propagation velocity here the max at pi over 2 when there's pi over 2 phase shift between the different lattice sites that's when we make a wave packet centered around those momenta components that's the wave packet that actually moves fastest through the lattice so it's quite different I just contrast this to a parabolic particle where with a parabolic dispersion relation in free space where a particle with higher momentum always moves faster right if you calculate d e d q for a parabolic a pasta particle a higher momentum particle always moves faster this is not the case in a lattice so velocities are bounded and there's actually maximum currents flowing at pi over 2 okay another phenomenon that's actually you should now be very easy to understand is the phenomena of block oscillations how do we actually manipulate the block waves how do we change from blonde block wave to the next so let's say we start with this ground state block wave but now I want to create a block wave at the band edge how do I do that well we can very simply do that by applying an electric what by a potential gradient this could be in for electrons an electric field gradient for atoms and magnetic field gradient where now we apply this potential gradient and now if these walls are decoupled you can think of these phases evolving each to their corresponding eigen energies right so each of those wave packets here evolves now with an energy e j given by this position in the gradient times t divided by h bar and that now if I look at the phase difference between neighboring site which remember was the quasi momentum that will change now as a function of time so if I apply a gradient then the phase difference between neighboring lattice sites will be just v prime times the lattice spacing times the whole time in the system okay because they actually evolve at different energies and now you can just kind of have the particle start out at q equals 0 lacking a little bit so you can start out with a particle at q equals 0 in this distance e versus q we start here and then we apply this electric this gradient this potential gradient and then the particle will move up here and what happens when it comes here what will it do here what's gonna happen when we when it arrives at the band edge that wraps around then we have a Bragg reflection right that's when we get a Bragg reflection the system particle enters here again and comes down here okay so if I neglect the lower band that's completely fine just to complete the picture if I indeed have higher bands available as well there's another thing that can happen if you accelerate through this through this point what could happen then if you think about the other bands as well could happen to that wave packet now think about a two level system here we go to an extended standard zone scheme now you could have actually avoid a crossing you have an avoided crossing here and if you change your wave function at the finite rate remember there's a Landau-Zena transition that's possible okay so for very slow speeds you will stay on the ground state curve this is what we've shown if you're adiabatic but if you're very fast if you're too fast then you can actually also start populating higher bands okay we don't want to discuss this we just want to stay in this adiabatic regime of very slow kind of accelerations that we have in the system alright so now let's let's all single particle physics let's make things a bit more interesting and go to interacting systems and look actually should ask do you prefer to have a break in between or should we just go I mean I'm not going to go two hour full don't worry that's just also too intense for me so either we have a break in between gives you a chance to ask questions also in between the break or we go through whatever you prefer so should we maybe we make a small vote who's for break okay seems to be already the majority but let's just make the counter test who's who's for continue okay so let's make a small break so we'll just go a little bit further on and we'll make a small break and you can you can ask questions okay so now now we everything we've discussed everything I talked about is just just single particle physics a single particle tunneling in the lattice so everything we've talked about is just the in a in a kind of second quantization description is the action of this operator which describes the tunneling of particles on the lattices and trying to find out which is the lowest energy states and in the system and higher energy states in the system now let's add interactions and let's add them in this very simple form of this onsite interactions of the Hubbard model and let's first do this also for the case of bosons will come to fermions in a second so for bosons if you put bosons and bosons on a site you and they either repel each other attract each other which is in atomic physics given by the scattering length a so this gives us kind of a onsite interaction energy which tells us how much energy does it cost if you put for example two three four particles on a site okay if the particles repel each other then of course this will be a positive energy cost something you have to pay if it's attractive then the particles favor to occupy that state for attractive interactions so let's imagine we have repulsive interactions for a second so this u is positive so let's say we have two particles so this formula tells us this is the number operator that counts the number of particles so I have two times one divided by two that's just an interaction energy of you if you have three particles you have three times two is six divided by two you have three u interaction energy okay so that's basically how much energy you pay for two three four particles on a given lattice site let's now just consider what happens if you have just these two systems use these two these two things you have kinetic energy given by the j term and you have interaction energy and it turns out that everything you need to know to understand this problem is solely determined by the ratio of the interaction energy to work to kinetic energy so it's a ratio how when we say interactions are strong we always have to compare them to something right otherwise has no meaning this statement we compare to is the other parameter in the Hamiltonian the kinetic energy and if interaction to kinetic energy is much larger than one then we are in a strongly interacting regime where interaction effects are very strong if kinetic energy is dominating then this parameter will be very small and we're back to the single particle physics we discussed so if I have n bosons for example and I'm in the weekly interacting limit and particles in the lowest energy state what's the ground state of the system so I give you n bosons in this lattice and I ask you what's the many body ground state of that system how do you write that just a bit louder initial condition is just n particles n bosons n bosons completely negligible interactions so interactions are zero and I want you to find right down the ground state of the system yeah post condensate is right so if I write it in the single particle wave function what what would I get localization so what's the ground state of a single particle we had that in the dispersion relation what was the ground state of the in our in our cosine and so it'll let us slow down you're you're too fast thinking of yourself so we said the ground state of one particle is the