I. Bloch - Probing and controlling quantum matter with ultracold atoms

thank you very much for the kind introduction and thank you very much for having me here it's a pleasure to tell you about some of our recent experiments over the past years that we've been carrying out in Munich on rather diverse range of topics so what I planned today we have to see if we get through everything but give you a little bit of a flavor of the things we have been doing and the questions physical questions we have been addressing in these ultra cold atomic systems so first I would like to start out with an introduction into the field talk a little bit about something that's of course very much connected to the most recent Nobel Prize in Physics topological matter and how we can actually probe that using ultra cold quantum gases different ways and either developing new interferometric techniques or using these quantum gas microscopy experiments that I'll show you next then there's a completely new topic I think which has been taking a storm the field of condensed matter physics which is called many-body localization it's in fact the old problem that Philip Anderson try to address already before but it very much connects a lot of actually fundamental questions also in quantum information theory and statistical physics and again I'll try to show you a little bit what we've been doing in the cold atom context in this direction and finally if I do have time but we really have to see I would love to talk about rittenberg atoms because we like atoms really this is the place to talk about rittvik atoms if not lkb where else and so I'd like to show you some recent progress on these rid progress quantum gases okay so let me start and motivate just briefly the research we're doing of course everything we're doing is driven very much by trying to understand quantum many-body systems and they of course complex they're different kind of things going on there we have very many complex interplay of phenomena masking several effects so very just stop for a moment all right that's okay and and of course you know what what is trying apart from all the technological things that you can do with these systems if you understand them of course our interest is more directed to the fundamental questions associated with these to the fundamental scientific questions and of course it builds a lot on work famous work you know going on in the field of quantum optics the most recent Nobel Prize in that field - Dave wine and inertia wash of course - being pioneers in controlling single atoms ions and photons and we want to take this to the next level in trying to build upon these advances in trying to assemble controlled many body systems in trying to now understand look at questions of complex questions of you know complex quantum matter and this is of course not pursued only in the systems I'm going to introduce to you these crystals of atoms but of course other people are doing this as well the ion trap community is very successful superconducting device community is very successful in doing that so I think almost everybody in the field of quantum computing quantum simulation is trying to address many of these similar questions it actually starts out with this famous paper from Richard Feynman but let me skip that and go directly to the heart of what we're doing and just very briefly introduce the system that we're looking at so we're trying to load ultracold atoms so we load ultracold atoms into crystals of light and here's a beautiful pattern that was actually taken by Ted hench just to show you the beauty of these crystals of light this is a very simple experiment he did he took a laser beam expanding the laser beam okay so there's a big laser beam and this big laser beam is hitting these apertures so there are five apertures here the laser beam passes the apertures and with the lens you focus the laser beams behind and of course what you get is the interference pattern of those five these openings that you get there the light passing through those five openings and this five fold case you actually get a quasi crystal structure as you can see so this is the beautiful interference pattern that emerges just from light hitting these apertures here and it just shows you know that this structure is not only aesthetically pleasing but we have very good control over them they are completely defect free and what we do is we essentially load break we place very gently cold atoms into these structures and try to observe the behavior in these artificial crystals of light so these atoms are held in these light structures and we try to look at their behavior in these light structures now how is this complimentary tour - what real condensed matter physicist is doing and actually there is something on this slide you can't even see it I can hardly conceive myself but I know it's there you will see it in a second actually it's here you can see it maybe so this let's imagine this is a real crystal real material with the spacings going on and in essence of course what people building better x-ray microscopes are doing they're trying to build very good microscopes to look into this very small crystal and what we're doing is something exactly the opposite we're trying to blow up this crystal by a factor of 10,000 instead of having angstrom distances we go to micrometer distances and now these systems become accessible to optical means that we can look into these optical crystals so we're essentially blowing up this crystal to large scales 10,000 times larger than what you have in in real materials and this of course has many implications and you might ask you know first of all can you even hope to observe condensed matter behavior in such systems but then of course you remind yourself that the only thing that matters to see collective quantum behavior is in essence the ratio of the debris wavelength over the inter particle spacing so if we on the one hand make this D 10,000 times larger then in a real crystal if we can make lambda 10 thousand times larger at the same time by cooling to lower temperatures we can be in the same quantum degenerate regime as we are in real materials so we can in essence observe many-body physics at these very dilute densities observed phenomena that we typically only know from real materials at kind of ten orders of magnitude larger densities in these systems of course the price we have to pay is that everything happens now at few nano Kelvin compared to millikelvin two kind of several hundred Kelvin here another advantage of the approach if we look at time dynamics when you go to these large scales with these low energy low energies involved this of course also means that everything that happens here on a femtosecond at a second timescale will happen on this case here in this regime here on a kind of millisecond timescale so time dynamics is very nicely accessible in these systems and we're making a lot of use of that in the experiments okay so let me start with the first topic then where to be analyzed this is ultra called topological matter and I want to give you two examples of how we can probe that in the experiments and this is of course very much connected as I said to this year's Nobel Prize to especially today with telus work which I'm going to first touch upon and later also connect to Duncan Haldane's work a little bit later in the talk so where do we see topological effects well the most prominent cases we know are for example