 We sent some of the things we are working on in Innsbruck. So we are trying to use superconducting qubits or circuits, I should say, for a couple different things. We want to couple them to micromechanical oscillators using magnetic fields. We are trying to use them to build structures we can use for amplifiers and single photo microwave switches. And the topic I want to talk about today is how we want to use them to build systems to perform analog quantum simulation. So here's a short outline of my talk. I want to first give you an introduction to circuit QED, sort of what are cavities, how do we realize qubits, how do we couple the two. And that brings me then to how we want to combine these systems to essentially build up something with which we can perform a simulation, an analog simulation of dipolar quantum magnetism on lattices in the long run. And then in the end, I want to show you some first experiments towards that direction. OK, so I guess most of you are familiar with cavity QED. And the idea in circuit QED is we essentially take the optical resonator we have and replace it with a microwave resonator. So of course, we have to then use, instead of optical photons, microwave photons. Now, that's not so different to what Sascha Roche has done for what he got the Nobel Prize for. The main difference in circuit QED is really that we don't use atoms as our two lever systems, but essentially we are using nonlinear quantum circuits. So we have a much larger design flexibility and some certain advantages. Now, over the course of the last years, this field has taken off. There's many, many groups around the world. This is, I think, not even a 50% complete list, meanwhile. So people are working on quantum information processing, quantum simulation, quantum optics experiments. They investigate quantum measurements, and so on and so forth. Now, as you can see here, a cavity QED system and also a circuit QED system essentially consists of two building blocks, the cavity and the qubit. So let's first talk about our cavities. So what we are using in Innsbruck are these waveguide microwave resonators. So what you can see here is actually the two halves of such a resonator, so you bolt them together. And essentially, this is a superconducting box. So this is aluminum, and we can send microwave signals into this box, and it comes out. So unfortunately, the microwave couplers are not shown here. And if you look at the final mode of this resonator, so it's about lambda-half in size, you get something which looks like what you would get from a drum. So you have an electric field maximum in the center. The electric field points in the C direction. It has to go to zero on the side walls. So the advantage of these resonators is they are very, very easy to build. So essentially, you can send one of your PhD students or you go yourself into the workshop, machine it out of a solid block of aluminum. You bolt it together, you cool it down, and you see that you can observe quality factors in excess of a million. So reaching up to 10 million is very, very easy and very straightforward in such a system. Now, what pretty much every microwave resonator can be described with is essentially, well, around the resonance at least, is essentially a parallel combination of an inductor and a capacitance. Now it turns out I have an associated charge and flux, and I can quantize those. I can write down a Hamiltonian by just adding up the charging energy and the energy stored in this inductance. Charter and flux are conjugate variables, so actually what I'm allowed to do is I am allowed to write down ladder operators, just as I would have done for a regular harmonic oscillator. And if I now take these expressions and put them up here, what I get out is just a regular quantum harmonic oscillator Hamiltonian, where instead of, for example, a mass and a spring constant, I have now my inductance and capacitance or alternatively, if you prefer, a resonance frequency and a so-called characteristic impedance. Now, typical resonance frequencies for superconducting circuits are somewhere in the 4 to 10 gigahertz range, and this characteristic impedance is somewhere between 1 and 100 ohms. So essentially what we have is we have this harmonic potential. We have equidistant energy levels. So in some sense, we already have a quantized system, a quantum circuit, but it's sort of boring. I mean, I can cool it down. I put it in one of our cryostats. I cool it down to 10, 20 millikelvin. It will be in the ground state, but if I now apply a classical drive, nothing so much exciting happens. All I will get out is sort of a coherent state, so a very classical quantum state. The key ingredient that's actually missing is the non-linearity, and that brings me immediately to how we do qubits, and then you'll see how we do the so-called 3D transmission. So the key ingredient is really this non-linearity, and the non-linearity in superconducting circuits we actually realize with a so-called Josephson junction. So the junction is nothing else, but a superconductor insulator, superconductor sandwich, so this insulating barrier is very thin about the nanometer, typically aluminum oxide. Now Cooper pairs sort of can tunnel completely dissipationless across this barrier, and you can actually look at the Josephson equations and then figure out what's actually the energy associated with this tunneling, and you find it's given by minus Ej, this is the so-called Josephson energy, which is essentially just a constant depending on what material of superconductor