 by Professor Melina Grifoni from Reckensburg. And she will talk about transporting DCAC driven interacting Josephson junctions. OK, so welcome everybody. And of course, I'm really grateful to be here. And thank you a lot to the organizers. So it was a big pleasure to hear all these amazing talks. And yes, so what I'm going to speak about today is something which is very much related to what Christian has just said to us and also what we have heard in the morning. So you see it is transporting DCAC driven Josephson junctions. And let's say a bit like Alfredo was saying this morning, we have a peculiarity that this is an interacting Josephson junction. And maybe, however, let's say the new thing I would say compared to all the talks that we have heard so far is that what I will try to convince you is that you can use a particle conserving approach to superconductivity in order to describe a Josephson junction. And then this is, let's say, the setup that we are going to study. So we have our superconductor with Cooper pairs, the quantum dot and let's say Cooper reservoir. And perhaps you can already notice here that I do not put a phase here. I don't put a phase phi l or a phase phi r. So in a superconducting theory with particle number conservation, in fact, the phase is not defined. This is maybe the main message. Right, now why should we look at the particle conserving theory? Because I think that actually in a superconductor there is no you want symmetry breaking. So I'm making a bit statement. But actually, if you think the Hamiltonian you are starting from is something which is due to essentially coulomb interactions or an effective phonon mediated perhaps coulomb interaction. And this coulomb interaction has a lot of symmetries. But in particular has a superselection symmetry, which means that it is invariant upon changing, say, the phase of annihilation operator by e to the i phi. So this is, let's say, the mathematical way of saying that we have actually particle conservation in our original problem. The second reason why you should perhaps do that is to think what is actually a current. So if you have a device like that, essentially what you measure is how much charge is leaving one of your electrodes. So for example, the left electrode, right? So if you have then a superconductor where you don't know how many particles you have, how can you define a current? So this is maybe another reason for this. The third argument that you could use is that actually when you look at the AC Josephson effect, and we have heard a lot about this in these days, you apply essentially a DC bias to your system, which means that you have a difference of chemical potential, new left and new right, between your electrodes. And the AC Josephson effect tells you that the super current is proportional, in fact, so the frequency of the oscillations, to the applied DC bi. So and why is this? Because you have a reservoir of cooper pairs which sit at two different chemical potentials, so new left and new are. Okay, so this is hopefully a good motivation to maybe look at superconductivity in another way. And here I would like to make you think. So now let us consider a free electron gas, so non-interacting electrons. And you know very well what is the ground state of this electron C. So what you should do is simply you take your vacuum, and then here you see you create a cooper pair like. So K up, K down, and you fill up all of your levels until the Fermi energy, right? So this is what we know from our course experience. This is quantum mechanics. What you can do is to rewrite this thing in a slightly different way, because let's say we are dealing with fermions, so you cannot create two electrons in the same state. So actually here by taking, let's say, the power of M of this expression, I'm adding a lot of zeros. And now you see already that what we have here is a linear superposition of states which all have the same total particle number M. Okay, now if we have a superconductor, we can imagine exactly the same thing. The only difference is that now I don't have the heavy side function anymore. I have a weight alpha K to these cooper pair states. And the difference is simply that while the Fermi C has a heavy side function, so either all the states are filled until KF and empty above or let's say now I have the possibility also to have other kind of excitations also above the Fermi level. All right, then we can go on and let's say we can postulate as I said this state and this is now our ground state. Is not a BCS ground state is a state which has a definite number of cooper pairs. You see? This is essentially a ground state which has been proposed by Leggett and Longago by Anderson. Now the very nice thing of this ground state wave function is that it has a property which is an electron hole symmetry property. So let us consider let's say our ground state with n cooper pairs and let us imagine that we destroy one electron. So we break one cooper pair if we wish from our condensate. This is equivalent up to some pre-factor to adding a quasi particles to a condensate which has n minus one cooper pairs. If you now use this electron hole symmetry, what you can do is the following. So what you can do is to introduce what is called the Josephson pair operator. What does it do? So essentially you start from your ground state with n minus one