 I can exactly model it with this LC resonator. So this works for all of those systems. So essentially, what it means is we now have our quantum system in some sense. So we have this harmonic oscillator. We have discrete energy levels in it. But it's sort of a little boring. We've already talked about that a little bit. I can bring it in a ground state quite easily. I can cool it down. But then if I actually want to prepare a non-classical state, say one of those Fox states here or something similar, it's actually quite difficult. So if I just apply, say, a classical drive, all I will create in this oscillator is pretty much just a coherence state, so a very classical quantum state. So this is sort of boring. So what I'm trying to do now in the following is how can we use what I've shown you so far and actually build a qubit out of it? Maybe before I do that, let me just say one thing, that even though, at first, maybe they are boring quantum systems, they are very, very useful, it turns out. And this is just a very small selection of papers where superconducting resonators are used for. You can read out superconducting qubits. We'll talk about that in a little bit. You can use them to mediate into actions as a quantum memory. You can use them for quantum simulation. You can use them for quantum optics. There's a whole bunch of applications where these resonators are really useful and very, very nice. OK, so how can we actually build a qubit? And when I say qubit, I mean, of course, a superconducting qubit. And more specifically, I'll only talk about one possible kind, which is the so-called transmon qubit. There's a whole zoo of qubits out there. I don't even know how many different ones, I guess, at least seven or eight. All of them are slightly different. I'll focus on the most simple one and also most widely used one. OK, so quick reminder, we have talked about Josephson junction already. And here is, again, a comparison to a regular inductor. So here's the voltage across a normal inductor is proportional to the derivative of the current times the inductance for Josephson junction. I have a very similar equation, but this time, I have this nonlinear inductance upfront here. The energy for inductance is here, the phi squared, whereas for the Josephson junction, it's this cosine of the phase phi. Now, we've already discussed that there is a relation between the phase of the superconductor and the flux that I have. So it turns out I can write this phase difference in units of flux quanta. So essentially what I'm doing is I'm saying, OK, I treat this Josephson junction as a nonlinear inductance and I assign a quasi flux to it, but I have to write it in units of phi 0 that it actually works out again. So what I will use as a circuit element to draw a junction is actually this, what you can see here. So it's a box and the cross inside, and that signifies the junction. So the Hamiltonian for that is actually quite simple. I mean, one part we have already up here. So this is really the term that comes from my Josephson equation, so from the tunneling of the Cooper pairs. But if you now remember back, this junction is essentially a metallic plate, an insulator, and another metallic plate. So this should remind you of a capacitor. So in fact, this is exactly what it is. So there's not only this energy coming from the tunneling of the Cooper pairs, but there's an additional just capacitive contribution, a so-called charging energy. So if you want to describe this Josephson junction completely, we have to take that in account. So now it actually turns out it's very, very easy to make a cubit out of that. All I have to do is actually add another capacitance in parallel. So I take my junction, and I add a big-ish capacitance in parallel, and I've created my cubit. And I'll show you in a second why this thing is a cubit. So this is the so-called transmon cubit, and what is nice is this is a so-called charge insensitive cubit, meaning even if there's sort of fluctuating charges around, because of unwanted defects and surface defects and so on, this cubit doesn't care about it. It will still be a very, very good cubit. So the total Hamiltonian didn't really change much from the simple junction. The only slight modification is instead of a simple C here, I have a C sum now, which sort of takes into account that I have this additional capacitor. And now what I can do is actually something very simple. I can actually start to tailor-expand this cosine term here. And what I then get is this first, so I've neglected the constant term. We are not interested in that. That would just give us an energy offset. But then we get this phi squared term, and we get a phi to the fourth term. Now, the phi squared term combined with the q squared term should remind you, again, of the harmonic oscillators we have just discussed. So first order, no surprise, we will get a harmonic oscillator Hamiltonian out, because we said this is an inductor. But we have a correction coming from the nonlinear part. And this correction is exactly this phi to the fourth term you can see here. So what we can now do, it turns out, is we just do the same thing we've done for a harmonic oscillator. We again introduce raising and lowering operators for charge and flux. And this is an exercise you can do yourself. Take the raising and lowering operators I've shown you, put them in there. And what you'll find is you'll get this Hamiltonian down here. So where, well, this comes up in a second where this omega q now is a combination of parameters. And also this E c is a combination of parameters. Well, instead of a's, I've used b's here to really distinguish the qubit from the