did I use, what's the barrier, what's the dimensions of the barrier, and how thick is the barrier. And then the cosine of phi, and phi is actually the phase difference between the macroscopic wave function of the Cooper pair condensate here on the top and the bottom. So this is the phase difference of those two wave functions. Now if you think about, and if you look at that a little more closely, it turns out that this is very close to actually what the energy would be if you write down an inductor, or so the current voltage relation would look very similar, so very often actually these Josephson junctions are also called non-linear inductances, to some order at least. Now if you look a little more closely, this is not only sort of, I cannot only have this tunneling event, this is also sort of a metallic plate, an insulator and another metallic plate, and maybe if I say it like that, you can see that this also looks a little bit like a tiny capacitor. So there's not only this tunneling energy associated with this element, but also we have a small charging energy, so this which is sort of given by this capacitance. As a circuit element, people actually use sort of a box with a crossing side to say this is, well this is a Josephson junction, which has this Hamiltonian. It turns out it's actually now very, very easy to make a qubit out of that. All I have to do is I add a bigger capacitance in parallel, so nothing much happened to my Hamiltonian with the exception that now this sort of capacitance has been modified, I have to add up the capacitance of the junction and this parallel capacitance to something bigger. And I've actually created a qubit, it turns out, and this is the so-called transmon qubit, which is at the moment one of the most widely used qubits we have in the community. So how can we understand what goes on here? Well, one of the easiest ways of viewing it is, let's take this cosine term here and tailor expand it. And now I've done that here in sort of units of a quasi flux and I've divided it by the flux quantum here, so what you then get out is again, a flux and a charge operator, so these first two terms you can see right there is again pretty much a harmonic oscillator Hamiltonian, nothing much. But now I have this five to the fourth correction term and so that's sort of where the magic comes in, because if I now do use the same replacement rules for charging flux like I did for my harmonic oscillator, I get out a slightly modified Hamiltonian. So again, harmonic oscillator, but now I have this additional b dagger b squared, so a non-linearity in my quantum circuit and the size of the non-linearity is actually given by this charging energy. So what has happened is that I've essentially gone from this harmonic, from this parabolic potential to a cosine potential and my energy levels instead of now being equidistant start to sort of bunch up. So meaning that sort of as I go up in this potential well that the sort of energy levels come closer and closer until sort of I can really say that sort of down here if I only shine in radiation at this energy difference I actually have my two level system. Typical resonance frequencies in this case are again somewhere in the five to 10 gigahertz range and the anharmonicity so the energy difference between sort of going from comparing the zero to one transition to the one to two transition is somewhere like 200 to 300 megahertz right around there. Okay, so that's how we do qubits. So let me for a moment stay a little longer in this sort of circuit language to think about how we can actually couple a qubit to a resonator. Drawing a circuit is actually quite easy so all I have to do is I have to add a capacitance between my resonator and my qubit. So I have my two level system, my harmonic oscillator and now sort of to understand the coupling one of the easiest essentially classical ways to think about it is if I have an excitation living in my qubit I'll have an associated electric field here in this capacitance and then I have this capacitive divider right here. So what will happen is that I'll also get an electric field over there in the resonator. So if one does that a little bit more carefully really figures out what goes on here thus a rotating wave approximation and so on what you'll actually get out is this interaction Hamiltonian. So on resonance the qubit and the resonator will exchange excitation. So whenever I annihilate an excitation to qubit I create one in a resonator and vice versa and sort of nowadays circuits can sort of typically have in the axon strength somewhere between 50 and a few hundred megahertz. And I guess that Hamiltonian most of you are familiar with this is the famous change coming simultaneously and essentially and we can use that to in the dispersive regime to read out our qubits. We can use the resonator to mediate interaction to drive the qubits and so forth. There's a little thing I want to add here is I can also actually replace this resonator with another qubit. I can write down the same coupling capacitance and there's not much that changes but it means that I can also capacitively couple qubits and sort of mediate interactions in that way. So my Hamiltonian didn't change by much instead of an a dagger and an a I now get a plus and sigma minus. So again qubits which are capacitively coupled can on resonance exchange interactions with this sort of exchange or flip flop interaction. Again interactions range can be around 50 to well a few hundred megahertz in this case. So how does such a system in our case