the integer quantum Hall effect topological insulators off we talked about interacting quantum Hall swandam systems for example the fractional quantum Hall effect and let's stick for a second with the simple systems the symbol band systems and try to understand recall again where this topology actually emerges from and for that we have to remind ourselves how we describe the eigenstates in such crystal structures well there of course Bloch waves and they give rise to scalar features which is just a dispersion relation it links the energy to the momentum of these states in the lattice but the other hand we have geometric features and these geometric features are intimately tied to the wave function so we can define on the one hand something that we call the Barre connection which is taking the gradient of this block cell periodic Bloch wave function overlap with the Bloch wave function itself again and if we take the curl of that or the generalized curl because this can be a multi-dimensional parameter space if we take the curl of this Barre connection we can define something that we call the Barre curvature in the system and these are the geometric features that can give rise to topological features and these are things we would like to measure in the experiment in new ways so how do you measure that that's the first question how can you measure these geometric topological features well the first thing you can do to measure these topological features of these geometric features is for example that you take a closed loop in your parameter space and look at the phase shifts that you pick up under such a closed loop evolution so let's imagine this is the Brianza of your system you start out with some case base point here and then you move along a closed trajectory and come back to the same point here so what happens now for the wave function of your particle is that on the one hand you pick up a dynamical phase shift of course but on top of that you can pick up a geometric phase shift and this geometric phase shift will be just the line integral of the very connection over this closed contour divided by 2pi now there's a very special contour you can take and that's given if you take the contour around the entire zone of your system of your crystal structure that would be written down here so you take the line integral of the berry connection over this path divided by two pi and you can actually show that this number this is a topological invariant and this is the so-called famous churn number of your bulk material it's about property because it just depends on these bulk eyeing states that we have in the system okay and this again if we apply now Stokes theorem you can see that this is equivalent to integrating the berry curvature which is just the curl of the very connection over the entire breanna zone and this is exactly what a condensed matter physicist physicist would do if you take an integer quantum Hall system you have an insulator you have a Fermi sea which fills all the states in your band so with this filled family see in your band you're effectively integrating over the entire Breann zone so you get the char number and this char number here if it's non non zero you we say that the system carries topological features this char number actually links the number of edge States you have in a finite sized sample and this is was the famous work that was one of the things that was responsible for getting the Nobel Prize to David tell us this was expressed in his famous paper in 1982 which linked this bulk topological invariant to the quantized Hall conductance that we people measure in the integer quantum Hall effect and you can simply say the number of edge States that you have is determined by this char number each of these edge States carries a quantized conductance of e squared over H and this explains why you have the quantized Hall conductivity in the integer quantum Hall effect now you see one thing we can do with atoms were not restricted to moving particles around the bronze own here we can move them basically around any trajectory and that's a big advantage that I'll make use of in a second but I just want to show you another method how we'd use this and this was actually done by by Monica Idols Berger who's also in the audience who's spending has been spending a year now in John Dalli Pass group how you can measure this this very curvature in another way and determine this topological invariant and this is coming back to the basic semi classical equations of motion how a particle moves in in a solid so you see on the one hand you have the force which is just the gradient of the potential here which just gives us the electric field and the Lorentz force of the particle and then you have the velocity of your wave packet which is just given by your group velocity this is the dispersion relation of your band so it's just a group velocity and you have this this term here too and this term here actually as a side note in history is interesting to note that this was already expressed in the nine famous 1954 paper of couplers and Letendre but later it was completely forgotten again so if you look at the standard text books of Ashcroft and lemon you will not find this term for example in the semi classical dynamics yeah but you see actually all these equations just become fully symmetric if you add this term and this is what we call the anomalous Hall velocity that you pick up when you expose a particle to a force in this solid and if there is a very curvature here and you apply a force in this direction this will lead to a transverse displacement to a transverse velocity the particle picks up given by this very curvature and that's one way how to actually measure this this very curvature in the experiments and this is what Monica and colleagues did in in the experiments at the time in Munich and to show that from this you can actually determine this churn number that I explained before which gave rise to this measurement of the churn number of one which we expect it to be one without making use of any edge States so it was the first time you could measure this Choa number in in kind of a system without making use of edge state transport and a non electronic system actually I want to move on and tell you about a second method which i think is even maybe a little bit more striking and makes really nice analogy between the Aronoff boom effect and what we can how we can interferometric we measure this very curvature in the experiment so let me remind you again of the errand of boom effect so you know it's one of the most foundational effects we have in quantum mechanics usually explain that if you take an electron wave packet you split it you move it around a region with a certain magnetic field and you recombine the electron wave packet again then you get a phase shift between these electron wave packets arriving at this point here and phase shift is going to be proportional to the magnetic flux enclosed in this interferometer area so that's the standard error norm effect okay so this is the vector potential of electrodynamics this is the B field we have and now I can write down exactly the same expression for the berry curvature that's present in a solid so we can now take the berry connection here and if we take an atom we're now in momentum space if we split the atomic wave function in momentum space we move it around a region with very flux then the particle the wave function of the particle is going