state with quasi-mantem zero that's the one where we had coherent superposition of the different gaussian wave packets this for a single particle now I just have n particles they're all identical so the only thing I have is the tensor product n times that's my many body wave function so the many body wave function my system is just n times q equal zero for the first particle tensor q equals zero for the second particle tensor q equals zero for the third particle and so forth right very easy n times just that wave function and it turns out we can actually approximate this wave function very nicely by thinking of it as uh... as actually a coherent state on each lattice side so if you have these n particles uh... on these m lattice sides for example and you write it out you actually find that you get a multinomial particle number distribution but we can actually for a given lattice side we can actually approximate the wave function anybody wave function on the lattice side very well by a coherent state so I remind you again what a coherent state is so you have this e to the minus norm alpha squared divided by two alpha to the power of n divided by square root n factorial n now where n is the occupation the number of particles you have on the end side so that's what you have so that means actually the particle number on a given side is actually uncertain okay even though we have a fixed number of particles in the total system when we look on the local lattice side we find actually that it has fluctuations in the particle number given by this kind of uh... coherent state in the system how do you calculate the probability of having n particles if i give you the state on a given lattice side what's actually if this is the state on a certain lattice side how do you calculate the probability in a measurement of finding n particles how would i calculate that maybe somebody just tell me what i need to write so i want to know what is the probability on a given lattice side of finding let's say n prime particles yeah exactly thanks thank you very much you just overlap it with the state which is our coherent state and then you take the norm squared okay that's a probability for finding n prime particles given that you have a coherent state and what you get here actually if you do that you get a Poisson distribution exactly so you get a Poisson distribution e to the minus lambda lambda to the power of n n factorial now where lambda is just this norm alpha squared and you can check this for yourself maybe on a piece of paper that this is indeed true so that means actually if i go and make a measurement of this post-Einstein condensate and i look locally how many particles they are even though my total particle number is fixed i will approximately find these uh... Poissonian Poissonian distribution of particles on on my given lattice side so you have a lot of fluctuations in the particle number but there's one thing that's actually very nice here uh... one thing we see actually from the interference pattern that the phase between the lattice sides the phase of the wave function of the different lattice side that's rather constant because we see a beautiful interference pattern for such a state okay so we have a state with a lot of particle number fluctuations but very weak phase fluctuations phase fluctuations are very small now there's the other limit let's just discuss this before the break when the interactions become extremely strong so when interactions become extremely strong then you can immediately see that having two or three particles on the lattice side is not a good idea energy-wise because it simply costs you a lot of energy so you might form another state out of that which is this state here which we call the mod insulator state where you have exactly one particle per lattice side and this exactly one particle per lattice side as you see minimizes interaction energy that's great for interaction energy because you have no interaction energy cost for that state because you always only have one atom per site but where where do you have to pay now a price there must be a price for making that state what is the price you have to pay in which no not entropy there's all zero entropy states is all the ground states of the system not an entropy yeah almost there yeah it's kinetic energy you have to pay so this state localizing a particle if I want to make a localized wave function of a single particle not an extended one how do I write that in terms of these extended block waves you remember that if you form so we have on the one hand we learned how we make the block waves out of the by superimposing the localized waves but now I want to know how do I make a localized wave packet out of this other basis of block waves in a given band yeah you superimpose basically in order to make one localized wave function here you have to superimpose all block waves of your of your of your lowest band and superimposing all block waves of your lowest bands means you have to occupy higher momentum states in terms of the single particle states of the system which means you have to occupy higher energy kinetic energy states so you're paying kinetic energy at the cost or at the winning of minimizing interaction energy now when is that trade off that's going to be favorable of course at the point where interaction energy becomes much larger than kinetic energy so when you is much larger than j then suddenly this state can become much more favorable than this state if j is much larger than you then it's always you don't want to pay kinetic energy because that's the larger energy you have in the system you favor this state but if interaction energy becomes strong you favor this state now here's a state obviously where we have zero fluctuations in the particle number I have precisely one atom palaticide and now actually if I do an interference experiment with this state what I see that the interference pattern is completely gone because now for these fox states you have maximum phase fluctuations for the matter waves emitted on those different sides you can think of it think of it as all those wave packets now emitted with random phases if you do calculate the interference pattern of N slits with random phases with a large number of emitters you see that the interference pattern actually vanishes so you have two limits on the one case you have a state where uh... delta n is large but delta phi is small the phase fluctuations are small on the other hand you have a state but delta n the particle number fluctuations are small but the phase fluctuations uh... are large so you can trade off and there's something like even uncertainty relation that I don't want to derive here like delta n times delta phi is larger than uh... and then one larger equal than one for these macroscopic wave functions that one can write down in the systems and here you're basically seeing two metals two sides of the system for strong and for weak interactions okay so let's make a break and make a break and or give you a chance to ask some questions and let's continue a bit more and then see what we can do to detect these different states and actually see them really really in the experiment okay here's the now it's an excited state now this thing has to do also with h bar k everything in adam gets if you're part of your very fast so if you separate your particles very fast through the bandage then all of them will transition to the higher state