harmonic oscillator. So essentially what we have done is we have now by using a junction instead of a regular inductance we have gone from this harmonic potential to this cosine potential. And now what happens is that my energy levels are not equidistant anymore, but as I go up in my potential well, they start to bunch up. So this means that now I have down here a two level system with a given energy difference, which is actually larger than the energy difference from this level to that one. So this really means I have a two level system I can address individually just by using the right energy photons, so the right frequency of my microwave drive to really introduce transitions between the ground state and the first excited state. So in essence, I can now take the semitonian and even simplify it further, neglect all those high-lying states, and just write this as a two level system. Well, because if you look here, I mean, if I drive, then it will connect. If I drive at this difference here, it will connect this level to this one, this level to that one, that level to this one, and so forth. I'll just walk up the letter, and I create a coherent state. There's nothing, as soon as I bring a part of the population from a grounded and very excited state, I can bring a part of that into the second excited state, a part of that in the third excited state, and so forth. So I just climb up. Now this here is not possible anymore, because I bring excitation here, but then the next photon, I mean, there is no energy level up here. So I can't go anywhere, so it just stops. Now this is, of course, a little simplifying. If I drive it hard enough, I can still do it. I can't drive it very, very hard, but still everything works. So this is what I wanted to mention here. There's this omega q, sort of the resonance frequency here. It's given by square root 8 Ej Ec, where Ej is the Josephson energy, and Ec is just the charging energy, so q squared over 2c of the capacitance. So now here, typical resonance frequencies for such a qubit are somewhere in the 5 to 10 gigahertz range, so convenient microwave frequency. And this anharmonicity, so really the energy difference of this 0 to 1 transition compared to the 1 to 2 transition, is about 300 megahertz, so a few percent only. But this is more than sufficient to really say, I have a very nice two-level system down there. Okay, so, oh, the gammas, state of the art, best gammas, a few kilohertz. So sort of that corresponds to coherence times of a couple of 10 microseconds, 100 microseconds, up to for some T1s of a millisecond. So here you can see actual pictures of a variety of transmon qubits. So up on the top left, these are the sort of initial transmons that have been built, you sort of see those large fingers here, that's your capacitor. Then in the middle there, right in this area, you sort of see this loop on another zoom in, you see here one metallic film and another sort of this finger lying up on top and there's an insulating barrier in between, so that's your junction. And actually in this case, there's two junctions within a ring, so this will be a frequency tunable qubit because I can apply a magnetic field and change the effect of Josephson energy like we've said in the morning. Another variant which has been very successful is what people call a 3D transmon. So, well, this guy here will be coupled to a coplanar waveguide resonator, so a two-dimensional flat structure. If I want to couple something to those box resonators, to those waveguide resonators, I need something a little different and this is then this so-called 3D transmon. You see the form has changed a little bit. You now have this big antenna, this big capacitor plates, so these plates do two things. On one hand, they form the capacitance that I need to actually make this transmon qubit and they also, if you look at that, it could remind you of a dipole antenna a little bit. So, this is essentially an antenna coupling the microwave radiation in the cavity to the qubit. I'll talk about coupling and how to do that in a little bit. And then here in the center, that's where a single Josephson junction sits. And actually what is quite surprising with these guys, here you see that thing is quite small. This is like 50 by 100 to 200 microns. That guy is actually really big. It's about a millimeter in size, so you can really see those guys with your bare eyes and it actually turns out that these qubits are so far the most coherent transmon. So, those guys have about 100 microseconds T1 and T2. Another variant very similar is the so-called X-mon by the Martinez groups. So, you've already seen that sort of names are always a little flashy somehow. So, the qubit actually sits here. So, you see this cross here and it has sort of a capacitance to this ground plane and then down here that's where the junction sits. Again, this is a squid, so two junctions in a ring, but nevertheless it's the same circuit. It's a Josephson junction in parallel to a capacitance. So, all of those are sort of described by the same system, although the actual realization is slightly different. Now, sort of coming back to sort of stating some numbers. So, actually we have something like a worse kind of law. I'm not sure it will continue forever for qubit coherence times of charged qubits. So, this is another classification. The transmon is a charged qubit. So, initially sort of the first superconducting qubit that has been shown like 1999 by Nakamura had coherence times of well a few nanoseconds. And then sort of there have been several different developments, several improvements. Actually, this was a slightly different qubit than changing to the transmon qubit