actually look like so not that one but a qubit coupled to a cavity. Well so this is a picture you can see here essentially we have this big microwave resonator this time it's not made out of aluminum but out of copper for reasons we want well we want to apply a magnetic field essentially. So and then in here you see a piece of silicon with this you know silver structure up here on top. So essentially if I make a zoom out that's what you can see. So the juncture sits here in the middle then you have those two wings. This whole structure is about a millimeter in height. So you can easily see it with your Bayer eye. Now these two plates here actually serve two purposes. On one hand it actually creates the right capacitance such that I actually get a transmit qubit so it's just the right size and it's essentially a little dipole antenna. So because the more volume of my cavity has increased so much that the electric field strength actually went down I had to pretty much increase the dipole moment of my artificial atom. And that's what we have done. So essentially if you think about electric field in the cavity it would point sort of in this direction it has a significant overlap with sort of electric field that would go from one plate of the qubit to another. So these couples quite strongly. If you sort of you know want to make a very hand waving argument you could say oh it's essentially sort of a cooper pair oscillating across this whole millimeter and such which means I could say okay this guy has a dipole moment of something like 10 to the 70 pi. So like five orders, six orders in magnitude stronger than any atom in molecule we have at hand. Now what is quite nice though is that sort of with these qubits sort of the best or qubits in these cavities the best coherence times we have observed in such a system is right around 100 microseconds. So T1 and T2. I would say more regularly in experiments nowadays sort of pretty much all the groups see a few tens of microseconds coherence times right around there. Okay so this is the essentially fundamental system these are the fundamental building blocks we want to use then to try and build an all in all quantum simulator. You can find more details in this physical review B where we sort of joined forces with Peter Solow and Margello sitting over there to figure out what we can and want to do with it. And there's sort of a couple more papers by Florian Marquard's group working on spin chains. Okay so what's the basic idea? I guess I don't have to repeat sort of the idea of quantum simulation again. So essentially we want to sort of build a simulator of a quantum system which is very hard or essentially impossible to simulate on a classical computer. Now this depends on very much what I want to know about that system. Is it a frustrated system? Do I want to know dynamics or just ground state? And our idea is really sort of to try and map some of those Hamiltonians onto our qubit cavity system because we have it much better under control and we can change parameters. Okay so sort of the sketch is the following. Let's take a cavity and let's put many, many qubits in there. We can arrange them on lattices. They have this large dipole moment so they can interact directly with another. And then we can sort of start up trying to maybe start with spin chain physics or maybe sort of have two chains in parallel sort of create this ladder system work our way up to full 2D lattices. And as I'll show you in a little bit, actually it turns out we can sort of create a quite unique system which is we sort of have these interacting few body systems so I don't know five, six, ten maybe which then talk via an open quantum system a waveguide actually with another such system. And I think we have a very unique thing here which I guess can create very interesting systems. So to really see if we can sort of realize these kind of systems we first sort of did some finite element modeling. So this is actually quite convenient for all of our circuits. We could, we can put them in a finite element solver and really figure out in the axis range resonance frequencies and pretty much all parameters for the experiment. So what you can see here is sort of a model of cavity with a piece of sapphire in here and then there's two qubits sitting on that piece of sapphire and you can sort of see the mesh and on that mesh this program actually solves pretty much Maxwell's equations. You'll, you know, get resonance frequencies out. You get electric fields out. We can vary parameters and sort of change the frequency of one qubit and sort of sees essentially avoided crossings pretty much completely classical of course and from that we can figure out what's the interaction strength of two qubits with another of the qubits with the cavity and so on. Now if we do that for two of those qubits inside a cavity then we find the following thing. So we start out with sort of two qubits being aligned like that and we push them apart and what you can see is we can achieve interaction strength of at about a millimeter distance of a few hundred megahertz but as you pull them apart actually since the actual strength goes down we have done this for three different antenna length so three different sizes of the dipole moment essentially and you can see that sort of all of those curves seem to meet in one point and then they sort of acquire this tail. Now this point, yes both. So I'll explain it, it just was about to explain it exactly. So the tail, let's start like that then the tail actually comes from the cavity