to appear have a phase shift here which we can measure and this phase shift is going to be proportional to the enclosed berry flux in this interferometer area and you can see actually the equations are precisely the same that you would write down in electrodynamics okay so you have exactly the same equations now you've built a momentum space interferometer that doesn't measure magnetic field that that measures these geometric properties of your bands that measures this very curvature in your system okay the test case we used to study this is graphene hexagonal lattice graphene has an interesting Brown zone so this is the Breann zone of graphene we have these gamma points the K point K prime point and if you remember graphene has this Dirac cone dispersion relations this relativistic dispersion relations precisely at these K and K prime points what it also has at these K and K prime points is actually a singularity in these geometric features the very curvature that I mentioned before so the very curvature the very flux is actually concentrated to values of pi and minus PI as Delta functions at these K and K prime points okay so I really have Delta function localized buri flux at these singularities at these Dirac points that you see here in the experiments and that means of course in a transport experiment you cannot measure that because you have zero measure in momentum space if you move particles through this region you will not be able to hit this kind of and see kind of some effect you have because simply this has just zero measure but in interferometric way we can measure it so what I want to show you now our results where we build this momentum space interferometer we split the wave function of the atom we move it around these paths and we buying it here and what I'm going to show you is when we read out the phase of this in the fra meter how that changes as I change the endpoint of the interferometer so we're going to start with a diamond like this no deer a cone included then we're going to move to a point which includes one Dirac cone and then we're going to move to a point where we have two Dirac cones included in the interferometer and we just want to see how this phase shift actually changes in the experiment and this is what we see so see this is the end point of the interferometer this is the end point of this diamond and you see as long as there's no Dirac cone included in the interferometer there's zero phase shift as soon as you hit the Dirac cone you jump by PI and phase this is this PI Barry flux located at the direct cone and as soon as you have the next Dirac Cohen entering it jumps back to zero again because you know of two pi or zero pi included in the interferometer area again what's really quite striking when we did these experiments is how sharp this jump is so here's the endpoint of the interferometer and just by changing a few percent in momentum the endpoint of this interferometer you see this jump from zero to pi occurs and that of course immediately tells you that this very flux must be concentrated to extremely small regions in momentum space and from the steepness of the curve we could actually deduce experimentally that the Barry flux has to be local localized to better than one millionth of the size of the Brillouin zone our system of course remember that's the best we can do in terms of approaching this Delta function that I said the system should have theoretically all right let me move on to another topic and come to topology in a different context of many particles in the system and this turns to our quantum gas microscope microscopy experiments and before I explain them a bit more let me first introduce the basic notion of what this actually means to do quantum gas microscopy on a wave function on a many-body wave function so imagine you have some complex quantum system you prepare some ground state or some many body state and let's say this many bodies state is a superposition of different spatial configurations of the particle of course there will be complex phase factors here but let's forget them for a second ok and now you make a photo imagine you could take a a fractional quantum Hall liquid and you could make a photo where you see all electrons in the system what what would you see okay well if that's the state of your system and you make a photographic measurement what's going to happen in of course is that the wavefunction collapses on to one of the Constituent configurations in particle space so in one shot of the experiment you might see this configuration of the particles but if you prepare up sigh again and make another measurement then you might see any of the other configurations also with the probability of course given by the weights of these amplitudes that we have here in front of these wave functions okay that is extremely powerful because it means now you can ask very complex questions you can start to ask questions how is for example this particle correlated with this hole and these two particles over here and that hole over here so you can really start to ask questions when you get access to these single snapshots of the wavefunction that are simply impossible to ask in standard condensed matter setups and I'll give you an example in a second of how this can be used to reveal something that you cannot reveal incident at condensed matter setups we also showed at least in other experiments ideas how this can be actually also done for currents so in principle you can also measure currents between bonds and thereby get snapshots of the currents the quantum currents going on in such a many-body system okay and that gives you complementary information about the phase of these prefactors in the amplitudes of the wave function so this is what this looks like typically in the experiments so we have a very good microscope objective with a high resolution that can take a photograph of the atoms in a single plane a single two-dimensional plane of our system and the way we start to see the quantum system is that we turn on resonant light and this resonant light will be scattered by the atoms that's exactly the moment where we make this measurement this projective measurement and then we see the atoms coming out for example like this and you see that this is really a fantastic signal-to-noise ratio you can see each atom each individual atom here lighting up and showing you that we can really see these individual atoms in these lattices we can also control them I'm not going to talk a lot about that we also have possibilities to control them here is just how we can manipulate bring them by shining a laser beam onto an individual atom and flipping the spin state of that individual atom so you can for example make a line of atoms a star and yell anti ferromagnetic a this just for fun we made this sigh out of twenty-six individual atoms in this optical lattice just to show you that we can really you can tell me I want to go into atom number fifty five forty three and I want you to wrote the spin of that atom by ten degrees and we can do that okay and this this just kind of the proof that we can do that recently this has been extended also to fermionic systems in in our munich team in marcos con st martins villains team stefan curse team so many teams around the world actually chose of tigers and also is here who've been doing this now also for phonic particles this is just a photo of a fermionic band insulator showing that we can do this as well the twist we