and so forth. And so, this is a graph again out of the science paper. Meanwhile, you, one has to add a couple points here such that transmon qubits now regularly or can reach up to 100 microseconds T1 and T2. So, and there's another kind of qubit which is somewhat related to the transmon. Unfortunately, they don't have the time to talk about it. And there people can now regularly reach T1 times of a couple milliseconds. T2 is not as good yet, but there are ideas on how to get it much better. So, here we'll then really entering a couple milliseconds coherence times. And these numbers you really have to compare to our operation times we have. So, typically to do a pipers or something like that requires 10, 20, 30 nanoseconds. Two qubit operation time is somewhere maybe closer to 100 nanoseconds, 200 nanoseconds. So, you see that with that we have like about three orders of magnitude in between our gate operation time and the actual coherence time. So, we are really sort of going into a regime where we can do many, many operations before an error occurs. And even sort of starting to enter maybe something which in the future can be used for false tolerant quantum computation. So, let me just flash this slide. I don't wanna talk more about it. Just to sort of tell you there is more. There are many, many more qubits. So, I've only talked about combining two of those elements. I could add another inductance. I could have many more Josephson junctions and so forth. And then you know, you can sort of have a whole classification of this qubit zoo. And I think one needs to add at least two points here. I know of other qubits that would go somewhere. And here is sort of a classification in terms of what's your charging energy compared to your Josephson inductance and what's your Josephson inductance compared to your regular inductance. And sort of every qubit somehow lands somewhere on this map. All of those qubits have some advantages, some disadvantages, some are more complicated to build but have like better tuning parameters, are more flexible, have different coupling schemes. It really depends on what you wanna do. It seems like at the moment I would argue the community has settled on pretty much two qubits which is the transmon, which is for example also used by Google and IBM because it's very simple to produce and shows these very good coherence properties. And there's a number of people also using Fluxonium which is much harder to fabricate but has very interesting properties and also quite good coherence types. Okay, so questions? There again, the operating temperature. So typically the fridge temperature, pretty much everybody uses, I mean those fridges go down to like 20 millikelvin or so and essentially you have to be below 50 millikelvin to ensure that your qubit is in a ground state at a couple gigahertz. Okay, so now we have resonators, we have qubits. How can we combine these two things and what do we gain from it? So this really brings me to circuit QD and also back to sort of these original ideas which have been sort of awarded in the Nobel Prize 2012. So this is not the violence part but really sort of the roaches part where they essentially got the Nobel Prize for experimental methods that enable measuring and manipulation of individual quantum systems. And this is really cavity QED where in Sir Sir Roach's case you have this microwave very, very high-Q resonator and you send little atoms through that interact with the microwave field inside that resonator and you can sort of use these atoms to manipulate and probe this microwave field inside here. And we'll sort of take these ideas and now transform them over to our superconducting quantum circuits. So essentially what we do, so this is this picture, resonator, atom inside, the atom passes through or is trapped inside, it interacts with the cavity, the two-level system KDK, the cavity KDK. What we do is we sort of replace optical photons with microwave photons which means of course we have to replace optical resonators with microwave resonators so that's what we have at hand and instead of atoms, as two-level systems we use our non-linear quantum circuits. So we use our qubits to otherwise build very, very similar systems. Well, and as I've mentioned already this has been used for many, many different applications. So now if I stay in this circuit language I've shown you so far, it's actually really easy to couple a qubit to a resonator. All I have to do is add a little coupling capacitor up here and I've copied them, why? Well, you can imagine what if I have, so something is going wrong with animation here a little bit, okay, what if I have an electric field over here in this capacitor? Now I have this capacitor, so which is for example a part of a qubit circuit. So I've simplified it here and just made it a resonator which is part of one part of the circuit. But now because of this capacitive divider whenever I have an electric field here it means I'll also get an electric field over there. Of course there will be some factors, some sort of capacitive divider that will sort of attenuate this field and this will tell me what the actual coupling is. Oops. So essentially what you have is that that an electric field in a qubit, let's say this would put a qubit circuit, creates a field in a resonator and vice versa. So this of course works in both directions. Now if you walk through all of this properly and do it correctly, really think about everything here what you'll see is that the coupling actually between those two circuits is proportional to this constant beta times B plus B dagger times A plus A dagger. So here this is sort of the excitation or the displacement