so you can just have a complete transfer well like this depends on its like exponential function this lambda z in a transition for a building but if you go fast enough you have 99.99% probability in the higher state and then they're there now you can use the mean way you want here with the gradient they're close enough to the next bandage yeah exactly then you could also so higher bands or you can directly make a transition for example the photon now depends on the photon also transfers momentum when you're driving a process it goes like this now if you drive a process you drive a process like this it goes like this that would give you that would give you two h bar k now depending on the standing wave the momentum transfer you get just to get to the contour again wait a minute so that first of all the delta q so you have two beams okay two beams in the bearing so there are two mics one that you get delta q delta k that determines here how much you move on the horizontal axis then now the frequency difference of them determines where you move the vertical axis so now you have to tell me when you say you have two counter-propagane views you have to first tell me is it the same wavelength like the lattice or is it the different wavelength these beams could be different wavelengths so the delta q and delta k could be different if it's the same wavelength then of course you're making transitions basically delta k is just going to be to the band edge basically alright and if you have two h bar k you just stay in the center sorry you just stay in the center stay in the center of the system you just have the same lattice which would basically mean the same periodicity of the lattice if it's moving you would have delta q equal to h bar k which would mean this is h bar k go back in zero there if you have a different if you interfere rama and beams like this under a different angle you know where the delta q that you get can be anything depending on the angle it could be large or very small but if you have a very narrow angle the delta q is very small then you basically make this arbitrary small and the delta q this is omega 2 omega 1 and the delta omega then just determine where you go in the vertical direction so you just keep this in mind I just think the balance is on the table yeah okay no no you can keep them in typically takes 60 seconds so I had to say that AO would really load maybe you can go anywhere where you want the AO the acoustic modulator can give you the frequency difference and the angle between the beams sets the delta q well a single frequency then of course you have a single photon then you have h bar q and then you cannot you're not driving a two photon transition then you have the full optical photon energy what you want is right you want a two photon where the energy difference is determined by the difference frequency of the optical fields if you have just a single beam single optical beam then you have the full optical energy that has to be absorbed so you want a two photon process that just the only thing that remains in the system is omega 2 minus omega 1 two beams two frequencies well even two beams but they're so close to the frequency difference basically okay okay very close okay very good question the one band approximation is valid as long as the u-term is small enough right smaller than the gap the gap between the bands that usually is quite high it can be high, it can be low it depends everything I'm going to talk about is in the region where we only talk about single bands single bands people also do experiments with two bands oh yeah sure sure we can do that with fresh bar resonances you can make this u we're tuning the schedule and then we can tune the A very large the u is very large and then you have to have a music background absolutely I have a very simple question if you made this grid and you showed that you have to have that okay can you simply imagine that's one of the how much effect on that yeah, of course when you have short exposure times you can of course of course you can for a long time you just have to have a long exposure time you can have short exposure time and of course the whole time and this has been used by people to get a short exposure time to measure the volume I think so and then we can go on back to the ratio of the split second to the periodicity how far could you push it but with this I think it's about 10, 20, 19 okay very good so let's continue so how long do I have much Hello until okay great good As I said, we don't have to make it through the whole program. I think it's more important you understand what I tell you. So again, if I get too fast, you tell me. OK. And then of course, what we said, we have these two distinct regimes for weak and strong interactions. And actually, what separates them is what people call a quantum phase transition, where basically you are not driven at temperature-driven phase transition, but only by a tuning parameter, which works even at absolute zero temperatures. And you probably maybe heard a lot about these quantum phase transitions in other contexts in the school. If you want to learn a little bit more introduction, Article is a nice one from Subya Sajdev and Bernard Keimer here in Physics Today in 2011 about these different phase transitions. OK, a very similar thing. Actually, we talked about bosons. Let's get back to the fermions, because I said this is the paradigmatic model for the electrons on a lattice that we really want to understand for material science. For fermions, basically this Hamiltonian looks very similar. Again, you have the kinetic energy part. Now we have, let's say, two spin states. So we have to label the fermions with their spin label, which is given by the sigma term here. So we have the sigma can be spin up or spin down. So now you can hop from side i to j, of course, not changing the spin. So this is spin preserving. So sigma is the same here. But now when two particles come together, we have an interaction energy. But you see, we only have for fermions an interaction energy when a spin up and a spin down come together, because as Pauli principle, we cannot put two spin ups on the same lattice site that's forbidden. And then actually, for the cold atom experiment, something I didn't discuss at all so far, we are breaking translational invariants in the system that our systems are not completely translation invariant, but there's a very weak harmonic trap below them. And that's something we can model with this third term here that keeps our atoms confined to a certain region of space. So this is this third term, which gives just the parabolic potential going like the lattice site squared. And this is just a number of particles that are on that lattice site in their potential energy, basically. It turns out, actually, that for fermions, we actually have a very similar phenomena of the system. And now I just want to plot this as a function of temperature. What happens for the system for strong repulsive interactions when you are at high temperatures, lower temperatures, and very, very low temperatures? And all three regimes we're going to see. At very high temperatures, you can imagine, now if KBT is on the order of the interaction energy, you can even have put particles on top of each other because KBT can be comparable to you. So you have a lot of fluctuations in the system. If KBT becomes much smaller than you were basically in the zero temperature regime