cooper pairs here, right? So sorry, the other way around. So you start with a ground state with n cooper pairs. You destroy one and you end up with this different ground state. Because of the symmetry that I was telling you before you can then define a linear combination of CK up and SC dagger up such that if I apply this whole operator on my ground state, I get zero. Why am I saying this is because we recognize immediately that we have discovered a quasi particle and the quasi particle gamma is a Bogolubian as we know from the conventional BCS theory. And its action on the ground state is that is zero, right? So essentially the superconducting ground state is the vacuum of the quasi particles. And this is the definition of a quasi particle. And the nice thing is that if you remember your Bogolubian transformation, usually you define your quasi particles exactly like this. So gamma is a linear combination of an electron with weight u and typically a whole with another weight v. In this theory, however, we have charge conservation, you see, because of the action in addition of this S operator. So this is the only thing that you should remember. So we have constructed a theory which is essentially very similar to the one you're used to with the difference, however, that now charge is conserved all through the system. And the weight u and v of the usual BCS theory are related to this alpha that I was telling you before by these relations. Okay? Now, if you are happy with this, so maybe, I mean, many of you already know these things. So when you have the S operator, usually you call this transformation the Bogolubian-Valatin transformation. Okay, now maybe to summarize the first part of the talk, what I'm trying to convince you is what we can do is essentially to start from a full interacting Hamiltonian, do a mean field as you're used with conventional BCS whereby, besides the usual pairing operator delta, we have here, you see, in addition an S operator, right? So if you now count charges, you see, we create two electrons, so we gain two charges. At the same time, we annihilate a Cooper pair, so the counting of charges is fine. All right, and if we now apply the Bogolubian-Valatin transformation that I was telling you before, then we arrive at our mean field Hamiltonian, which has the form that you already know, right? So this is the quasi-particle part, E is the usual square root delta square plus chi-square, and also we have the Cooper pair part. So essentially, our mean field Hamiltonian has two parts, the quasi-particle contribution and the contribution of the Cooper pairs which sit at a given chemical potential. Okay, fine, hopefully. So this I think it's the summary of this first part. All right, then we are able to go on, and simply I would like to mention that what I told you is something which is really old because this was already in the original work by Josephson 1962. So 60 years from now already, and if you look careful to his work, he has already introduced the quasi-particle operators, he has the S operators, and he also has the Bogolubian-Valatin transformation. So this is really, really, really old stuff, only simply revised if you wish. And actually, I mean, the main idea of Josephson was that because this S operator is creating a Cooper pair, when you calculate the commutation with the Hamiltonian, essentially you get two times lambda k, which is the chemical potential for him, and an S dagger operator. So if you look, this is Heisenberg equation of motion, which means that the S operator have an evolution in times which is provided by the chemical potential. All right, and then he concludes essentially by looking at the current. So he's exactly what I was telling you before. So the current is the commutator of the tunneling Hamiltonian with the particle number, and he gets his famous Josephson current, and their important point is that the oscillations are due to the time dependence of the Cooper pairs. So, so far so good. Now, what is new at this point, at this point, what I would like to discuss in the time which I've left, is how our Josephson ideas generalize to the case of interacting nano junctions. And the second thing is what happens if we don't only have a DC bias, but also we apply microwaves, AC drive. So this is the topic of the conference, so it's important that we have AC drive. Right, so now let us really shortly define my setup. So this is a, let's say, single impurity under some models. So the quantum dot with, say, the onsite interaction and the Hubart interaction new. And then we have, let's say, the superconductors, and then we have the tunneling Hamiltonian. Now, what I would like again to mention, so if we do the Bogolubo-Valating transformation, we can express the creation operator of the lead L. So this is a real electron. We can decompose it into two parts, right? So one part is, let's say, I create simply a quasi-particle here, or in this other part, I destroy a quasi-particle and simultaneously I create a cooper pair. Again, charge is conserved during this tunneling process. Okay, and on top of this, what we do is that we apply, let's say, a chemical potential, which has both a DC and AC component. All right, the current, once again, is given by, this is the current operator, and the current for us is obtained as a trace of the current operator over the total density operator