 Also, she would like to know whether you can have fish at the conference dinner or whether you prefer vegetarian food. And yeah, so that's about it. Thank you. Good morning again. Welcome back for the next talk. So Wolfgang Belsik from Constance will speak about higher dimensional topology in Militavinal Josephson Matter. Thank you. So thank you very much and thank you to all organizers for inviting me and organizing this great event. Yeah, it would have been nice to meet already two years ago, but anyhow, now we restart everything like this again. Okay, so I would like to talk about our work on higher dimensional topology and actually also lower dimensional topology in Militavinal Josephson Matter. If outlined just to explain the structure that I will first introduce, they say some general words about topology and the geometry. And then I will introduce the synthetic dimensions in Militavinal Josephson Matter. Also some people might know it very well here. And then I will talk about our results concerning microscopic wave spectroscopy, which allows to address the quantum geometry and will spend a few slides at the end to show you that there are some interesting phenomena in higher dimensions which one can simulate. Okay, you all know that one of the paradigm systems of displaying topological effects as a quantum Hall effect, which shows the display quantized Hall conductance. And this is surprisingly accurate. I wrote it despite this order, but actually also due to this order, we have these nice plateaus, but they are extremely sharp and that's kind of quite amazing. So there is a relation of this effect, which was pointed out, for example, by Laflin, to pumping adiabatic pumping by putting the system on a cylinder and this will become later important. So you have some magnetic fields and perpendicular and you drive a field flux through which is an electric field. And this gives you, for example, in terms of this pumping parameters, very easily this exact quantization of E squared over H for the conductance. Okay, and then this was later related to topological effects. So the quantum Hall insulator, if you want, is a topological insulator and this was pointed out by Soules, but I don't want to go into much details here. Yeah, I also wanted to point out another thing which was recently interesting or found out in the context of topological insulators, doing actually completely different kind of an experiment from a completely different field with angle resolved photo emission spectroscopy and spin resolved, so spin circularly polarized spectroscopy, so they kind of can detect actually the spin polarization of the bands and the spin polarization of the bands locally are related to the topological properties of this Hamiltonian. So this is the Hamiltonian and then basically I just want to point out they measure, so to say, a difference in the absorption between right circular polarized light and left circularly polarized light and this gives an access to the polarization which is here. So this can be actually connected to local geometric properties and I will basically in the end also or in the middle of the talk show you how polarized light absorption, what this has to do with local geometry or topology. Okay, a brief intro to quantum geometry because not everybody might be familiar with this but it's very basic and actually I think now I should basically teach it in my basic quantum mechanics course which I didn't do before but now I have the feeling it should go there. So if I have a quantum state with a Hamiltonian which depends on some parameters and if I basically the quantum geometry just as in classical geometry is a difference between a vector which is transported by some infinitesimal amount in parameter space and then I naively can kind of say, okay, so this will be something like this derivative taken product with this psi but this is not a gauge invariant quantity but when can define make a gauge invariant quantity which is called the quantum geometric tensor which is defined here so it's defined in terms of these derivatives and some projector maybe exact definition is not so important but one it's a Hermitian matrix so it has a real symmetric matrix which is called the metric and it has an off diagonal term and this is actually the very curvature okay this is called the Foubini study metric and this is a very curvature okay and so yeah so this for example is somehow a measure how much the phase changes if I change my state and not only the absolute value is so to say orthogonal okay so that's geometry and then we know very very phase is if you move along a closed circle in parameter space we collect geometric phase so it's slowly which is then given by this by this very phase which is yeah this geometric phase which has a certain value independent of the value of the path and it's actually also related to the topology so if we define the first turn number in this way and integrate over the whole many fold we get an integer this is a global property and it was shown that this is only an integer and actually related for example to the if you want so the very curvature of the Landau levels if I integrated over all over the face over the space of parameters I get the turn number and this is exactly the quantized conductance of the quantum Hall effect okay so this was already the intro deduction to this geometry and I want now to introduce topological Josephson matter and Andrea bound states again this doesn't really need an introduction here so Andrea bound states are quasi particles in junctions usually discussed between two superconductors so they are connecting by Andrea reflection and normal scattering and form such bound states which which live inside the junction but also somehow inside the superconductor and it's important that one obtains the it's Andrea reflection is phase dependent and therefore what obtains a phase dependent energy quantization condition and therefore the energy depends on the phase and this means that we have said these states so to say carry a current and arc can be microscopically related to the Josephson current okay then this formula is already well known I forgot it's also going to know so well known that I forgot whom to site here for this and so this is the energy of an Andrea bound state for a junction which was a certain transmission T and depends on the phase and I calculate the phase difference and so in the transmission probability so the spectrum is well known so it's just a kind of two level spectrum