electric field in this circuit and this is sort of the electric field in this circuit over here. So this is the coupling term between those two circuits. So what we now really have is, okay and now I've actually replaced this with a qubit but nothing much changes. So I really have my two level system, my harmonic oscillator and now I also have this interaction term because it's a qubit I replace my B and B daggers with a sigma, this should be a sigma minus with a sigma plus and a sigma minus. So I can just excited or de-excited by the console to walk up the letter. And so you might have already seen a Hamiltonian like that. An interaction Hamiltonian. I guess in Christopher Wunderlich's iron trap talk he might have talked exactly about a very, very similar Hamiltonian because if I now do multiply these terms out do a rotating wave approximation sort of only keep the terms which if you wanna say so conserve energy. So I have to annihilate an excitation in the qubit to create an excitation in the resonator and vice versa. And this is then exactly my interaction Hamiltonian and this is sort of this famous change Cummings Hamiltonian that is also used in trapped iron systems or in cavity queuing. So essentially what it means if I couple a qubit to a resonator capacitively if my two systems are on resonance they will exchange interactions with each other. So what this really does in terms of physics I'll discuss in a little bit. Let's first have a look at how do we actually do that? How do we achieve this capacitive coupling? It turns out it's not too hard. So here's this transmon qubit I've shown you already. So big capacitor junction here in the middle and now you see there's an additional structure up here. This is this coplanar waveguide resonator so this center conductor. And all I have to do is I have to park my qubit close to it because now this part here to that one will form my coupling capacitance and I've actually coupled the two devices. What is nice is that here we have a very small mode volume. I mean, compared to my wavelength this is only a few tens of micrometers. The sort of extension out of the plane is maybe like a micrometer of the electric field. So a very small mode volume means you have very strong electric fields, a very large coupling. So it's really easy in such a system to achieve coupling strength of 50 to a few hundred megahertz. Again, if you compare that to the coherence times we have or the line width which are a few kilohertz or so then this is orders of magnitude larger. Something like, I don't know, 10 to the five or so. So this is again this transmon qubit and now you can actually see that this here was the transmon from Google and up here, this part here, that guy is actually the coupling capacitor which couples me to this coplanar resonator here. So now it's really sort of this thing down there that creates the capacitance. In this 3D case where you have sort of this sort of one half of this waveguide resonator it's a little bit more subtle to see because this forms the capacitance for the transmon but it also if you think about how the electric field lines would go it would go from one capacitor plate to the other one and this would couple to the electric field in the cavity which I've shown here which sort of points in this direction. So a slightly easier way to think about this is really think about it as a dipole. So this guy really has a dipole moment and the size of this dipole moment of this qubit is something like a cooper pair sort of oscillating across this whole millimeter. So this is something like two E times a millimeter and if you calculate what this means this means it has a dipole moment of something like 10 to the 70 by. So if you compare that to atoms, molecules or something like that, this is something like five or six orders of magnitude larger than anything you could do with a regular atom or molecule. So essentially I mean these 3D resonators we have seen have much higher quality factors so that's nice but of course they have a large mode volume so one would naively expect that the coupling has to go down but we can compensate that by just making this thing much bigger and we can again achieve coupling strength of something like 50 to 200 millihertz. So all of those systems are very, very similar with just the typical advantage that these guys here have sort of the best coherence times at the moment. Okay, now, yes, that's sort of in units of the, for more magneton sort of for the dipole magnetic dipole or dipole moments essentially. Now sort of what does the cavity bring us? Well it brings us a couple of things. One very important aspect of the cavity is it actually protects the qubit from sort of the bad environment you have around. So I've already said I mean these qubits coupled very strongly to electromagnetic fields so essentially also coupled very strongly to vacuum fluctuations. So if I would use one of those qubits, especially those 3D ones out in free space and calculate what would be the lifetime just because of the coupling to vacuum fluctuations I would see that it's something like a picosecond or much less. Now what the cavity really does is it essentially reduces the density of states. So there is, if the qubit wants to decay and it's not resonant with the cavity it doesn't have any available modes it can decay into. So it can't get rid of its energy so it will sort of stay in its excited state. So and this is what's called the Purcell Protection. So really sort of I have this resonator and sort of if my qubit is very very far detuned the only possible electromagnetic fields in the resonators are really close to its resonance frequency. But now if my