regarding to density fluctuations, and we remember that for strong interactions, we don't want to spread the particles out till they localize in this phase where we have commensurate filling of particles. So if we have 100 lattice sites and exactly 100 fermions, we have one fermion per lattice site. Let's say we have 50 spin up fermions, 50 spin down fermions. In this strong interaction regime, we want to be in this regime where density fluctuations are suppressed because they cost kinetic energy, remember? And we don't want that for kind of large repulsive interactions. So again, we form this mod insulator state with actually one fermion per lattice site. But there's no kind of spin ordering the system. What we'll talk about tomorrow in quantum magnetism is how we can understand this other transition when we go to even lower temperatures, how this anti-ferromagnetism emerges. When we go to even lower temperatures, we see that we get actually a Heisenberg anti-ferromagnet, which is pictorially depicted here by this classical state up, down, up, down, up, down arrangement of the spins. That's something for tomorrow. So we want to focus on this part a little bit and how we can actually see that. So let me move on now to something that's exciting you, I think, in our field is how you can actually get real space images of all the phenomena we talked about, these interaction phenomena. How can you see single atoms on a lattice? And this is actually quite a remarkable feat because what you're trying to do is kind of sketched out here on a kind of symbol theoretical level. So let's think we have this many body wave function, which is a superposition of different configurations of the particles on the lattice. Could be a very complicated state of fractional quantum. Hall state, very complicated, strongly electron state in our lattices. And let's imagine we make a photo of those particles on the lattice. What happens then? And we do that with high resolution. So then we can see how many particles we have on a certain lattice site. So the wave function will collapse onto one of those configurations. And this is the configuration you're going to measure in the experiment. So the one you get here is completely random. That's just chosen by the collapse of the wave function. If you prepare psi again, you take another photo, you might see this configuration. Or the other one. You get all these different particle configurations. Now by getting access to these particle configurations, you have a very powerful analysis tool because now you can ask very subtle questions about the particles and their correlations. If you have all the probabilities of these different configurations, you can now ask questions like, how likely is it to find two particles here, one particle over here, and an empty lattice site in between? So how are these three particles this thing arranged? How likely is it to get such a configuration? Or that's possible only when you get access to these single snapshots. You can actually also, I'm not going to show this, you can also extend this to a current observable. So you could also measure particle currents in the same way on all these different bonds. So you could get single snapshots of the current distribution, of the currents in the lattice system, and then also talk about, let's say, current-current correlation functions that you can get in this system. So how do we do that in the experiment for the, for the lattices? Remember, we have these large space lattices, so we can look into them with good objectives. So let's take a very good microscope objective with a very high resolution, which has this numerical aperture here in our case of 0.7. Let's make a 3D lattice, a three-dimensional lattice. Let's just look at this 1D lattice first. This cuts the system into two-dimensional planes. So you make a one-dimensional lattice, the one we discussed before. So you confine in the vertical direction the particles are confined to these harmonic oscillator wave functions, but in the transverse direction, they're completely free to move. Okay, so you made a 2D system, actually. And if you impose X and Y direction lattices as well, then you can make a three-dimensional lattice or a 2D, if you just have a single plane, you have a 2D lattice in a two-dimensional structure. Okay, so let's imagine we have the atoms loaded in this one plane and now we want to image them in the system. We want to see where they are. What we do now is we shine in near-resonant light from the side, which makes the atoms fluoresce. Okay, so we scatter light off the atoms. That's the process where we actually take the photo. That's the process where the collapse of the wave function starts and we see a certain particle configuration. And this is, for example, shown here. This is a picture, actually a thermal gas, so it's not so interesting, but nevertheless very nice because you see each bright spot here. That's the fluorescence of a single atom held in our optical lattice, okay, that we see. So we get in situ images of all the particles in the lattice. And that's really quite remarkable because if you would think about the same experiment for a solid, you would have to make photo of all the electrons in your solid system. And that's, of course, impossible, but here we can do that. So a little bit how we actually do that. So we have a system where we do the physics, so we prepare the physical system. Okay, the many body state we're interested in, typically maybe at lower lattice steps. And when we image, we want to make a faithful image of the position of the particle. So we don't want that the particles during the imaging move around because we want to image where they were before, right? Not if they are perturbed during the imaging. So what we do to image, we make the lattice extremely deep, very rapidly, very deep. And then we scatter this light of the atoms in such a way that they basically stay put. The lattice is so deep that they can't move anymore at that point when we image. Okay, so we really get a faithful image of what there was before. And so an image like this has about a few thousand photons per atoms and we can see that gives us a really very nice signal-to-noise ratio where I can clearly tell you this there's an atom on this lattice side here. One little bit nasty thing we have to deal with the most experiments today, and one later one I'll show you tomorrow or later today, we circumvented that, is that when you have two particles on a lattice side, there's a very nasty light scattering process which releases so much energy that those two particles are lost. So without going into the details of that process, it's like a small bomb going off whenever there are two particles on a lattice side. That's what we call a light-induced collision. And luckily it only happens for pairs of atoms. So if I would have an original particle distribution like this, atoms get lost in pairs. So if I have a single side, everything's fine. If I have a doubly occupied side, that side is completely lost. If I have a triply occupied side, one atom will remain and two will be lost. So in the images you're first going to see now, we're limited by what we call parity projection because effectively what you measure is not the original occupation, but the original occupation model two. And that's what we call the parity of the original occupation in the system. Okay, and then we go a little bit further in our kind of experimental