which hasn't avoided crossing if the transmission is finite and so we have these two levels and one can calculate also the corresponding currents which are well known relations so we have the standard Josephson relation for small transmissions and non-synosodial relation for higher transmissions for example transmission equal to one actually it should be some absolute value here it's a bit yeah and I want to point out so this current so if you integrate the current over the phase obviously you get zero here yeah so that's and that's kind of so if you are debatically change the phase you get on zero current on average and and that's so to say well known now it was found it actually this is totally different in higher dimensions so this is basically one league in the face and this was found by Nazarov and collaborators in this work 2016 so they considered now scattering matrix connected to n superconductors and in that case the Andrea bound states kind of yes it depends on n minus one superconducting phase differences and if you want superconduct conducting and superconducting phase differences n minus one form and I went minus one dimensional Briand zone yeah and now we can play all the things we know from topological insulators three dimensional topological insulators while semi metals and so on all this can be so to say synthesized in such a Briand zone okay and it's quite interesting that there is a possibility to have this table while nodes which sort of say cross at a certain points and are not destroyed by us by certain perturbations this is different than one d where you get so to say this under crossing for any kind of perturbation unless it's protected by some special symmetry like time reversal symmetry and in this topological in Josephson matter these sort of say while nodes exist without any special symmetry so they are kind of constructed by this okay so basically we look for zeros in the spectrum and this was from this paper playing around with scattering matrices they found some bound states and then there's one which has zero crossing and these points are then located in this Briand zone at different different points actually so this is probably some kind of while semi metal thing I'm not really expert in this but around each point you sort of say have a three dimensional dirac cone and some there of course connected if you plot the energy I have some energies later where you can see this okay and then there's interesting another interesting prediction is that it's in the topological phase so if you do not have the crossing yeah so if you then you're so to say in a let's say possibly topologically non-trivial phase with a gap and then they have shown that they're using exactly the similar arguments as I did earlier for the quantum Hall effect that the current so to say in a time dependent if you consider the time dependent Hamiltonian it has a so to say adiabatic contribution which is just the energy but it has also a non adiabatic contribution which exactly contains the berry curvature right and then the interesting thing if I integrate the berry curvature over all phases we get an integer right so the first term vanishes if we integrate them over both phases but the second term not necessarily vanishes and this means we have using similar arguments a quantized transconductance in units of four minus four e squared over H and okay and this means this was then also calculated and each time we we so to say in this area in this kind of plane and if you change the phi one phi two plane then we each time the so to say plane crosses such a while noise that channel number changes by one and therefore the transconductance okay and there have been many works following that in in the group and also by others and some people are also here in a kind of a different system which I will also mention later so and okay so what motivated us was two things so first of all we wanted to so to say kind of try to have a more engineering approach to find this and if bound states and the other way is to see what one can do with microwaves this process back cross copy because we know that this is very very good to because you do not connect your system to leads and avoid certain noises which come in and so on okay so this and and then we have started to to look at systems and try to figure out when do they have a non trivial turn number and also the first system we found was rather complicated for superconductors some kind of dot region no Coulomb interaction here and then some kind of tunnelings between all these leads there's another version of this which has three superconductors and a flux passing through it as a search parameter then we investigated also linear systems of this type in this work and all of them show basically some kind of effective low energy Hamiltonian which is of this form of 2 by 2 Hamiltonian which has a d vector which is so to say related to it's like a synthetic magnetic field which is controlled by the phases okay finally we found the simplest model we just need these two dots and adjust this number of connections and so this also has a topologically non trivial phase but it's not published anywhere we just said okay I mean you can play around with this and create tons of models now but I think we first wait until we see something in reality what is what can actually be done yeah but anyhow so this was final the simplest model okay so just to I mean we described this with the standard non equilibrium greens function take technique also we don't need non equilibrium here but greens function techniques integrating out tunnel couplings and so on and so on and then one has this kind of simple result in lowest order in this t zero so this is some kind of perturbative result which shows nicely like you have an effective magnetic field vector which you can in magnitude and in direction tune but by playing around with this fluxes and now the question is does it have a topological phase and then when first looks is there a while note and yes so if you if the absolute value is zero we have a while note for this set of parameters and the dispersion relations are shown here I don't want to so we have seems like a single while note here but we have also situations where we have two while notes so this is a bit like this while semi metals which have two while notes which are somehow connected okay and it turns out that there are so this also for while notes and they too are on one plane in the in the one to plane and then we can calculate the