mediated interaction that's essentially very long range because it's given by the cavity mode. That left side here is actually dominated by the direct capacitive interaction and at the zero right there actually the two cancel because they have opposite signs. So I can have actually two qubits sitting in this cavity right around like three and a half four millimeters in this particular case and they would not interact with each other at all because sort of those two interaction strength completely cancel. So we can also sort of then take the qubits and sort of rotate them from this parallel interaction sort of around each other until they are aligned and we see that the interaction strength goes from minus a value to zero to two times that but positive. Now maybe to emphasize the open circles are actually the final element simulations and I haven't really told you what those lines are. The lines are actually just a very essentially pretty much one parameter fit to using this model and this is essentially just a one over R cubed and cosine of the respective angles so pretty much just a dipole-dipole interaction and to sort of go from the blue to the red to the black curve, all we have to do is actually rescale by the length of the qubit squared and that's pretty much it. Of course we have to add this cavity mediated term which sort of gives us these tails but we can get those parameters out sort of independently and we don't have to fit them. Okay, so these qubits really behave like dipoles, we can change the interaction strength just by rotating them or placing them at different distances so the idea is really we think we can build up this system like that, tailor the interactions to the Hamiltonian we like, we can then also rotate them in the cavity such that we can only read out the qubits we are interested in and really try and measure correlations. Now trying to do that on a system like I've shown up here which is something like I don't know 25 qubits or so is maybe not that realistic so if you really think about we could build such a system but then we actually couldn't learn what goes on there because it would be way too complicated and we don't have enough sort of knobs to really turn so maybe something like this central eight I've put here in this dashed box, that would be feasible. So how can we scale this up to something a little larger and that's actually where, oh yeah so we can maybe do some small instances of like spin chain physics in such a case or these sort of X, Y model on a ladder. Now actually if you wanna scale up the system the idea is to combine this with the waveguide because now we don't have this length restriction of the cavity we can just make this waveguide pretty much as long as we want we can have many, many, many qubits in there but now we don't have the cavity for the read out so what we actually do here is we introduce these black hue shaped lines and those are little lambda half resonators which can sort of interact with the traveling wave going through this waveguide which actually allows us to have a much more fine grained read out exactly at the locations where we want it. Again, I don't wanna do state tomography or anything on this whole system but I wanna measure enough correlations, excitations, maybe global magnetization or something like that to extract what phase I'm in and the physics what goes on. So yeah, sort of this waveguide is actually very convenient and it actually gives us a few more things we can investigate. So we have these sort of short range diaper like interactions but we have something where these qubits can also talk to the waveguide and have sort of a long range photonic mediated interaction. There's band engineering possible, I mean these waveguides, this waveguide naturally has a low frequency cutoff but we can do something more complicated like build in bandsaw filters and so on. Now especially we have an inbuilt per cell protection in this case because I just park my qubits below the cutoff and even though they are very strongly coupled to the resonator, they won't decay so they're protected in this case and one can even think about dissipative state engineering. So and that really should allow us to build systems of that kind where I have sort of an interacting few body system talking to another one much further away. So here you can see a sort of a first incarnation of such a waveguide. Microwave would be sort of coupled into here, again this is one half and in there there's a little qubit we can sort of talk to and the wave would just propagate through and come out the other side. Now for these systems there's a couple open questions. So how do we best characterize them? What exactly is it we wanna measure? How do we in the end verify and validate our measurements? You know, is it really meaningful what we get out? And I think one of the big questions is also how would I do that in this open system case? So what's the observers I wanna see there? Now for that reason we actually paired up with Marcello and Peter to at least for one of those models try and figure out what we can do and what we wanna do. And so this is this X, Y model on the ladder so essentially two rows of spins. I have one interaction strength J2 within a chain and between the two chains I have this J1 interaction strength so we can build up the system like that. So the Hamiltonian is essentially just this exchange Hamiltonian now instead of sigmas here we use S and on top of that though we have this term here which is essentially coming from the fact that