analysis just to show you how we do that by knowing a little bit more about our imaging system, how the lattices are oriented. We can take an image like this one and basically get out from it the occupation from the different lattice sites. So light blue points mark all the different lattice sites. At dark black spot marks, there wasn't atom detected at that lattice site. So this is what we can then have and we can do analysis of all the images of just this matrix, which tells us where there were lattice sites that were occupied or unoccupied in our system. And that's of course great because we can also give it to a theorist who just wants to know the occupations who doesn't want to deal with these fluorescence images and now you can run your favorite correlation analysis over that, yes. When the light gets scattered, when the first photons get scattered from that system. Sure, yeah, that's actually a subtle process and Jean could tell more about that process but that's true that you get this very subtle process but we don't record the direction of the photons. We lose all the information about the light. So basically what happens, you get a projective measurement on the particle number of the system in the end. Of course, the light atom state, if you would just scatter very few photons, could become highly correlated light atom entangled state but in our detection, we're not recording where individual photons go, we're losing that information and in the end, what you do in this fluorescence measurement, it's just a fluorescence imaging technique, is a projective measurement on the particle number basis. Okay, more questions? Yeah, in different layers or in the... So what I can do, what I will show you, this is now with bosons first. We can then look at the same experiments with fermions and see what we get with fermions but what you are asking is what, if can you have combined one system bosons, one fermions or, yeah, in principle that should be possible although you are, you know, in principle but in practice these things then turn out to be often quite difficult to realize but yeah, in principle, I don't, maybe think about an arrangement where that's possible too so we would have to talk about the details of what you have in mind, okay? Yes, yes, good question, yeah. So we prepare this psi in an idealist way, of course, we in principle have a density matrix but because we're not at zero temperature let's say we prepare this psi every 20 seconds in the lab. So it takes us 20 seconds to cool down, go through the whole cooling cycle, ramp up the lattices, prepare the many body state we want. When we make a photo, of course that state is destroyed. Remember, it has to be destroyed, right? It's not a bad thing by making this projective measurement on the particle basis we are taking this state here, this psi, that was the original state. It was a coherent superposition of different particle numbers and in principle there are also complex numbers in front of these things here. When we see a certain configuration we have collapsed this state onto that configuration so that's not the original psi anymore, right? So it's a destructive measurement. The photo, taking the photo is inherently destructive, there's no way around that, okay? That depends on how high your correlation function is. So if you want a 20 point correlator I will need a lot of data for me in the experiment. If you want, so we have made measurements up to eight point correlation functions that can take a week to get enough statistics to get reliable information. But every, you know these experiments run almost automatically and overnight also so you can just keep them running overnight. So in a week you can get a few 10,000 data shots, pictures that you can then analyze. Okay, so now let's look at this superfluid to mod in the boson description again where we had these large particle fluctuations here for the superfluid and for the mod insulator, the strongly interacting bosons, we had this unity occupation. And actually here you can see that we can beautifully see that indeed in the experiment with these quantum gas microscopes. So here's the picture, the raw data. So the top line is just the raw data that we see the fluorescence images. Here you see the Bose-Einstein condensate and you see it has a lot of particle fluctuations. The density is not constant. A lot of fluctuations in the particle number. Remember what we see here is the parody projected particle number. So you never see more than, you either see only one or zero but you see one and zero are fluctuating a lot in the system. When you make the interaction strong and this is the only thing we changed here. We made u of a j large by ramping the lattices to larger values. Then you can see you get this homogeneous plateau, fluorescence plateau. If you analyze that in terms of the occupation of the particles in the system, you actually find that now you really have exactly one atom pallet site. The atoms have rearranged from this configuration to this one without us doing anything. We just made the interaction strong. That's the only thing we did in the experiment and took pictures. You see also when you put more particles in the trap you see there seems to be a donut here. That's just an artifact of the detection because actually what the system has evolved into is kind of what we call a double mod insulator where you have an n equal one outer shell with one particle exactly here and then n equal two core here in the center. But remember due to parody projection this two core appears as zero atoms. So it's an artifact of the detection that's really more like what we call in Germany a wedding cake distribution of those particles in the system. So here are nice pictures of this showing this again, BC mod insulator mod insulator with n equal one or n equal two. What's also nice actually if you look at this sometimes you see things are going wrong. So sometimes you see there are two holes next to each other where they should actually be one particle, one particle if it would be the ground state of the system. So what you're actually seeing here is an individual thermal fluctuation of the system. So if you're a little bit residual temperature that means you have a little bit of residual entropy in the system, how does that manifest as a low number of excitations in the system and the lowest excitation of this mod insulator is a state where a particle one one configuration where one particle hops onto the neighboring site that costs energy you and leaves a doubly occupied site and a hole behind. Now if we detect that with parody projection remember the doubly occupied site we see as a hole. The other side also is a hole. So we see kind of these two holes that we have next to each other. So and that's also how we can measure temperature. So I can now do very basic thermodynamics on the system. Just use a grand canonical description. In principle what we're doing here is just we're counting the number of these kind of defects. And from these number of defects these average number of defects I can tell you what the temperature is of the system. I can tell you what the chemical potential of the system is. So by just doing very elementary kind of thermodynamics here on the system calculating what the probability is to find n particles at a certain distance we can use just a grand canonical description fit it to our data and you see we get a very nice match both in the