have in our famiiy onyx system is interesting compared to the other experiments is that we combine it with a way to see all the spins also in these photographic measurements and let me explain a little bit how we do that so imagine you have a now fermionic particles spin ups-and-downs electron spin ups and downs you can have holes there you can have doubly occupied sites and imagine this would be the configuration of your electrons on this one-dimensional chain so how can we see that in it's been resolved way we just take a photo we would just get something like this where we see maybe there's a hole here there's an atom here but we don't know is this a spin up or is this a spin down atom we just don't know okay how can we still resolve that how we can still resolve that is that we take this chain we split it into two chains with the double well beam splitter and we do this under the action of a magnetic field gradient so in essence what we're doing is we're making a controlled sangala separation we're making one line into two but we're separating it under the action of a magnetic gradient field such that the spin ups move into the upper chain and the spin downs move into the lower chain so now you have for example all the spin up sitting in this chain all the spin down sitting in this chain and now if you make a photo of these two chains you have a complete reconstruction of the original occupation of that single chain for example let me you picture like this one this is the upper chain the lower chain okay this would mean no particles a hole no particles a hole the spin down the spin up spin down a spin up and spin down a spin down a spin up and the W occupied side that would be the configuration we have on that single chain and that you can directly see in these experiments and you can fully kind of get the information of all the particles and holes and dublin's in that single shot so what can you do with that or the first thing you can do if you have for me on ik particles you can look for anti ferromagnetic correlations a standard thing that also a normal condensed matter physicists could do and if you look at this in the 1d setup here look at this correlation function two-point correlation function actually this should be I plus D sorry here then you see you see nicely this anti ferromagnetic staggered configuration that you expect for a Heisenberg model okay so that's pretty much stand up but now and actually sometimes you get the beauty of seeing shots like this okay well it's really up down up down up down up down but keep in mind that for Heisenberg I need for a magnet even at t equals zero this is a very rare configuration the ground state is not at all like that this is like the classical nail state but it has a little bit of amplitude in your wave function so sometimes you do see that state okay what else can you do let me show you something that's more complex that you cannot do in a condensed matter setup and that comes down to the phenomena of spin charge separation so imagine you have this anti ferromagnetic chain here and you remove one particle let's say we kick out this particle so it leaves a hole behind what can now happen is that can this hole can move freely in this anti ferromagnetic chain and at the same time it leaves the so-called spin on excitation behind and this is what we call a hole on excitation the hole can move freely and it leaves behind it's been on kind of excitation you can also ask if you have a hole in the system what is actually the ground state the lowest energy configuration around that hole is it a ferromagnetic alignment of the spins or an anti ferromagnetic alignment of the spins around this hole and it turns out that the lower energy configuration around a single hole is an anti ferromagnetic alignment of the spins so if you would do this in a ground state of this whole doped chain you would actually find that around a hole the spinster order antiferromagnetic lee and not just what i showed you before that you kick one out and their fur magnetically aligned so what you see is what happens in the ground state of this one D for me how about model if you dope it with holes is that each time you have a hole you flip the parody of this anti ferromagnetic background by by sine so let's consider let's say this would be a plus one configuration down up down up and now when there's a whole this is slip two down up down up down up okay whereas if we would just kick out the hole it would be just the reverse so each time we have a hole we get parody kinks in this anti ferromagnetic background and now let's see if we can now I show you an example of already something that you cannot measure in a solid this we can do that we can measure precisely that with our quantum grass microscopes we can look how is this how are the spins oriented around holes so we look at the spin correlator at site I and I plus two conditioned on the fact that there's a hole in between how are the spins aligned conditioned on having a hole in between that's a three-point correlator already right something you cannot measure anymore and if you do that you see compared to the undoped chain where you have this positive correlation for distance D equal to you see the sign indeed flips in the case when we have this hole in the ground state configuration of the problem so indeed when you have a hole the ground state here gives rise to anti ferromagnetic correlations around around this hole you can actually see this even more striking by just looking how the spin correlations present over larger distances conditioned again on having a hole in between those spins or at the same side of the spins so here's the kind of anti ferromagnetic lee ordered system so you see it goes from up down up down up down and whenever you have a hole in between you see how this pattern is just shifted by one side how this anti ferromagnetic patterns just shifted by one side because this hole just flips the whole anti ferromagnetic domain by parity sign minus one here's just the same data where I just took out this minus one staggering of the antiferromagnet showing you clearly that when the the hole is between the two spins you get a different sign of the anti ferromagnetic parity whether the case when the hole is on the same side as the two spins when it's not sitting between these two spins you get the opposite sign of the anti-ferromagnetic parody was actually also pretty interesting what was striking for us is that this anti ferromagnetic correlations even persist even if you have many holes between the two spins so this show this is shown here so this is a number of holes between the two spins so let's have like five holes between those two spins okay and you can see that even for these large distances the anti ferromagnetic correlations remain so even if you have many many holes in between those two spins they still want to remain anti ferromagnetic li correlated in this strongly doped fermi hobbit chain in our system okay so let me connect this now to the notion of topological order that was now actually important for for banking Haldane's Nobel Prize how that is connected so we said each time there's a hole there's a priority flip in the anti ferromagnetic background right so if we somehow could remove the holes on the other hand what we would actually find that this is a perfect anti ferromagnetic in a fictitious space that call that's called this space squeeze space it's the space where we would have removed all