turn number and everything is in some sense already now as expected so the turn number becomes jumps at this point and it can jump to minus one plus one and back again and this all depends on parameters and if the parameters vary a bit and we have investigated this for some examples this points move around but they can only be destroyed if they all if they somehow move together yeah okay yeah and then let me now come to the point how to relate a spectroscopy in quantum geometry now I'm gonna tell you something you already know but you didn't know that it's related to quantum geometry so we all know Fermi's golden rule is so let's start with two levels then but two levels actually is this tense or simplifies because you can use a completeness relation so it's related to such matrix element with final state and initial state and for example the diagonal elements are justice and if we for example drive one parameter exactly this parameter J here is some amplitude with a certain frequency then we we kind of we know we kind of wiggle it is so to say the energy if you want and then this is just so to say what we do then to calculate golden rule so this is golden rule for this parameter for this perturbation here and we see this has a usual matrix element which to contain the derivative of the Hamiltonian with respect to the parameter and then there is such a relation because these are eigenstates which basically gives the energy difference and because now we have the delta function here this energy difference so to say cancels out big busy omega and actually we see that the pre-factor so the line the line intensity is proportional to the diagonal elements of the quantum geometric tensile yeah so this is again you everybody knows golden rule but you didn't know that the line intensity is related to some kind of geometric property of the system okay and now comes the point now we want to access other quantities and this can be done by again looking at the two-level approximation and now we do not change one parameter in time but we change another parameter in time so very with the same frequency but with a different with a phase difference yeah so that's kind of in my experimental friends told me this is straightforward to implement I look at who told me this first time so and but in principle yeah okay it's the same frequency but with a controlled phase difference yeah and so basically you depending on this phase difference you make certain lines or loops in your parameter space yeah and now you do exactly the same golden rule calculation and as a result you find that the pre-factors align with now depends on this phase parameter phase shift and it contains other elements of the quantum geometric tensor like g jk if j is equal to k and the Berry curvature right and this means that allows us in principle to access these quantities by comparing different let me call it artificial synthetic polarizations right because if for example if I take phase shift pi over 2 this is like right circularly polarized light or whatever and whatever say that minus the ratio at minus pi over 2 I obtain this this quantity here yeah so then this the difference in line widths will be exactly proportional to the quantum to the Berry curvature and if we take so to say linear polarized microwave light in this parameter space we get the off-diagonal elements okay and then it has to be normalized and so on but in this way one obtains the Berry curvature and all other quantities and if you want integrating this one has an independent measure of the churn number okay so that's that's quite nice and that's in some sense already the main story here of this part let me check 20 minutes perfect yeah so that we we have some recipes and I have in the end I have a list of all the the references or you just archive my my name for example or so you will find all the papers with lots of details about the different systems we consider and what are the parameters and the requirements okay now let me come to the so to say original title or the main word in the title higher dimensional topology I it's a bit and kind of exotic phenomenon because it's I don't know of any practical use of it but anyhow it's a very interesting physics phenomenon and you see there's a very recently there was a kind of highlight article in physics today so this is the cover page and then this is a page by Hannah Price I hope you can read it who did some major works in this so yeah it's about simulating four-dimensional physics in the lab and yeah so and it says in the title experimental methods to imitate extra spatial dimensions reveal new physical phenomena that emerge in higher dimension in a high dimensional world so there are certain phenomena which are existed to say only in a higher dimensional world and so I think this was for the first time probably studied in in string theory for example because as you know strings live in some kind of a higher dimensional world but of course it's very difficult to access all these dimensions which are crumpled into into the into the strings so here we have much better tools and so this one example is a 4d quantum Hall effect and I don't really say anything here about the real physics behind this and so on so the first this was studied purely theoretically 2000 Watt by Shu Cheng Zeng and I forgot the first one of the second also just so how to generalize this idea of the quantum Hall effect to more dimensions yeah and it was purely theoretically discussed what are the quantities you have to consider and so on and what is quantized and so on okay and and how to how to set this up properly okay then let's say first ideas to realize this in systems of ultra cold atoms we're actually by here in this prl for example and then it was realized experimentally in two back-to-back papers in in in nature one with cold atoms and one in yeah sorry I forgot some kind of photonic generation and so the idea is in some sense I told you that you can think about the quantum Hall effect as electrons living on the cylinder and then you pump around the cylinder and so this was realized basically by creating also for at least in the ultra cold atoms the two-dimensional atom system for atoms and each atom has itself so to say an internal degrees degree of freedom which is again has a topology of of this cylinder so that's the idea so that's basically for each point you have another cylinder of internal coordinates now here called y and w yeah and then you can play you and okay then the the real trick is