all of our qubits are man-made so we have a slight disorder in the resonance frequency so all of them have slightly different resonance frequencies so we have to take that into account and really see if this disorder is a problem. Now for this X, Y model on the ladder you for a certain ratio of J2 over J1 you actually get this so-called dimer phase where you have triplet states sort of on all of those pairs across this ladder. So actually Marcello has run numerical simulations for us and sort of to for example see if we can find or get the fingerprint for this dimer phase and one way to do that is look at these so-called bond correlations or bond order parameter so essentially I wanna measure correlations of the form sort of as Z as Z as X as X and that would sort of give me a very definite fingerprint of being in this dimer phase so if I sort of reach this negative value of one quarter so spin half times spin half gives you a quarter exactly right around there for J2 over J1 right around one half that's exactly when we are completely in this dimer phase and that should be a very easy to sort of see fingerprint in our case so it can sort of of course you don't necessarily wanna measure all the correlations across the whole ladder so it turns out only measuring a couple of those correlations should actually be enough to give you a very nice and solid indication you are in this phase. We've also looked into what happens if we introduce this order to the system and so we can sort of see that this characteristic fingerprint vanishes but we think in experiments we can realize something which is sort of this dark blue curve or maybe even slightly better so essentially all the physics with the disorder we have should stay alive and we should really be able to investigate that. Now yeah and sort of because I really wanna talk about some of the experiments I think I have like a little more than 10 minutes or so. Yeah okay so we've also looked at to what happens if you add dissipation how would you use adiabatic state preparations to prepare this dimer phase and so on and all of that looks pretty, pretty good. Okay so let me talk a little bit about the experimental progress so really the very, very first steps towards that direction. So we have single qubits in our labs we have full single qubit control so we can drive Rabi oscillations, do a single shot measurement, we have T1 times for single junction qubits or not frequency tunable ones of right around 40 microseconds, T1 and a few 10 microseconds T2 we can make the frequency tunable if we want. We have looked into whether two of those qubits inside a cavity really interact in a way we think they should do whether those simulations we have done actually make sense and for example we can sort of see this avoided crossing between two qubits so here sort of this line here is actually so this is doing spectroscopy on those qubits so this line here this is a fixed frequency qubit and I actually take another one which where we can change the frequency by applying magnetic field and I can tune it into resonance so at first please ignore this central line here you can sort of see this nice avoided crossing and another thing is if you look carefully it's not perfectly visible actually this lower line here seems to vanish and this vanishing is actually an indication that right around at where they are on resonance actually it's not qubit left and qubit right anymore but it's really symmetric and anti-symmetric state of those two qubits that talk to the cavity and because of symmetry reasons the anti-symmetric state of the qubit actually does not talk to the cavity and it vanishes in the spectroscopy and actually if you look more carefully so it's not that easily visible actually if you look at then the line with here that guy actually doubles because it's coupled stronger this central line here is actually a two photon process actually where you go from the ground state to the double excited state of the qubits so that's why that guy actually shows up here but the splitting here this 2J is pretty much exactly what we would expect out of our simulations okay so we have these qubits in the resonators they seem to obey the simulations we have done we can actually turns out quite easily build up a whole chain of them we can sort of just fill up all of those slots in the cavity one of the sort of important ingredients that was still missing is that while during the simulation I want to have all of those qubits as perfectly as possible in the same resonance frequency but then after I've done my simulations I want to be able to measure these correlators so and that means though that these qubits can't all be on resonance because then they will interact strongly so what I have to do is I want to take those qubits and detune them fast out of the resonance and then try and measure correlations on a pair of those on the other way around maybe in the beginning I'm interested in trying to bring in excitations in the system to prepare a certain state maybe also to look at dynamics or something like that so essentially what we need is fast flux tunability now it turns out doing getting magnetic fields or time varying magnetic fields in a metallic box is not that trivial even if say the box is just regular copper because of the any currents you'll see that this acts as a low pass and you can maybe get time varying fields of a few hertz in there but not like hundreds of megahertz if it's made out of a superconduct it gets even worse because then you can't even get the magnetic