the holes from the system we squeezed the system together and then we would find actually a perfect Heisenberg anti-fur magnet as in the undoped case okay and this is actually the essential notion of spin charge separation so if you look at it from a wavefunction level and these are actually beautiful results from beta and Zod solutions from voinovich in the 80s and Ogata and Sheba which showed that if you write down the wave function of this state it factorizes into a product wave function of spinless fermions which basically define the position of your particles times a Heisenberg antiferromagnet living on this fictitious squeezed space where those particles are and this factorization of the wave function into this charged part and the spin part is really the true fundamental essence at the wavefunction level of spin charge separation now you can measure that you can actually do this pulling out of the holes if you can see them in the experiment like in ours you can do that by introducing the notion of these string correlators so what this string collator does is that each time it encounters a hole it flips this parity of the anti ferromagnetic background by minus one so if you now correlate the spins on-site as Z and as Z a distance D if you take into account all the spin flips you actually find a perfect anti-ferromagnetic N and this should give rise to this anti ferromagnetic spin correlations of this dope dope chain and that's actually very nicely written down if you want to read about this there beautiful papers by Young's on which explain this connection in a very very nice way so again let me now connect this to the whole of to a whole notion of order parameters of how we characterize many bodies States and typically we do this through this Landau paradigm of phase transitions where we take these observables at to size x and y and as we take the distance between those two points to infinity this approaches a constant value so that's very common scenario for example for a ferromagnetic and anti ferromagnetic ABC a condensate wave function of BC a superconductor for example in a ferromagnetic if you tell me the position of the spin at some point here I can tell you what the spins will be pointing everywhere in your system okay the notion of long-range long-range order ok now you can ask are all of the phases of matter characterized by such kind of Landau paradigm order parameters and of course by now you should answer no because we've seen I've already introduced you an example in the last case where this is not the case we have more general correlations in kind of strongly correlated many-body systems where in order for this call a that to have a finite value of measuring the particle observable at point X and observable at position Y we need to also know what's going on on all the particles in between those two points okay this is the idea of what we call a string correlator and if that goes to a constant value that we say this has a hidden kind of order parameter given by this string correlator why do people say this is hidden people say this is hidden because this was completely unimaginable that you could measure this okay that's why people thought this is entirely theoretical concept to kind of characterize correlated states of matter because typically you cannot measure what's going on on all your particles between the two end points where you actually want to do the measurement okay but what I showed you in the quantum gas microscope we can actually do that we can precisely measure such spin correlate as because in a single snapshot of the experiment I'm not sone Lee seeing what's going on at point x and point y I'm seeing what's going on at all the points in between and therefore I can evaluate these string correlate as an experiment so actually there's a famous example in the case of the Haldane spin one chain where this indeed shows up and this is kind of I've sketched it out here for the case of both sob out interactions but it doesn't matter imagine you have a spin one chain here and in the case of the how they in spin one chain you have this again this hidden anti ferromagnetic what you've seen in the Fermi Hubbert chain I've shown before so there you have spin zeroes spin minus ones and spin plus ones now what's characteristic to this how they in chain is that whenever you have a minus one here the next spin which is not zero is going to be a plus one and when you have a plus 1 the next spin that's not zero is going to be a minus one the problem is you don't know where the zeros are the zeros can be everywhere on the chain okay so that means in a two-point correlator you cannot keep track of these correlations you need to know the where the zeros are to reveal this hidden anti ferromagnetic structure in the system and by making use of these string Kroll adders you can precisely do that and the string correlate again just removes those zeros and then you see you just have the minus one plus one minus one plus one structure in the experiment okay so now finally I have to show you that this really works in the experiment so this is all good but let's now show you that this works okay so this is a spin correlation measurement now for an underdog chain and we see the nice and different magnetic correlations and now we have a heavily doped chain averaged over different doping values ranging from 40% to actually from 0% to 60% doping and you can see that beyond nearest-neighbor spin correlations all the correlate has just vanished which makes you believe there's absolutely no magnetic correlations anymore present in the system beyond this distance but now if you evaluate the string correlator what you can show is if you pull out the holes all these systems now you see you you see these emerging anti ferromagnetic correlations again you see how you can restore these anti ferromagnetic correlations living in the chain it's just that they weren't visible to you because you weren't keeping track of this hidden anti ferromagnetic structure hidden in these zeros in the system in the holes in the system but if you can measure the holes you can pull them out you can go to this squeeze space and in squeezed space you see nice and affirming netic correlations that you see here that four different doping levels we can recover this anti ferromagnetic restructure beautifully in the experiment okay let me move on how am i doing in time actually good okay so let me move to the third topic another topic so if you haven't been if you can start again now and join me again so the next topic I want to discuss is on many body localization and this is again very much connected to statistical physics and very much connected to the fundamental question how come that we can describe a system you know by thermodynamics statistical physics so well that we can usually do in many body systems so so let's take for example this coffee that's served to us milk coffee okay you wouldn't be happy right you wait a bit then it's all mixed and you get you end up with this and the system we say would have thermalized that's and after after it's stabilized from this state we have no way of telling for example that it actually came from this original state here okay so what I want to tell you now is actually a case where this does not happen and we want to look at scenarios where this typical thermalization does not occur in the system and the system actually can be prepared in a way such as if you prepare such a non equilibrium