average measured particle number and in the fluctuations in the system and from that we can determine all parameters temperature, chemical potential in the system. So that's how we measure temperature in that system. Okay so let me extend this to fermions because this is what we want to also go more tomorrow for these fermionic quantum gas microscopes. The idea is in principle the same so you can do this fluorescence imaging now on fermionic species of atoms that would be like for example potassium 40 or lithium six which are favorite candidates for us to do these experiments with and let me actually skip how we do that in practice the detection. But let me just show you some images of these fermions and you see now if you take a fermionic gas of lithium six particles and add more and more particles you see how this band insulating state emerges where you have the band filled each block wave with one particle or at the same time that also means each lattice side filled with exactly one particle. So now just Pauli blocking Pauli principle makes you a uniform occupation of particles in that system that you can see. In that experiment we actually avoided parity projection so in that experiment you can take an image like that and reconstruct even when there are holes or doubly occupied sites you can actually distinguish and here's a picture of about 800 atoms on these 2000 lattice sites that we picture. And if you take many, many of these pictures again then you can calculate different quantities you can just average these pictures and you get the average density distribution in your trap. You can also calculate the fluctuations what is the fluctuation of the particle number on a given lattice site. So this would be the fluctuation of the particle number and you see that actually there's a very strong suppression of the particle number fluctuations in this regime where you have exactly one atom per lattice site we call this a squeezed state. And you can actually directly use your elementary formulas from statistical mechanics for the entropy for example which is just the probability of finding this particle zero one or two particles on lattice sites I and J and just use that to calculate an entropy map of the system. So now we have a two-dimensionally resolved entropy map of your system just based on your very basic kind of elementary thermodynamics formulas. Okay, now we've seen how we can observe we'll come back to that tomorrow with more interesting scenarios how can we manipulate the particles? Well by basically doing just the reverse we can just now send through our very high resolving objective we can send a laser beam and focus it onto the atoms and then we can change for example the energy separation by this focused laser beam between two atomic states. These are for example two spin states of the atom hyperfine states and by this laser beam hitting the atoms we could change the energy separation and now if we shine in microwave radiation for example that's just resonant with this shifter transition then we can rotate this spin into any direction we want without touching any of the other spins. Okay, so you can tell me go to atom number 5543 and rotate the spin into a certain direction. Okay, flip it for example. So we can flip it and then we can do that with the other particles as well we can flip those as well and then in order to see those flip particles we have to remove these other spin particles then we can see all those states or these atoms where we flip the spins. Here's just an image that this actually works really very nicely. Here are just different patterns that we made out of that small insulator just by going in like a pencil you know one by one flipping the spin. So this is like a square, a line, an anti-ferromagnetic star and this quantum physicist we like the psi of 2026 individual atoms that is composed of and you sometimes often see this individual atom floating around somewhere we actually like to put that because it gives us a way how we contract the phase of the lattice. Okay, so why do I say that? So remember we have this lattice that's created by optical interference. Now you know optical interference is a very subtle phenomena if you change anything your mirror mount moves a little bit in the lab this interference pattern will change. Okay, so actually what happens in the lattice that this lattice drifts around in space on an hourly basis if we're doing a good job it drifts around very slowly but we wanna know where the lattice is and in order to know where the lattice is the best thing is just to put an isolated atom because that fluorescent spot will mark the position of one lattice site and then you know what the phase of the lattice is in the X and Y direction so that one atom is just a marker atom we like to put. We can actually achieve this resolution much better than the lattice spacing of down to 15 nanometers in this addressing so we can really have very high, high spatial resolution of the particles. So nowadays we don't want to do it like atom by atom but we like to use a more parallel approach using these digital mirror devices that I'm sure Jean will also talk about in his talk. These are very nice kind of arrays of small mirrors that you can turn on and off. They are also in the projector up there most likely and they allow us to project arbitrary light patterns onto the atoms. So we can just shine a laser beam onto this 2D digital mirror device and out comes for example a line of light or an exotic lattice or a square box potential. Or something that Tillman Esninger has used very successfully in Zurich in a kind of similar setup. You can think of two reservoirs, two boxes connected by a very thin wire and the wire is just a wire of light. And now you can put particles in this one box and study how these particles flow through this wire into the other box. And you can add this order, you can change interactions and now you can study all the range of transport phenomena you might be interested in. Okay, you can also make spin impurities so we can start with a kind of entirely spin polarized state and then we can basically just take a line of light focusing onto the atoms and rotate the spins on that line and create deterministically a spin impurity in that line and we'll talk about what happens to that when you release it for example into this gas. You can also shape the cloud. This is what I call the cookie method. It's like really Christmas cookie baking or whatever country you're in when you bake your cookies. You take a profile here on your DMD which removes all the atoms outside of this profile and shields all the atoms inside this profile. So it's like, you know, chuck, taking a stamp and making, now you get a perfectly round cloud out and you can change the size of that cloud. So you can now study finite size scaling, for example, of the system for different sizes of the system and you can study very different geometries that you also have for those systems. So in terms of initial states and single atom control it gives us a lot of flexibility of what we can do. We can cut down a single large mod insulator down to a single atom, also make square box kind of shape cloud so it gives us a lot of flexibility in the atom control. Okay, so let's look at a single, a simple problem again of tunneling of those atoms in the lattice. Let's go back to the non-interacting regime and let's try to understand that problem. So we start with a single