field through so what do we do? Well there's ideas you can essentially just sort of get a wire in there that's not ideal because then your qubit has a way of coupling to that wire and essentially the coherence time would go down so our way of doing that is actually was inspired by collaborators in Barcelona where they developed the so-called magnetic hose so that essentially allows us to bypass the superconductor such that it doesn't see the magnetic flux and get the magnetic field inside the cavity so the basic idea coming out of transformation optics pretty much applied to static magnetic fields is I want to create a material which has a mu R in parallel to the propagation direction which is infinite but I want to have a mu R perpendicular which is zero and if you think about that this would really sort of whatever magnetic field I have here in the input route perfectly to the output well it turns out of course well we have perfect diamagnets we can use our superconductors so these would pretty much fulfill that of course we don't have materials which have mu R infinite but we can make something which has a very large mu R so and then it turns out using real materials it's actually better not to just have one central rod and just one shell of superconductor but sort of layer it up and there's some more details that I don't want to go into essentially what we use is we use stainless steel wrapped with the superconductor around so in the experiment this looks somewhat like this so this is again one half of our cavities there's two qubits this time on sapphire in there and back here you can actually see this magnetic hose it's well this is sort of still how should I call it prototype like 1.0 or something 1.1 maybe so it's pretty big still so whether we think we can even make this smaller in this case for these qubits we have also measured the coherence times of a few microseconds slightly smaller but that's actually not the fault of this hose but we have actually over coupled the cavity so they are percent limited by the cavity so what we can do now to see if this hose really works is we can apply the following sequence so we can try and read out our qubit but before that we apply a pipe hose to see if we can excite it and we have a flux pulse before where we sort of change the frequency of the qubit from one rally to another one then do a big step and then bring it back and we sort of sweep our pipe hose through and we do that for varying frequency to figure out what happens to the qubit and what we can see is the following so we start out right around 6.3 gigahertz then do a jump to higher frequencies a bigger jump to lower frequencies and then come back and this is inside the superconducting box so you can see that the right time here is something like a few hundred nanoseconds or so so it's actually quite fast and it seems to jump quite nicely there is no hysteresis it looks like it turns out we can actually measure this transfer fraction and do sort of a deconvolution essentially what I can do is I can overshoot in this initial pulse and then come back down and sort of then go from this frequency to another one even faster so if I maybe sort of drag out what happens to the resonance frequency you can see you have this slow rise but if we do this sort of overshoot we can actually see we do a very fast step and sort of from here to there this is sort of 50 nanoseconds and so this means this is even faster than that and here we haven't done this deconvolution perfectly so you can see we actually have this little overshoot and then it comes back down so this was sort of just a quick try from my PhD student to try and realize that we have to do that properly but we know how to do that so this means with this hose we can really tune these 3D transmons inside this big microwave resonators yeah I'm pretty much done and really have all the building blocks at hand maybe just one thing we've also been working on implementing these waveguides so we actually have realized waveguides we have this lambda-half microwave resonators they show very nice queues for these sort of 2D structures right around a million and the advantage here is those are super simple to fabricate and normally if you want to see like a queue of a million for a CPW you have to apply very very fancy micro fabrication techniques but it's sort of again the sort of specific architecture why they are so good and sort of this is our waveguide where we can sort of have many many slots with this loading system where we can sort of put in qubits resonators and sort of build up a very very complicated system to pretty much our liking and so we can really sort of have the resonators talk very efficiently to the propagating waves use them to measure it and the qubits sort of sit wherever we want below the cut-off if they shouldn't talk to the waveguide above the cut-off if you wanted to have them talk to the propagating modes so we have the full flexibility there okay and with that I'm actually at the end so I hope I could give you a sort of brief idea on how we think we can sort of build up these sort of interacting well few many body systems however you want to call it and that we sort of made the first steps in this direction that we have all the fundamental building blocks essentially available in the lab and we hope we can sort of make much larger structures within the well next couple months or so and let me really finish with a picture of the group and the people who actually do the work here and yeah thank you very much for your attention