state like this one it can in principle persist for indefinite amount of times okay so that means that no thermal state is reached because the thermal state has no memory of an initial kind of configuration by definition and that means actually that if we're not in a thermal state all our statistical physics approaches fail okay now that's why I think why people are so interested in this field because it means we really have to derive many things from scratch and think about the notion of phase transitions and statistical physics in this context completely new way let me put this a little bit more on formal grounds what I'm saying let's imagine you have an isolated quantum system prepared in some eigenstate of the system let's say this is our total system it's isolated from the rest of the world it's in some eyeing a state of the system and what you typically find is that if you take a subsystem of size L which is smaller than the entire system here you will find if you trace out this remainder of the system you find that the subsystem typically looks like a thermal density matrix and that's of course the reason why typically if thermodynamics works so well so the remainder of the system acts like a heat bath for this subsystem giving rise to this thermal density matrix that we typically encounter that's and that's what I realize what people call the eigenstate thermalization hypothesis but as the word hypothesis says this is the hypothesis we don't have a proof for that but it actually looks like for most generic quantum systems this is true of course we know cases where this is not true one prominent example where this is not true is our integrable quantum systems and integral quantum systems typically have so many constraints on their dynamics that this cannot be fulfilled but integral systems are typically always fine-tuned so they might not be relevant so much in nature because if you go epsilon away from the integral point in your Hamiltonian you immediately come to a thermal Ising system again but the question is are there more robust scenarios where this is indeed also the case and this is now where the field of many-body localization comes about that basically there is a notion of emergent self emergent integrability occurring in these systems that gives rise to non-thermal izing behavior and failure of statistical physics for interacting many-body systems and this goes back to pioneering work by ultras group in 2006 and I think good reviews if you want to start reading into this and connections to quantum information theory are these two review papers here from David use group Andy would admins group shown here now for a long time this has been a purely theoretical field now more and more experiments are emerging on cold atoms called ions and B centers so it's really getting a very exciting field also experiments to work on so what can we do to make such a system what ingredients do we need well we need an interacting many-body system we need something that's isolated from the environment because if we're connected to a thermal reservoir we're going to thermal eyes by definition and so cold atoms are good in the tourists to these two first respects but we need a third ingredient and that is disorder okay we need also disorder to be present in the system to realize these many-body localized systems and then I'll show you ways how we can probe for them so the way how we create this order is by using these kind of digital mirror devices so these are just probably the same thing that's in this small projector down here just small micro mirror arrays which we can turn often and often we can we can use to make basically arbitrary patterns of light like these two reservoirs for example connected by a single wire light wire or arbitrary light structures but in the case I want to show you now we want to use it for even something much more simple we just want to make a disorder pattern okay so we just turn on these pixels these DMD pixels in a random fashion they are random kind of pixel structure is convoluted with the point spread function of our imaging system and that's what our atoms see as the disorder distribution that we can now fully control and project onto the atoms so here's the probability distribution of the disorder at a given lattice site that we have on our atoms and it's an hour full control now how can you show that the system is many-body localized well let's think again of this coffee and milk example and trying to show what you want to show for a non-thermal izing behavior it's actually easier than showing thermal izing behavior what you want to show that you find one observable which is non thermal for long evolution x okay so what we set up in in light of this we take this initial mod insulator which had unit occupancy in a circle of the radius that you can see the contour here and then we removed half of the particles to this side okay so we created this very sharp domain wall it's like this milk coffee interface but now just particles and no particles vacuum and unit filled lattice structure and this is a single shot with the quantum gas microscope and this is averaged over hundred shots okay and now what we can do we can let this system evolve and then we can do this over 250 tunneling times for example and you can see how this for a very long time lling times it revolves basically into this symmetric cloud and we can actually fit this with nice thermodynamic fit functions which would tell us that this has become kind of a thermal system I can characterize it by a temperature which very well agrees with the initial energy that we have in the initial system here so everything looks nice and thermal izing here however if you add this order you can be in a situation where this occurs for long evolution times where you evolve into this structure where you see this did not evolve into the entire system it did not become symmetry there's still some memory of this initial domain wall even for long evolution times and if you have this memory of the initial state it cannot this cannot be a thermal state and it tells us something about and gives us an indication that we've entered a a many-body localized phase now the way to characterize this is just to take the number of particles to the right of this domain wall minus the number of particles to the left of this domain wall divided by the total number of particles which we call the imbalance of our system and if that imbalance is 0 the particle have spread out completely we're in a thermal izing ergodic phase and if this imbalance is nonzero then we're in this many-body localized phase where the particles basically do not still have this memory of the initial state and do not thermalize and you see actually in the experiment there seems to be a rather sharp point where this transition from their gothic to the MBL phase happens surrounded disorder strength of 6 to 7 we see this transition from ergodic to MBL phase we've also looked you know what happens if you do this for different interaction strengths in the system and you actually find that for different interaction strengths or densities effective interaction strengths as you change the density of your particles going to more non-interacting systems this critical point seems to shift to lower lower values of this order and I want to the experts of you and Amazon localization for non interacting particles this is what you expect for the non interacting curve this is the white points here