qubit's very far away then it can't transfer any of its excitation into the resonator because it's so far detuned and it will sort of stay alive. And this so-called Purcell Effect is then given by how strongly do I couple to the resonator divided by the detuning this all squared times and this is the decay rate of the resonator and this will tell me what the limit is for my qubit decoherence time. So essentially I can just detune my qubit far and the resonator protects it from decaying. Now here one would say okay so I want an as high as possible queue for my cavity but maybe that somehow then limits how fast I can do operations and so on. Well it turns out it doesn't because of resonantly the cavity rings up much faster. It actually rings up if I try and drive my qubit through the cavity. Oh I only show that in the next slide really. Then all my microwave tones and everything rings up as one of a delta so this is really not a problem. So in high queue cavities I can still do fast operations with the qubits inside them. This is not a problem. So how can we do single qubit gates in such a system? So essentially we have our qubit which is coupled to the cavity and again this cavity is coupled via its port to the outside world. So some coax cable that comes down into the fridge. So we can now send microwave pulses where we control amplitude and phase through this coax cable. So the drive is really resonant with the qubit but off resonant to the cavity so it will only effectively excite the qubit. We can actually realize single qubit operations. So what we really wanna do is we wanna shine in a microwave tone resonantly from the zero to one transition. So if you now sort of look at this at the block sphere sort of initially my qubit starts out pointing spin down then I shine in this microwave for a certain amount of time and what will happen is that my block vector sort of starts to slowly rotate up and what I've done now here is a pi over two points. Now how far I rotate here is actually given by so this is this angle theta is given by the amplitude sort of by the whole pulse area. So something like amplitude times time of the pulse whereas whether I rotate around the x-axis or y-axis is given by the phase of the pulse. So if I can control amplitude and phase of my microwave pulses I can fully control qubit rotations on that block sphere and it's actually quite easy. I just have to send them in through my cavity and the qubit will sort of react to this drive. So yes, in fact you could it's nothing preventing you from doing that. It's I anyways, but I wanna make sure that whichever way this microwave radiation comes in it's sufficiently weakly coupled to the qubit that the qubit doesn't have sort of a path for its excitation to go out, okay? So because if you just attach directly a cable to the qubit and say, oh that's how I bring in my microwave radiation that's also a way for the excitation of qubit to leak out, nothing is preventing it from it. So what I do though is with this cavity I build in a filter. So my qubit can't decay out because there are no available modes, but in my input field what this means is it will be heavily attenuated because of this filtering but that's easy to overcome. I just add another amplifier to my microwave generator and just pump up the number of photons that go in the essentially electric field of my microwave drive to compensate for that. But if I do this right, the qubit can't leak out while I can still drive it. Now for your additional, so I don't necessarily have to do that through the cavity. I could do it through another port, but in this sense I'm just being efficient because I can use the same cable to drive the cavity, to drive the qubit because we'll see in a minute I wanna use the cavity and drive on the cavity to read out the state of the qubit. So it's just a means of being sort of efficient. I don't wanna have like too many cables going down in the fridge. So here I can get rid of a few. Maybe one quick thing is, so how fast can I actually go with these single qubit operations? Well sort of what's the limit? Is there a limit? So if you think about it, if I start to drive the qubit here, I'll also get the similar drive there but we have said okay it's not resonant but sort of what I've neglected there is, so this is sort of the same picture. I have one sort of transition of the qubit. Here this is F01, it has some line width because the qubit has a finite lifetime and somewhere the tuned by alpha I'll have this transition from one to two. So here this is frequency space. Now if I start driving harder and harder I make shorter and shorter pulses and I look at the Fourier transform of my pulse it will actually turn out if I drive this guy on resonance I will get something here on the wings just because I make a very short pulse, a very intense pulse. So what it means is even if I intend to excited on this zero one transition I will maybe do a small excitation also on the F12. So that essentially gives me a limit on how fast I can do operations. So this means for the anharmonicity it means that my operation time has to be actually much larger than one over alpha where alpha is this anharmonicity. Another limit of course comes from my coherence time. I can only do as many operations as I have time for my qubit staying alive. So the coherence time is given by one over gamma. So that means ideally my operation time is actually much, much smaller. So this gives me sort of and this tells me okay go slow. This guy tells me go fast. So I have some sort of to find some middle ground and so for example, what this means is if I sort of take the ratio of the two anharmonicities are for example