of our quantum dot. So what we are going to use now in order to calculate the transport characteristic is not a Green's Functions method as this morning, in the case of Alfredo. It's a reduced density matrix formulation, which many of you know from quantum optics. So this is really a problem of quantum dissipation in fermionic environments. All right, then maybe, again, this is the total density operator, so it is the total density operator of this full time-dependent Hamiltonian. Okay, now the full total density operator is, I mean, dynamically defined through the Liouville-Fernoy-Mann equation. And here, for simplicity, I express this commutator in terms of the Liouvilleian. So every time you see a Liouvilleian, this means just a commutator of an operator, or with a Hamiltonian, H i. But this is a detail, so forget it's not important at all. What is important is that we have here a problem of having a time dependence. So when you have a time-dependent quantum problem, and here is a many-body time-dependent quantum problem with interaction, is pretty complicated. And what we want to do is to eliminate, through a unitary transformation, the difficult part, which is the Cooper pair part. So essentially what you do, it's, again, is a detail, but maybe some of you know this trick. So it was proposed for the case of Hamiltonian systems already long ago by Quevas and also Alfredo in this paper of 2002. Essentially, by this transformation, just look at the final result. You have a transformed total density operator where you don't have the Cooper pair explicitly anymore. You see, quantum dot quasi-particles, and all the effect of the Cooper pairs is embedded now in a family Liouvilleian, which, because of the transformation, is time-dependent. Okay? So this is, let's say, a trick which is quite important, and then at this point, I would like to stop. I would like to stop and let you think. So what do we have to do is to evaluate a current. A current is given by a total trace over all the degrees of freedom of the system. So Cooper pairs, quantum dot quasi-particles. So the thing that you want to do is to trace out all of these degrees of freedom, one after the other. And the first thing that you typically do is that you trace out the environment. By doing this, you transform a unitary evolution, which is the unitary evolution of the Liouville von Neumann equation, to an irreversible dynamics, which is also the reason why we have a steady state or a stationary current, okay? So the first question is what is our bus here is the quasi-particle bus. So the reduced density operator is obtained in our case by tracing away the quasi-particles and getting then a reduced density operator row prime. Once we do this, we are left with a reduced density operator which has degrees of freedom of the quantum dot and still has the degrees of freedom of the Cooper pairs. Okay, this is very important. Once we have done this, what happens is the following. So what happens is that the S operator enables us to transfer Cooper pairs from one part to the other. It means that the total number of Cooper pairs, left plus right, is conserved. But the relative number of Cooper pairs is not. And it is this delta n which varies, which is associated to the delta phi. So in this theory, the phase is not an absolute phase. This is the take-home message of today. It's not that I have here phi left and phi right. I have a definite particle number. But because I can transfer Cooper pairs, I have a conjugate variable delta phi, which knows about this transfer of Cooper pairs from one part to the other. And this is what gives rise to the Josephson current. Okay, if you understood this, it's perfect. The rest does not matter much. How much time do I have? Okay, now, given this, I tell you how we do and I show you some results. So the first thing that I want to say is that if you trace out the quasi-particle degrees of freedom, you can do this, for example, using the Nagar-Chimatsansi Projector Operator Formalism. This gives you an exact generalized master equation for this reduced operator. The important point is that, okay, this is the free dynamics of the quantum dot and then you have a tunneling kernel, which now depends on two times because of the driving. And this is the object which will drive the system dynamics to the steady state. And the current, in turn, is given by an exact integral relation, whereby I have another kernel, the current kernel, which acts on the density operator. And here, as I was telling before, the quasi-particles have been traced away, but the Cooper pairs and the quantum dot are still to be traced out. Now, in general, these objects, these kernels are very complicated, but you can do a perturbative expansion in them, in principle, to all orders in the coupling to the leads, gamma, gamma squared, and so on and so forth. What I'm doing today is simply the easiest one, is the weak coupling, so sequential tunneling regime. If you want to have a super current, you should go at least to the order gamma squared. I don't do this today. Now, how does it work in second order? So, in second order, you see we have two tunneling new billions, second order, one, two. Then we have the free dynamics of the Cooper pairs and of the quantum dot acting on this density operator. So, it's not so important. What I want to simply to show you is that, of