that many systems have internal parameters but you have to you know apply a flux in this and apply an electric field to drive in this parameter and so on so it's it's rather involved but you see that it worked out and experimentally was was shown that there is some kind of quantized response seen then if all these parameters are varied simultaneously so you have to look at some kind of higher order um response function yeah not like five more minutes okay yeah that's perfect okay so let me tell you about what is this higher this kind of typically invariant okay internet is unstable but it's still there so if if we talk so we we now look at the system which has two for the general state depending on some parameters and in that case because we have a we have so to say multi multi more states the very connection if you want is a matrix right because it can be calculated between each of those states and this means that if we move our state around adiabatically we not only change the phase of the state but then visualize a spin we also kind of rotate the spin if you want okay this gives the matrix value of Berry curvature which has an extra term compared to the other one and then one can look at topological invariance for example the first turn number would be just integrating over two parameters taking the trace of this matrix Berry curvature and then there's an invariant which only exists in four dimensions this is the second turn number which is this object here okay and then one has also in this context and also non-abelian very phases so if I go around the loop in parameter space I also change the direction of my spin if you want so this is a simple picture okay and now we we try to simulate something like this or not simulate but kind of engineer this in a 4d Josephson matter and I don't even show the system here it's very complicated and I hope we can simplify it also in the future so it like two quantum dots with some spin splittings and couplings between the levels and between the dots and it's a bit complicated to search for it because one needs a four-dimensional integral which is not so fast to do so one we kind of use to construct it by say construct the Hamiltonian in terms of matrices which have this Clifford algebra and then they have this eigenvalue so they automatically have the two dimension two degenerate states if these matrices are four dimensional and then that's what we have to do okay we need five of those matrices and and okay here they are and in the end we succeeded to find some kind of second turn number which so to say in certain parameters can take some integer take it has to be an integer value and here we could calculate how it depended on the parameters in this this internal field which is however taking a different direction for the two dots here so maybe you can simplify this in the future okay this also shows a non-Abelian a berry phase which means that you just vary one parameter for example phi one by two pi and then you rotate the spin and the corresponding rotations are here so in some sense if it would work it might even have a use as a platform for so-called holonomic quantum computing which is discussed by some people but I'm not so sure if it okay it's at least it's worth to explore I think okay the last thing I wanted to say before coming to the end is one word about tensor monopoles yeah because so what is a tensor monopole it's a generalization of the magnetic monopole which is a generalization of so to say the electric monopole and I kind of don't have time to go through it but this is the field of a magnetic monopole and for a tensor monopole one has a one-order higher dimension in the Hamiltonian which depends on four parameters and not here like in this case depends on three parameters then one can define a tensor a berry curvature and calculate it and then there's again a dd invariant I forgot now the names dd stands for two names I forgot the names now invariant which one calculates and in that case one finds that there's a tensor berry curvature and we found two ways to construct it I show both because they look rather similar so this is the first one here is by Andreiov states with three dots the second one is using this method of coupling Josephson junction I think you will hear more about this later in the conference and then one can use this so this was called the bisquit right and and so two coupled bisquits and then you with three islands and this also can have a tensor monopole okay with this I would like to summarize maybe not go through it in detail I just wanted to say okay microwave spectroscopy of this Josephson matter might be interesting to get access to the quantum geometric tensor and berry curvature we have some ways to realize higher dimensional topology but not yet kind of maybe not yet in the practical proposal and it's anyhow also in the platform for other kind of interesting effects higher which higher dimensional effects which only exist in higher dimensions and before closing I would like to thank the collaborators and here's the promised list of papers which is also again some kind of summary and I want to particular highlight the first two persons Raphael Claes and Hannes Weisbrich who were two extremely brilliant PhD students or are so Raphael finished last year with now a postdoc in Würzburg and Hannes Weisbrich is finishing the in the end of this year and we also enjoyed a much collaboration with Gianluca Rastelli and Carlos Cuevas in the course of this this works and here you see the list of papers we published in the last year and year before okay thank you very much thank you very much for going for the very nice talk and thank you very much for being exactly on time now the talk is open for for questions sure relatively simple so from Amit Basu online he asks whether the well he asks how is this d-vector in the Hamiltonian calculated for some of your models I think that's the idea of the question yeah yeah it's it's it's similar to a tight binding Hamiltonian but be aware that we have superconducting contacts so which which basically means so in principle this corresponds to some self-energy and so on so what we really do is we calculate the Hamilton the greens function of the system if you want and then the greens function of the system is energy minus the minus some effective Hamiltonian plus some self-energy term or say bare Hamiltonian plus some self-energy terms and this bare Hamilton plus the self-energy terms the self-energies contain the superconducting leads