 Okay, so first why quantum information? So this is a small classical computer. In 2019, I remember Professor Mazari had said that you can run quantum espresso for silicon on the phone. So this is our very small computer now. And this is what a quantum computer might look like. So why bother to move from our small classical computers to this quantum computer that looks very cumbersome and inconvenient? Well, the answer is that quantum computers can potentially provide new solutions to unsolved problems and also simulate quantum phenomena faster than the classical computers. So for example, quite long ago in the 1980s, Peter Shaw had proven that you can use quantum computers to find the prime factors of large integers. And this is something that a classical computer cannot do. And it's also the basis of the RSA cryptography that is the most widely used public key cryptography now. So now if the quantum computer can solve this problem, it means that there's a problem with the cryptography that we use now. So we also have quantum cryptography that will allow secure transfer of information across long distances. And even a more interesting quantum teleportation where you transfer the wave packet from one place to another using quantum methods but still within the speed of light. And how do they do this? Well, here on the left you see the classical bit of zero or one. And the qubit, which is the quantum bit, is a superposition of two states, zero and one. So you can see that here that the qubit has a lot more information than the classical bit. But a lot of this information is hidden in the sense that you can't measure directly what the coefficients of zero and one are. Once you measure it, the state is destroyed. So quantum information is about how one can manipulate this hidden information to perform amazing tasks. So one example here is the bell state, which is the basis of quantum teleportation. So the bell state is what one might call an entangled state because you can't factorise it into a direct product of two different states. So that's entanglement in your title. And how do you physically realise the qubit? So you saw that humongous a quantum computer on the first slide. Well, it's not easy. So these are some conditions that would have to be satisfied. The first you need to have a robust representation of quantum information, which typically means you have a finite number of possible levels, such as in a spin three-half particle. Then you need to be able to control the quantum states, and this is very tricky. And you need to be able to prepare initial states with sufficiency long lifetimes. So this is challenging if you are not in the ground state or not in the completely random states. And also you have to measure the output result. Most of the commonly studied qubits nowadays involve either spin, charge, or photons. And in this session, we are going to focus on spin where we look at localised spin centres in atoms, molecules, or condensed phase. So what's the role of first principles calculations here? Well, we can elucidate the quantum state in localised spin centres. And the challenge would be to look at, understand the excited states. And also because localised spins often have strong correlations that you have heard in the second session. We would typically might need to go beyond DFT using a DMFT for fluctuating spins and to capture intracite correlations. And as you saw in the previous slide, lifetimes are very important so it would be good if you can predict the lifetimes of excited states as the first speaker would be talking about. And we also want to be able to predict experimental observables for measurements and understand how the system interacts with the environment so that we can control it. So without further ado, I would like to introduce the first speaker for this session. Prof. Christian Van Dual from the University of California at Santa Barbara who is telling us about modelling points defects for quantum information science. Thank you. I am very pleased to be here at this excellent conference and I'd like to thank the organisers for giving me the opportunity to speak here. It's the first time I have a chance to speak at the Total Energy Workshop and it's a real honour for me. So, let's see, we need to switch over to my computer somehow. So I'd first like to acknowledge the people who have been involved in this work first and foremost Aldrius Alcauskas and various other people who are currently or formerly members of my group. A number of these people have moved on to other positions in the meantime and I'd like to thank David Afshalom who used to be at UCSB for introducing me to this field. And by the way on the programme it lists my first name as Christian. I don't know who put the programme together. They must have had access to government files or something like that because that's the only place where my name is actually spelled as Christian. I'll go by Chris, okay. So, our chair introduced the topic very nicely. We want to manipulate quantum information and kind of in the simplest possible picture of that. What you would like to do is take localised wave functions the way you have them in an atom, the way we study them, for instance, in the hydrogen atom in the introductory quantum courses and manipulate those wave functions. Okay, so you would want to somehow trap that atom and then address those wave functions somehow. And some implementations of qubits or single spin centres actually try to do that with trapped ions or molecules or you can think about quantum wells or quantum dots but actually a very attractive way of simulating such a trapped atom is to do it with a point defect in a semiconductor or insulator, okay. Point defects used to be referred to as colour centres. They were widely studied in alkali halides but of course in the context of semiconductors they have been widely studied because they are usually bad objects that degrade the performance of a device but here we are actually going to be using them to design single spin centres or in the case you want photons, single photon sources. So these will form our solid state system where quantum information can actually be manipulated at room temperature. So in my talk I will actually not so much talk