something like 250 megahertz gamma is something like 10 kilohertz. If I stay within those bounds it tells me I can do like a few 10,000 operations something like that. Still an error occurs for sure. Now it turns out I can play some tricks. So here this Fourier transform picture I can actually do some optimal control pulses and so on that I actually cannot, don't need a smaller here but actually can make an almost equal. So I can do a little better. For example, this is outlined in this paper. So I can really approach the limit where I can do with current coherence times I can do a couple of thousand single qubit pulses and they still work. Okay, so now let's come back a little bit to the physics. So we have seen we have this two level system coupled to a resonator and we have this change Cummings interaction. So I don't exactly know how much you have already seen that but here at least otherwise it's just a reminder. So I have my qubit ground state and excited state and then for I have sort of the number of photons in my resonator. So I have zero G, one G, two G or if I mean the excited state of the qubit I can also add a photon there. In this case what I've actually done is I've made the cavity and the qubit be the same resonant frequency. Now if, so this would only be sort of this part of the Hamiltonian. If I now add this change Cummings interaction what will happen is that these two degenerate energy levels will actually start to split up and what I guess there is this famous dressed states. So what will happen is because of the interaction I will not have a qubit state and a cavity state but actually this plus and this minus will be an equal superposition of the qubit being excited and the cavity being excited just with a different phase. And this is this so-called this famous dressed states between light and matter. Now actually the splitting here how far those two energy levels are apart is actually given by this coupling strength here and as I walk up this ladder this actually scales a square root of N. So here it's square root of one, square root of two where N is really the number of excitations in the system. So here's for example two photons no excitation in the qubit or one photon and an excitation in a qubit. So that's why it's square root of two and so forth. So what does this mean? Well if I bring my qubit on resonance with the cavity what I actually see and I measure for example transmission through my cavity originally I would expect one peak one Laurentian line like we have discussed from the cavity right there in the center but now that my qubit actually couples to it this line will split. I will see those dressed states and this splitting will be something like two G. So in this case here this is out of this paper down there. The splitting is something like 350 megahertz and you can see that this is much, much larger than any of the kappas and gammas I've mentioned so far. So we can really go in what's called the strong coupling limit where our qubits really coupling much stronger to the resonator than any of the decoherence rates we have in the system. How to when? Seven minutes, okay good, perfect. Okay so this is the resonant case. So qubit and cavity are on resonance. So what I have then is really sort of a combined system. I can't really talk about oh this is the qubit part and this is the cavity part as we have seen before there's only sort of a combination of qubit and cavity. So this is really similar to sort of you all know the classical example of that. Take two harmonic oscillators and couple them. If I have them coupled I can't talk about say the left and the right pendulum anymore but I can only talk about the say for example center of mass motion and the stretch motion of those two pendulums. So it's sort of the new normal modes of my coupled system and this is the same is true here. It's really I only have then these states here which are a combination of both. So what happens if I actually start to slowly detune my qubit from the cavity? Then I'll actually get a picture like that. So what is done here is I apply a magnetic field to my qubit so it actually sort of the qubit resonance frequency changes from down here, comes into resonance with the cavity there and then goes out and I've again measured transmission of microwaves through the resonator. So bright means I have a lot of transmission. So if I would take a line cut here this would be just a nice Laurentian line. If I take a line cut right there where they are in resonance I would get the picture back I have just shown you. So this is also called vacuum rabbi by the way. So where I have this dressed state. But now what happens in sort of this you know regime sort of a little bit out here. So how do I properly describe this system in this case? What actually happens there is this is then called the dispersive regime. So oh, here should be a G. So this detuning is actually much larger than the interaction strength. So in this case, I've written it in quotes here. So the qubit is then mostly a qubit and the cavity is mostly a cavity. So this perfect hybridization I had before goes into a tiny little bit of hybridization. Meaning an excitation of what I call the qubit lives in the structure which is originally the qubit and a little bit in the cavity. And the cavity excitation lives mostly in the cavity and a little bit in the qubit. So essentially instead of having these states with equal weight I would have like say 90% this and 10% that or 10% this and 90% that. So it's a small, it's a small just you know each qubit sort of the qubit gets a little bit of cavity character and the cavity gets a little bit of qubit character. And