coming from a simulation because we can't turn interactions in the experiment unfortunately in this experiment but you see how strikingly different Amazon localization is from the interacting case that I show you here okay understand localization occurs for arbitrary small disorder depth you localize in two dimensions so for epsilon disorder strength you localize and you get immediately the finite imbalance that I show you here whereas for kind of the interacting system you see that you get this completely different behavior that you need to go to rather strong disorder strength to localize the particles in your lattice and to see this many body localization phenomena so something completely different from the Amazon localizing localization case that I've shown you before one thing maybe one final thing that I would just like to show you what's become a hot topic in the field is how stable is this localized phase actually to the presence of thermal baths I told you in general a big infinite thermal bath will kill localization immediately okay but what if we have small baths do they kill the system do they kill the localization phenomena or is the system still stable to this localization phenomena and this has been I think to date an unsolved question whether what happens actually but the question is kind of sketched out here so you have your localized system you couple it to a thermal bath and you ask now is this localized system going to localize this guy here or is this thermal bath going to destabilize de-localized this system up here in when these two are finite have finite size so the way how we can address this question is actually a nice setup in our disorder potential we can tune this disorder potential that I showed you to this magic wavelength of seven hundred eighty seven point five five nanometers and that means that actually the light shift the polarizability of our atoms is very different for these two spin states so for a spin down particle we have a zero crossing in the light shift meaning that whatever light we project on them they just don't see it they just don't see this as a potential so it's completely flat whereas the particles in this other spin state actually they can be exposed to the disorder potential so we have an interesting configuration where one spin component sees the disorder and the other spin component does not see the disorder okay and now we can ask what happens now in the experiment when we do experiment like this so we set up an initial what we call charge density wave where we occupy only even sites in the experiment and then in order to populate the other spin component we can just do Rabi flops and just create a controlled fraction of these spin up particles that do not see the disorder potential so that's nice because I can change the ratio of spin downs to spin ups in a completely controlled way I can change the ratio of particles that see to this order and I don't see the disorder in a completely controlled way and so far first experiments that we've been seen first of all for if we have no kind of no kind of be localizing particles in the system we see that again for low disorder strengths this charge density wave relaxes to zero we have an ergodic phase whereas for larger disorder strengths we see we get this plateauing behavior and now we can observe this even out 2,000 tunnelling times okay so this gives rise tells us there is an indication for an MDL phase and what we now find that as we change the spin down population is a Ghatak component that we have in the system it seems to be the case that actually this just pushes the MAL transition to higher disorder strings so if we start to have a certain delocalize in a Ghatak component in the system we're still able to push the whole system into a MBL phase we just need a higher disorder strength to do so now whether that's always the case we don't know so what the phase diagram actually looks like is an open question so we just let's say have here let's say an ergodic phase and here's the MDL phase all right and now I'm having here my disorder strength on the vertical axis in my polarization on the lower axis and whether there's always a point whether I can always increase my disorder strength above a certain threshold of the polarization to reach the MBL phase or there's a certain point where I just cannot do this anymore is an open question which we hope we can answer soon with these experiments all right I think I should not go into the last topic but because I'm exhausted I'm exhausted too actually okay so let me let me just let me just skip the Riddler topic I can tell you more in private about that let me just come to the conclusion and just hope I could tell you a little bit of the most recent experiments we've been carrying out with these cold atoms and lattices on the one hand looking for topological effects in the single particle systems topological effects in many particle systems using these new ways of quantum gas microscopes to characterize these many body States in completely new ways and actually exploiting that to look at really fundamental questions in a completely new field of physics in this many-body localization field and of course we hope there are many more connections we haven't explored yet but we can explore in the future and with that I would like to thank you for your attention [Applause] it's what super well I mean that some state you only see the atoms but also measured clearance no not at the same time you have to make a decision what you that would be a problem no no no so what we do if we want to measure the currents we put let's say we have a lattice okay and let's say I want to measure the Y currents what I do is I partition I cut these these bonds here and make double wells isolated double wells now out of this out of my whole system but I do it in a sudden way okay and then I look at the Josephson oscillations in these bonds and the phase of the Josephson oscillation will tell me who be tell me what the what the current operator what the measurement value of my current operator is if I look at a specific point in time okay so the idea is to start with your full lattice separate it into double wells make use of Josephson oscillations in these double wells and these Josephson oscillations will tell you the measurement value of your current operator if you observe then for example the particle occupation at the right point in time so that's that that's the basic idea but you have to make a choice of course then you of course you made now your current measurement it's not compatible with the particle measurement anymore again of course another question I mean you able to measure a lot of things but behind that you assume that you know the amazonian no not really so tell me how do you measure the amazonian i'm the measure of amyl Tony well I'm not saying how much that's my question I am NOT measuring daemul Tony what I'm saying I'm measuring basically the wave function of the density matrix okay so actually so in the what I've shown you so in this let's go back to what we measure okay so actually what we measure this one this is the slide okay so what we actually okay we we measure we make a photo it collapses onto a certain concept so we see in a single shot we see this particle configuration then I repeat the experiment I see another particle configuration so in essence we're going to what I can measure then is the probability distribution of the different particle number