about the actual quantum computing aspects or the manipulating the quantum information but the actual implementation in the context of materials of these single photon sources or single spin centres. And the prototype of this of course as you probably all know is the nitrogen vacancy centre in diamond which I will talk about as an example. So just to elaborate a bit on the importance of this if you actually have access to such single spin centres or single photon sources you can create, control, manipulate non-classical states of light and matter and that will really enable us to go beyond classical limits. Not just in computing but also in quantum information science for instance with quantum cryptography and another very interesting application is actually to do extremely sensitive imaging and sensing. So in the case of single spin centres such as the NV centre in diamond where you want to manipulate a spin it turns out that to initialize and to manipulate that spin you want to use light. So that's already one manifestation of the fact that you want to really understand fundamentals of light absorption and emission. Another way to use these atomic centres is to generate single photons which are then used for for instance quantum key distribution and again clearly you need to understand fundamentals of light absorption and emission in such centres. So as I said in the NV centre in diamond this has already been widely studied in diamond you can remove a carbon atom create a vacancy if you have a nitrogen atom next to that you create a point defect that has extremely interesting properties it can function as a qubit at room temperature or even at elevated temperature it has a long coherence time it's optically addressable and so it has come to serve as kind of a prototype for such single spin centres and the electronic structure is actually reasonably straightforward on the left hand side I illustrated in a single particle picture you have a wide band gap which means that the defect levels associated with the centre which occurs somewhere in the middle of the band gap are well separated from the band edges which is important for the coherence and then these levels in the particular symmetry of the NV centre are A1 levels and E levels they are occupied with electrons in this way in the particular charge state that you want for this centre which is actually a negative charge state you have an S equal to one ground state and then by exciting an electron optically out of this A1 level to an E level you can actually get it into an excited state as is illustrated on the right hand side here and the energy of that optical transition is 1.945 EV and of course then you get also luminescence around that energy now if you look at this in more detail and look at the magnetic structure you actually find that there are a number of possible transitions between these magnetic sub levels possible and it turns out that there is actually another possibility instead of this optical transition to go from the excited state to the ground state via this intermediate level this is referred to as the inter-system crossing and these transitions actually are non-radiative transitions and they are very important because they actually enable you to initialize the system in this MS equal zero ground state so to have a particular spin state for the centre so I'm using this as an illustration to show that both non-radiative and radiative transitions need to be thoroughly understood on the radiative side here's a luminescence spectrum measured for the NV centre this 1.945 EV actually corresponds to the so-called zero phonon line which I will tell you a little bit more about soon but as you can see there is actually a very wide sideband which is caused by vibronic coupling coupling to phonons which you need to thoroughly understand to really understand the properties of this system so that's what we set ourselves as a goal to study these line shapes and also to understand non-radiative recombination so we use our formalism that we developed some time ago and this is actually a review article that can kind of serve as a manual for how to do these first principles calculations for point defects in solids we calculate the formation energy of the defect I won't say much about that but that obviously gives you information about how likely it is that a certain defect can actually be formed in the solid in case you're looking to design new types of single spin centres the specific example I'm giving here and that I'm going to be using throughout the talk is actually not for diamond but for gallium nitride which is a semiconductor that we have studied in a lot of detail partly for the quantum information science applications but also because it is the main material used for solid state lighting which is an extremely important technology in its own right so for this particular example my quote-unquote point defect is actually an impurity I use the word point defect in a generic sense it could be an intrinsic native defect or it could be an impurity the formalism is general I focus here on carbon sitting on a nitrogen site in gallium nitride this is the definition of the formation energy and this can be generalized for any point defect or impurity so we can calculate this we use density functional theory we don't use the traditional LDA or GGA functionals because they have severe shortcomings the most widely known one is the band gap problem if you're interested in calculating defects levels in the band gap obviously if you have underestimation of the band gap it becomes very difficult to get quantitative information and we have been using hybrid functionals particularly the screened hybrid functional of Heidskusserie and Enzerhoff to overcome that problem I'd like to point out that there is actually another major advantage to using this hybrid functional namely that you get a much more accurate description of charge localization which LDA and GGA fail to provide and particularly when you're looking at point defects where particularly the wave functions of holes tend to be localized on particular orbitals the hybrid functional turns are to give an accurate description of that as we have verified in a number of benchmark studies so coming