localized particle, let's say on site 20 in the Gaussian wave packet and now we ask what happens when it kind of tunnels in this lattice. How would you solve that problem? How would you solve the quantum dynamics? That problem, you wait to now calculate the time evolution of this initial state and calculate the probability distribution after a certain evolution time in the system. What would you do? You have your initial state, one atom in the center and now you want to look at the dynamics. But what's always a good approach? Always a good approach is to start decomposing the state into eigenstates because each eigenstate you know exactly how it's going to evolve in time. That's just e to the minus E i t over h bar the energy of that eigenstate, right? So if we know how this state is decomposed into superposition of the eigenstates of our lattice then we know it's time evolution. What are the eigenstates of the lattice? They're the block waves. Oh, the block waves. So we start with this localized wave packet which was a superposition of the different block waves and from that we can actually very simply calculate how the time evolution of this is and what you see is what happens when you do the calculation is this is what comes out. That's the probability distribution of that single particle tunneling in the lattice after a certain tunneling time and you see it's very different from a Gaussian spreading of a particle in free space because for a Gaussian particle you always have the maximum probability distribution in the center. Here for the lattice turns out you always get the maximum probability distribution here at the edges of the system. We'll come to that in the system what determines those edges. What's the velocity, the propagation velocity of this wave? How fast can this move? How fast can this move? What did we say? What's the group velocity? What's the maximum possible group velocity? So what was it? But what was the value of that? What's V max? J times A, why? J times A because it was the derivative minus two times J cosine QA and now you take the derivative with respect to Q gives you minus two times J A divided by H bar. That's the maximum velocity that you can have and that's the wave front propagation velocity that you can see here of this maximum wave front that you have here. So this initial state again we can think of as a localized state which is a superposition of all block waves and what we see actually these maximum velocity states they are the ones here propagating with the maximum velocity out here. And now you can measure that you can do the experiment. Let's say I call this the quantum horse race you line up all your single atoms on here and you let them tunnel only in this horizontal direction and then you make a photo of where they arrive, okay? After a certain time. And this is what you get of course this picture is deceiving because you never know where the particle which one is first before you made the measurement, right? Only when you make the measurement you know which one is first and which one is lagging behind. So we start with the particles here on the Y line we let them tunnel in the X direction for a certain amount of time and then you can indeed see after a certain evolution time you get a picture like this or like this or like this. Totally random. Every time we do the experiment again starting from here we get a random configuration. When we average many of those pictures we get the probability distribution and we see it actually very nicely agrees with the one of the Schrödinger prediction that we just calculated before. What you also see this is slightly asymmetric here. This is not symmetric relative to zero where we started with the particles. Why could that be that it's slightly asymmetric? What could make this slightly asymmetric? Here's gradient, yes, you're almost there, yes? Sorry? Sorry? Rheophile, yes, almost also. So actually what happens here what happened here that the trap, this harmonic trap was not centered perfectly on line zero. So indeed there was a small gradient to push the particles a bit more into one direction and that's what you see here. So if you would have done a better job in aligning the harmonic trap on the zero line that we prepare here centering it then we actually would have seen a completely symmetric distribution. Here's actually a longer time evolution which I like a lot because it just shows over such long times that we can see kind of this coherent evolution. This is a longer time evolution. This is the probability distribution we measure, the bars, and the orange curve is what we calculate from Schrödinger theory. So this is really coherent tunneling of those particles over in this case 20, 25 lattice sites with perfect kind of reproduction of the coherent evolution that we have in the system. Okay, so that shows you really we have this nice isolated quantum system where we can study coherent quantum dynamics of single and multi particles in the system. Of course, yeah, what we have to discuss of course how actually interactions change this picture. This is a topic I will skip although Pascale will probably talk about it. Not, okay then, but maybe we come talk about it maybe too difficult. No, now actually I want since I don't have so much time 10 minutes, no? Okay, let's just talk about something fun for the end. Let's finish today at least with some, a fun topic that gives you something to talk about tonight and think about tonight. Let's talk about, now that we know the latter stuff, we know interactions, we know what insulators, we know Bose-Einstein condensates coherent states. Let's talk about this negative absolute temperatures and for that, let's start very simple with thermodynamics and just reminding you how we define temperature and thermodynamics. In thermodynamics, one of the temperature is just the derivative of the entropy of the system with respect to energy. And typically this number is of course positive because when you add energy to a system, for example, you heat it up, the entropy increases of the system, okay? But there's actually nothing wrong with this being negative, actually turns out thermodynamics holds as well if this is positive or negative, both regimes of thermodynamic rules apply in the same way, but that's just our fundamental definition of temperature. One warning, sometimes you might have in mind equi-partition theorem, for example, temperature is somehow linked to the energy content, that's not true, that's not always true. So especially when we get into the quantum regime, the energy regime, this is not true anymore and this more fundamental relation is what we should have in mind when we talk about temperature. Okay, so now for positive temperatures, if we have a many body system, the probability of the ith many body state to be occupied is just given by the exponential Boltzmann factor. That's the energy of the many body state, that's just the temperature and in order for this formula to make sense to work, we need to have a lowest energy state in the system, otherwise we cannot normalize this probability distribution. That's okay because typically we have a lowest energy state in any system, right? There is a minimum energy state in the system. Now let's just formally put KBT to minus T, then what you get is of course an exponentially increasing distribution and for that to work out, you see actually you have to have a regime where you have a maximum energy in