arrangements that's basically the prefactor squared of those particle configurations answering my question could you think of measuring in some way I mean at least some matrix element of the image because I mean you agree if you measure all this thing but you would have no idea of what is emitted naturally you have IDs but you would have no ideas about what is a meltonian this will be terribly interesting so could you could you check in some sense as much as you could on the unobtanium yeah I think of course typically in cold atoms the problem is the other way around that we know that our Hamiltonian very precisely or we believe at least we know it very precisely and we don't know what the states the emergent states are what you're asking I think should be possible also but it requires of course different measurements on transition as you say matrix element measurements between between different states so I haven't thought about carefully about that but should be possible to but I would say typically this scenario we deal with cold atoms that our Hamiltonian is extremely well defined actually you don't believe it you know it's true well here there well here there are many examples where we don't know the answer for example in this MBL dynamics that I showed you in two dimensions there's absolutely no numerical method that can handle this to date we have absolutely no clue in numerix what's going on here okay even 1d dynamics is so difficult that already in the 1d dynamics experiment at the echo Theory actually Fabian a Laius group in Toulouse actually they're doing one of the best exact diagonalization simulations largest scale simulations but that's also limited to 1415 particles and then you always kind of find that size limited in the system today I want to understand one point here probabilities related you have it the lattice but actually the atoms are not the ones that making the lattice so when you see in the crystal if you take out an atom that potential changes well no no that's a very important point you know that's eventually is made by our light fields so this comes back to my first slide here so these so here so this the lattice this is the lattice we can be this is a fancy lattice take a square lattice simpler this is the light field distribution and we control it just through the laser beams but now so the atom has no role in creating this lattice structure exactly yeah so so we're so if you remove an atom it's the lattice structure is still there yeah the lattice is there yes but you don't like that or any crystal yes in some of the cases you discussed actually the afternoon it's involved in Bellotti V or sure self yeah so it's when it moves and it has a defect it changes the absolutely the whole thing that's true you can yes you can you can have situations like that also in the cold atom context not in this context that I showed you here but I think this is first of all a good good situation also to be in that you know your lattice structure is always fixed but you want you want a back action of the movement of the particles on the lattice you can create different ways how to do that you can put your particles either create the lattice in a cavity for example or you can have another species another atom component that forms now it's interaction with the other component a lattice structure and now of course if these atoms move around they will feel this lattice structure actually I didn't comment on this but maybe you is actually an example of how this how this looks so here is remember this relaxation dynamics I showed you where we stood with the to spin components one sees the disorder the other one does not see the disorder okay so one does not let's say see this so we have kind of a initial stripe structure there through the atoms where both has been up and down atoms are and this creates a positive imbalance here that we see here now we see the component that does not see the disorder potential it immediately wants to spread out but actually goes into a negative and balanced value and the only reason why it goes into a negative and balanced value because it sees a lattice structure through the other atoms and wants to move into the main minima of the other atoms so the density wave of the one component that sees the disorder that tries to stay localized tries to stay in this charge density wave creates an effective potential for this other component this iron actually just you had fun about short-range interactions that give rise to this s wave scattering that's right so there are ways to do what you want you want to do but a priori for many of the situations we have it's actually interesting to just have the lattice fixed by itself and no back action and then just look what happens if you dope the system if you change parameters I was wondering for the MBL phase that you've shown you at this interface which is sharp if you look at the statistics the time study yeah that's a good question not yet but I think that's very what kind of distribution is it connected to is you know fluctuations are a lock out after he's a year yes whatever yes very good questions you know we have not looked at it but what you're asking is basically this we have the single shots so these are the single shots how this domain wall evolved so we can now make a histogram of let's say for example actually see it's interesting this particle here made it very far from the domain wall but the others didn't make it so far so we can have a statistical distribution of how the particles have moved away from from the domain wall yes we have not looked at it but I think that you're completely right there are many powerful ways how to analyze this data even better and extract maybe more interesting features out of that yes so far indeed I think the one of the metrics which is very interesting is to know how the entanglement between Part A and arrest is evolving in time yes especially it should be slower you look at make in time I think compared to linear in time for the ergodic phase so do you have any future ways to measure this entanglement you know so there's a way Mao Markos kind of showed in his experiments how you can do this for very small systems how you can measure entanglement entropy I just don't believe it's feasible for a system of the size we're talking about here or hundreds of particles it's just extremely extremely complex to measure extremely many measurements that you need to do that but I think there are other quantities that we can look at and it comes back maybe to Christoph's question that are related to the entanglement entropy growth that is for example how the fluctuations in your imbalance trace for example in your time trace how they evolve as a function of time so it's not the average value of this imbalance which gives this density wall but the fluctuations how they evolve as a function of time so people have shown that these are also linked to kind of at the fundamental level to this phasing dynamics which gives rise to this logarithmic t entanglement entropy growth and those are things I think with the quantum gas microscopes we can indeed we can indeed should be able to measure in experiments absolutely there is no follower question like thanks again [Applause]

Channel: Département de Physique de l'ENS

Views: 2,522

Rating: 5 out of 5

Keywords: cold atoms, quantum simulation, condensed matter, quantum matter

Id: jPRxLrDHxog

Length: 57min 33sec (3453 seconds)

Published: Fri Mar 23 2018