course, you can have an analytical form of the tunneling new billion, and you have various contributions. In particular, you have your S operator, you have the D operators of the quantum dot, and also, due to the unitary transformation, I was telling you before, we have the full time dependence which is given by the bias, which is due to the fact that we tunnel in and out the Cooper pairs. Okay, and if you explicitate this phase, you will see that at some point, your Bessel function will pop up, and you will see also that you have the DC frequency and you have the AC frequency. Details not relevant. And maybe the last thing that I want to say is that this kernel has a very nice interpretation. So, once you trace out the quasi particles, you see that you have these two contributions. So, the normal one and the anomalous one, and because you have two of these tunneling new billions, you have four possible combinations. So, contributions where you don't have Cooper pairs involved, contribution where you only have one Cooper pair involved, or contributions where you have two Cooper pairs involved. And all of them together will give you essentially the steady state current. And importantly, the quasi particles give us these fermionic lines, and these are responsible for the reaching the steady state because give a finite memory time to the kernels. All right, now to the results, this allows us to get in exact form the stationary current with all of the harmonics that you should have. The DC current can also be calculated explicitly. And in analytic form, and I only show you again that we can get out the contribution of both the AC and the DC, and the DC is in this new. So, these are Bessel functions which have argument inside, which is also oscillating in time. So, pretty complicated, but it's really exact to the second order. And now, two minutes and I am finished because these things, I think all of you know. So the, let's say, theory for driven Josephson junction is again very old. Days back to Thien and Gordon 1963. And what they had there was a superconducting, insulating superconducting junction. And they wanted to explain an experiment where, let's say, upon application of microwaves, the VI current characteristic was showing peaks and these peaks were spaced by a frequency that they could attribute essentially to the applied bias. And they were explaining this feature by the famous Thien-Gordon theory, saying, okay, if I have a current characteristic at zero applied microwave, I can have all of these replicas essentially by shifting the density of states by photon channels, which the energy is shifted by n h bar omega and they are weighted by the squared of the n-special function. So this is the Thien-Gordon theory. And actually, we have seen already today, but you did not go into the details anymore that there are beautiful experiments where, again, you have this kind of junction. So S-I-S junctions where it's very nice to see, I mean, this is exactly what we saw before. Sorry. No, I, okay. That you can really see in a beautiful way these Bessel function patterns, which was already predicted by Thien-Gordon long ago. The fact that we want to discuss now is what happens is we don't have, let's say, S-I-S junction, but a quantum dot inside a superconductor. Again, here there is some work done in 1996 by the MIT Group of Van and Orlando and they again wanted to explain an experiment where you had a quantum dot and subjected to microwaves. What they did there was, again, to use the Thien-Gordon theory, this time saying that what is shifted by the microwaves are the tunneling rates in and out of the junction. Okay, so, and in our case, it works exactly the same and maybe because I don't have so much time anymore, I will show you, let's say, three slides with our result. So, the first is the DC component of the periodic current at zero applied AC drive. And what you see here, this is a typical stability diagram of a quantum dot nearby the zero one transition. And what you beautifully see is the superconducting gap. And what they want to focus is on these three points, one, two, and three. So, this is the current, it's not the differential conductance. So, in point one, we have current and this is simply because you can nicely tunnel, say, from the source to the drain here through a level in the quantum dot. So, this is quasi-particle transport. If you are in the situation two, there is no transport because essentially you have Coulomb locate and if you are in the situation three, actually you have the possibility of having in-gap transport due to thermal quasi-particles. In this case, you see, you tunnel from this either upper side of the density of states to the other upper side of the density of states. And you can see very nicely this quasi-particle peak if you do some scans. What happens if we apply the microwaves? Then we have all of these plethora of multi-photon assisted channels and you can again interpret them quite nicely in terms of photon assisted tunneling. Maybe what we can focus is on the three and in this three regime, you have a current inversion. So, even if you have a positive DC bias, you have a current not from the source to the drain but from the drain to the source because you use if you wish a, let's say, a photon in order to go from here to here and then you can go, yes, then you can go up. All