what this means actually for the Hamiltonian is I can now take this guy and go in this so-called strong dispersive limit so I have a detuning which is much larger so detuning of qubit and cavity which is much larger than the in the actual strength and this in the actual strength though is much larger than any of the decay rates. What I can then do is I can do perturbation theory for sort of small G over deltas and without going through the math you can find this in a couple of textbooks actually. I've mentioned is exploring the quantum by search a Roche, this is a book where I can for example find it. You can go from this James Cummings type interaction to this strong dispersive interaction. So my A dagger sigma minus goes into something which is A dagger A times sigma Z and now this pre-effector here, this chi is actually given by G and delta to be more exact G squared over delta. So what I now get is a slightly different form of interaction which we can actually use for many different things. One sort of footnote here, what I've written down here is true for a true two-level system if you actually do the sort of everything right for the trans-moniturns out that this chi changes form and instead of a G squared over delta it becomes a G squared over delta squared times the anonymity, but wow, we can neglect that. So effectively what happens is that in this dispersive regime my energy levels are not in resonance anymore and they're only just a tiny little bit shifted. So this is this tiny hybridization but there's not a complete splitting really there but just a tiny little shift. So what can we do with that? Well it turns out that if you talk about our systems with superconducting qubits doing quantum information processing they are all in this dispersive regime. So why is that? Well we can use that to read out the qubit state using the resonator. So here I have my combined qubit resonator system. I can send pulses to my qubit and try and manipulate it and it turns out I can send pulses to my cavity to read out the qubit state because if you take this Hamiltonian and now I've just slightly reordered it so I have now here's the qubit alone, this was the original resonator and here's this new term we've gotten out of this perturbation theory. What you can see now here is that actually, this prefactor actually determines the resonance frequency of the cavity, this a dagger a, meaning that now all of a sudden the resonance frequency of my cavity, my resonator depends on the state of the qubit and it moves over by chi but it's dispersive shift. So if my qubit is in a ground state I find my resonator on the right. If my qubit goes in the excited state I'll actually find this left curve. So I have this dispersive shift and I can use exactly that to determine the state of my qubit because for example if I shine in a microwave tone right here on resonance with the ground state I'll find a lot of transmission if my qubit is in the ground state and very small transmission because my resonator has moved over if my qubit is in the excited state. So again this is very similar to for example how detection is done in a trapped ion system. I have a lot of fluorescence if my ion is in the ground state and no fluorescence if my ion is in the excited state. Very similar here but instead of an auxiliary level in the atom I use the cavity to actually read out my resonator. Now it turns out though for many reasons I'm actually only allowed to drive my system with very very few photons so only like say up to 10 photons and now if you think about all the amplification I have to do and so on it turns out I can't easily detect the signal straightforwardly. So far up until very recently we could only detect expectation values for our qubits but not sort of a single shot is it excited or not. That has changed with actually Josephson parametric amplifiers becoming available so people have developed essentially quantum limited amplifiers which now allow us to do a trace like that. So this is essentially a voltage signal don't forget about the units so if it's down there it means my qubit is in the excited state if it's up here it means my qubit is in the ground state and this is using this quantum limited amplifier and you can see here that initially I measure my qubit in excited state it stays there for a while it stays there for a while and then I have a quantum jump down to the ground state. Now here this jump back up is because of that particular experiment so they thermally excited qubit which sort of you know does random jumps between ground state and excited state but here this is really a quantum jump my qubit changing from the excited state to the ground state and back. And sort of nowadays what we can achieve is within like a few hundred nanoseconds we can achieve detection fatalities of more than 99% so this really allows us to do a single shot detection of our qubit using this dispersive readout on the resonator so really I have for example low voltage okay so this is flipped around but this doesn't matter low voltage or negative voltage if my qubit is in excited state high voltage if it's in a ground state just for three more minutes or done okay so just and I'm done just to show you in comparison how this would look if we don't have this quantum limited amplifier then you see that trace so somewhere in there there should be quantum jumps but I don't know where so this is really in this case we could only measure expectation values. Okay and yeah so with that I'm done I'll show you the flip side of that Hamiltonian what it does to the qubit because you know it has this symmetric form then tomorrow morning thank you very much for your attention