the system, okay? Otherwise the distribution is again not normalizable. So we need a bounded spectrum, we need a system with a bounded spectrum from above. There's actually, if you want to read about this, there's actually a beautiful paper from Nobel Prize winner, Norman Ramsey, which he already formulated in the 50s, so you see this is actually a very old topic, but he already saw that negative temperatures, typically the conditions for negative temperatures are typically so restrictive, they rarely met in practice except with some mutually interacting nuclear spin systems. So why spin systems? Well you can immediately see why this condition of minimum and maximum energy works out. Imagine you have a spin one half system in a lattice, in a B field, okay? So this is the lowest energy state, so you have a bounded energy from below, and this is the highest energy state. So there's no state that has higher energy. So this system naturally has the lowest energy state and the highest energy state. So it fulfills the conditions for positive temperatures as well as for negative temperatures. And this is indeed the first system where this was realized in its beautiful experiments by Purcell and Pound where they looked at nuclear spin systems and realized negative temperatures in that systems. This was then continued in the famous Finnish low temperature labs, also in experiments in ultra cold atoms. But what I want to raise now is the question is, can we do this not in spin systems, but really in a moving gas of particles? So what does it mean to have negative temperatures in a moving gas of particles? So let's go to our moving gas of particles, our Bose-Habad model, our bosons on a lattice in a trap. This is what we have, right? The kinetic energy, interaction energy, the trapping potential. And let's first just look whether the bounds, energy bounds are fulfilled who can have negative or positive energies. So let's look first in a regime where we have repulsive interactions and a positive confining trapping potential. Well, kinetic energy, if we're in the lowest energy band, we saw that's good. That has a lower energy bound and an upper energy bound. That's okay. Interaction energy for repulsive interactions has a lower energy bound when the particles are separated, but there's no upper energy bound because you put more and more particles on top of each other, they have a higher and higher interaction energy. Okay, so this is good for positive temperature, but not for negative. Potential energy in the trap, well, is bound from below because when the particles have here, they have minimum potential energy, the more they go out in the parabolic potential, the higher potential energy have. So again, you have a bound from below, but not from above. So this is good for positive temperatures. And now what would we need to do to have negative temperatures? Well, kinetic energy, we said is fine, but for interaction energy, we'd have to make it attractive. When we have attractive interaction energy, then we have a bound from above in this state and no bound below for attractive interactions. And we would actually have to make the trapping potential anti-trapped. Okay, if that's anti-trapped, then you have maximum potential energy here and then lower potential energy when you go out. So you have the bound from below above in that case. So this is something that can support, at least in theory, negative temperatures, and this is something that can support positive temperatures. So now let's look at how we should think about those negative temperatures in just the kinetic energy time. So here's entropy of the system as a function of energy for this kinetic, only looking at the kinetic energy of the particles in the band. If everything is the lowest energy state, zero entropy, okay, we have zero entropy state, we have minimum energy. When everything is in the highest energy state, we have also a zero entropy state, but we're at maximum energy of the system. And in between, when we distributed the particles continuously over the band, we have a maximum entropy state with the energy given just by half the bandwidth of the system, E max minus E min. So the natural shape what this is is just a curve like this. So if you think of entropy now versus energy and derive the temperature from that, you find this has zero temperature, this state up here is plus infinity temperature, then it jumps to minus infinity and goes to minus zero again, okay? So this looks a bit strange, but this discontinuity is really just an artifact of our definition of temperature. If we would have defined temperature by a beta, you would see you would have continuous value of minus infinity to plus infinity. So this infinity is not so troubling. So how should you think actually of these things, are these things hotter or colder than this? Is a negative temperature state hotter or colder than a positive temperature state? Hotter, who's, who's for hotter? Who's for colder? So then somebody has to explain why is hotter? Why is, why is negative temperature hotter than, how do we, how do we define hotness? What does it mean even to think of hot, what does hot mean? Yeah? Yes, yes, exactly. Hotness, when we say one object is hotter than the other, we say that if we bring them into thermal contact, energy will flow from the hotter to the colder one. That's how we classify hotness in thermal dynamics, okay? And in that sense, you're completely right. This negative temperature state is hotter even than an infinite temperature, positive temperature state, because whatever positive, infinite positive large temperature you have, if you bring it in contact with a negative temperature state, energy will always flow from the negative state to the positive state. So we solve that question. Okay, how do we get there? How do we get there is the following sequence. So now you know, we start with our bosons in the lattice with weak interactions. You see they occupy Q equals zero. They form our bosons, they condensate. We're in a trap, we have repulsive interactions and now we make the interaction stronger and stronger or the lattice deeper, which is equivalent. And then we make our mod insulator. In the mod insulator, remember, we occupy all the single particle states of the block waves. We have one particle per lattice site. And this is, drive this to the ultimate limit of kind of almost flat band unit occupation, one boson per lattice site. And now we do a trick. Now we just switch interactions from U to minus U and we switch suddenly the trap to an antitrap, very suddenly. If we do that, the eigenstates of here, remain the eigenstates of here, but effectively what we've done, if you look at the Hamiltonian, we went from our original Hamiltonian to minus the original Hamiltonian. The eigenstates remain the same, but we just inverted the Hamiltonian. And then what happens now, we're not in the lowest energy state anymore. Now we're in the highest energy state of the system. And if we now ramp down the lattice and make interactions weaker again, we again get a super fluid, a Bose-Einstein condensate, but a Bose-Einstein condensate at those band edges. And we'll see tomorrow this condensate is completely stable, lives very long, as long as a repulsive condensate, and has kind of interesting features that we went a little bit over time today that we're gonna talk about tomorrow, okay? All right, so thank you very much for your attention today. Thank you.