right. And what we also were doing is a comparison to TinGordon. So if you would apply a simple TinGordon ansatz, this is what you would obtain here but in reality, it's much more complicated. So for example, due to the fact that you can have this current inversion that TinGordon would not give you. And finally, also in our approach, we can see the nice fence where if you plot the IDV as a function of the DC and the AC bias, you will see that the coherence, I mean, it's not the coherence peaks as we saw before, it's really, let's say, we are essentially here at zero gate and we see what is going to happen. And again, if we, let's say, have some traces as a function of the AC amplitude, you recognize very nicely the Bessel function pattern. Okay, so with this, I conclude. So I hopefully have demonstrated that it is possible to describe the full, the IDV characteristic of a AC-driven Josephson junction in a particle conserving framework. And in this sense, the coexistence of two complementary views is possible. So here you see these roses which have been climbing into these trees, so you don't know what is what anymore. And I think that the particle conserving approach can open new perspective and we are preparing two manuscripts, so one on this stuff, hopefully, we shall submit it in two weeks to the archive and soon also the work on the Andre reflection and the Josephson current. Thank you a lot. Thank you, Milena, for the very interesting talk. Any other questions? Yeah, thank you for the talk. I'm curious in the, how compare the two approaches, this particle conserving approach. So in some limit, if you consider a non-interacting case with the usual BCS approach, you can solve the problem exactly by regards to Green's functions. And if you, in this limit, you take the coupling very small, you should be able to compare one to one with this other approach. I'm curious about this comparison. Okay, so first of all, also in this approach, if you have no interaction, you can solve this problem. So, and also what we did is that we were using a Green's function approach with a non-interacting level, particle conserving, and you get the Andre bound states nicely out of the theory. So the approach, I mean, I believe, yes, absolutely. So, I mean, the problem of the Green's function formulation, but we can discuss this afterward, in my opinion, is that it works pretty nicely at zero bias, you have big problems when you have finite bias. And when you have interaction, you have even larger problems, but you have to do some tricks and you don't know if the current is conserved or not, for example. So the typical case is the case of an NSN junction, which very often you have to do some tricks in order to ensure it can't charge conservation, not in this approach, at finite bias. Okay, thank you. Thank you very much, Milena, for the nice talks. Just to comment this, maybe it's fair to say it's particle wave duality somehow that we, it's just the other seas, things the other way around. Yeah, I mean, my question was, what do you think is the benefit of working with these particle conserved states? First question, second question is more technical questions. Question, you have presented results for the weak coupling regime, and how do you think to extend this to strong coupling? Okay, as I was saying before, the theory per se is exact, and you can do in this formulation, so you have two things. So if you don't have interactions, then you can sum up the series, if you wish, so it's no problem. If you have interaction, you have to do some resubmission of some subclass of diagrams, and for example, this allows you to have also, I mean, not the condefect down to zero temperature, but you could also have the condefect in this formulation. So, and I think, let's say, in my opinion, a big advantage of this approach is that I have under control every time what is the charge in my system. So, and as I said, I believe that, I mean, of course, if you have a phase bias, Josephine Junction is no brain, you will do the phase approach, it's completely fine, but in this case, it's not phase biased. So, and in there, I think one has to be a bit more careful. Okay, now my question back to you. You see, people have been, when there is this, as you were saying, the standard theory of Josephine Junction, then there is Coulomb Locate, and we all know that this is basically two ways, two perturbative treatments of the Junction, and then the question has been how to connect actually these two regimes and theoretically, and of course, you can formally write things down, but it's very complicated to actually combine these two limits. And do you think that the methods you have been developing, there is ways to do so, yeah? Absolutely, absolutely. I mean, again, I was saying before, so again, so we have done this for normal leads, and let's say this approach can go down non-perturbatively, and the problem is that what is in this formulation in Andreev-Bound state? So, the Andreev-Bound state is a pole of the tunneling density of states, so in order to have it, you have to have some self-energy, and the self-energy only comes if you do this infinite reformation. And this we did, so, and of course, it's not exact the self-energy that we get, but still you get all that you had to get. Okay, any other questions? So, I see no other questions, so let's hang behind again.