the super conducting phases and in particular for example for energies much smaller than the gap they they they get they get much simpler and and energy independent for example and this kind of creates an effective effective Hamiltonian by basically inverting the greens function and subtracting the the energy if you want so but the first model is a simple binding in the second for the for the 4d josephson matter we also took an account coulomb strong coulomb interaction so we are basically restricting ourselves for example to a space there's only one electron in the two dots but it has phase dependent hopping terms so this is like crossed crossed and rare reflections or things like this which is so to say otherwise the Hilbert space would be very large but so one can take into account many body effects which we only did in the simple way to say that let's take some u2 infinity so that we have a fixed number of electrons for example in the system and so I'm gonna start with a stupid question just to be sure I understand and then maybe something a little less stupid so the first question is all the dimensions here are compact right yes okay because by definition basically so is there the possibility of an extended dimension in some sense and related to that maybe the the less trivial aspect is what's the effect of a weakly coupled environment because all these things what's robust here right right so to the first question I mean for the superconducting phase there's no way to I mean but of course you could think about making long arrays of Josephson junctions and then you sort of say have this dimension as a and you can fully transform it and have some kind of okay it's still in the end it's always a finite system yeah so I'm not so sure in some sense in these phases are always so to say compact absolutely but I think you could so to say by some tricks extended so that you effectively have a non-compact phase yeah your order exactly you need something like this like an order of magnitude difference okay the environment yeah of course of course we did the idealized calculation here so the superconductor is fully gapped bcs superconductor this is why we get the sharp lines and so on right so this is why we get if you want a delta function in the in the golden rule and so on so of course in practice the lines are broadened and but but I think one thing is that the widths of a line okay might depend on the phases the coupling to the environment and other complicated things which we did not take into account and I think one should take it into account yeah but you know something like for example one simple thing is that if you if you just add an incoherent broadening right then of course the weight of the line if you want doesn't really depend on the broadening but as I said the the broadening parameter might depend on phase itself and then you have to figure out how this yeah how this in practice looks like so one would need a certain model and then include it but I think it could be included in the approach well I mean the topological invariance you always have a gap in the system right and then of course the size of the gap matters yeah so it's kind of so in the topological phase you have a gap so we have these closings but then this is the point where the topology so to say can changes from either one to the other or from from one to trivial but otherwise we always have a gap in the system and how large this gap is and of course it's important if this gap can be overcome by some kind of fluctuations then of course we are in trouble with the quantized with the quantized value and I think I this is I mean if we have so to say a gap and we have additional noise so to say on the on the phases we haven't studied this how this would for example exact effect the quantization where the question is does it average out simply and I don't know I must say I don't know yeah but probably we should also go back to works who studied very phases and the noise and so on and and and look what they did because I think it's rather related we have time for one more quick question so I was wondering if you think we will find other invariance and if if you can just comment do you think there are some also applications of these invariance in some way I mean for example as not as a I mean the first in the first turn number can be used for quantized conductance yeah and the second turn number is a is a quantized nonlinear contact nonlinear it's not a nonlinear but it's sort of say a response to to three different param perturbations and then you look at the fourth quantity okay I must say I don't know about any practical use for it and and this other invariance we so far found in it I think it depends on even and odd dimensions and things like this so one has to dig a bit into topology to to find this maybe I can ask a question then um is there some limitation to how much you can do with loops of two terminal Josephson junctions to simulate or to realize the physics of multi-terminal Josephson junctions I mean theoretically there are no limits there are no limits theoretically right I can couple one thousand of them and they are all coherent I think if you so in practice I think you will hear more about this in other talks but I because these are charge I mean I think charge fluctuations are probably not so good for this so they they will certainly limit this somehow and you know in principle they are also higher states and so on and so on so you have also some gaps and if you want to make systems larger the gaps usually get smaller right and so there are certain limitations and on that but I this is really question to our experimental friends I mean what do you think how many you can couple together and still have so to say fully coherent coupling and does it make a difference for the way the microwaves couple to the to the device you mean now for the Josephson junctions for for micro spectroscopy of the for instance the the vial metal you think about cross talks or things like this or yeah I'm sure that of course in in the ideal way you you kind of excite one flux in one junction and the others don't see anything this might be there might be some cross talks and we haven't investigated something like like what what limits this if we make this you know if we make this with two microwaves and we want to have a definite phase difference between them then of course you probably want to avoid cross talk as much as possible it's a good point I I don't know let's thanks for long again