down to the issue of calculating radiative and non-radiative recombination so in the examples that I'm going to give, I'm going to assume that we have a recombination process between a carrier in a band edge, either the conduction band or the valence band and a localized defect level everything I'm saying can also be applied to so-called internal transitions where you would have a transition between two levels that are localized on the defect in a radiative process obviously the energy that you gain by recombining for instance an electron with a hole on the defect level is emitted as a photon in an non-radiative process the energy will be dissipated in the form of phonons so how do we calculate line shapes then we will know how to calculate the strength of an optical transition by calculating the dipole matrix element but these very broad line shapes that in the case of carbon and gallium nitride were already measured back in 1980 this requires a description of the interaction with the lattice interaction with phonons so I briefly want to show you the underlying principles of this which are very general we think about this problem in the context of a so-called configuration coordinate diagram we can do a calculation for carbon sitting on a nitrogen site in gallium nitride in a particular charge state and when we just let our modern electronic structure codes run we would immediately optimize the structure and give us the minimum energy position as a function of the atomic positions but in reality atoms move even at zero temperature zero point motion and when transitions are going to take place the atoms are going to move so we can actually map out around the minimum energy surfaces illustrated here in one dimension as a function of what I call a configuration coordinate which is kind of a common term for describing a whole set of atomic coordinates so I do that for one particular charge state, I can do it equally well for another charge state for instance the neutral charge state and I've added an electron here between these two transitions and then if we look at this curve here it has a similar shape but its minimum may occur for a different set of atomic coordinates because you've now changed the charge and so the atoms may want to assume different positions optical transitions will take place for instance absorption out of this defect level and putting it into the conduction band would take energy vertical transition would occur with this amount of energy and that vertical transition will probably be the strongest signature that you will have in an optical line shape after the transition has taken place the system of course relaxes to its minimum and then you can subsequently have emission or luminescence and you immediately see there is a difference in energy between where you expect the peak of absorption and the peak of emission to be so that's the well known stoke shift so that's the principle of it, we can calculate that in practice again carbon and gallium nitride we see what the energies are of these various transitions the transition that would take place between the minima of these two curves is the so called zero phone online as I said before the most prominent peak in an optical emission process would occur for this vertical transition which for this carbon impurity would be at about 2.1 EV now if we want to get the actual optical line shape out of this we need to couple this description to a description of vibrations in the system and here's our configuration coordinate diagram again and then we need to take into account that for each of these curves for the ground state and for the excited state since they describe the energy function of atomic position there will be vibrational modes in a one-dimensional picture and if these would be roughly parabolic you can think of it as simple harmonic oscillators with equally spaced energy levels which actually turns out in many cases to be a remarkably good description of what's going on so these horizontal lines illustrate these vibrational levels and then it immediately becomes obvious that the transitions in the space are not just from these solid dark curves here but are actually going to be transitions between these various vibrational levels which then ultimately leads to a luminescence spectrum that can be quite broad okay the energy that needs to be dissipated after making a luminescence transition and going back to the ground state will be dissipated in the form of vibrations and a very common measure for that is the so-called one risk factor which is basically this relaxation energy divided by the typical vibrational frequency for these transitions okay if the two curves for ground and excited states are pretty much aligned with each other in terms of where their minimum is then this one risk factor will be very small okay that's the case for instance if you have transition metals in oxides chromium and aluminum oxide in that case all of the energy of the transition is essentially concentrated in the so-called zero phone online most transitions are not that way in many cases you have significant relaxation energy, significant coupling to vibrations and in addition to the zero phone online you have this fairly wide phone on sideband which may have significant structure if the one risk factor becomes very large in case the relaxation energy is really much larger than the typical vibrational frequency for instance in f-centers and all cal halides you mostly get to see this wide sideband and the zero phone online becomes actually very hard to detect now from the point of view of computations this last case is actually the case that is most readily described because for that case we have shown that a one-dimensional model in which we describe the vibronic states in terms of a single coordinate actually does a remarkably good job and then we can we can just calculate a normalized luminescence intensity within the Frank Condon approximation calculate these vibronic overlap integrals in one dimension and for instance for our carbon impurity and gallium nitride we've done this for a number of examples but you can see the agreement between our theoretical description and experiments which are the circles is very very good which really gives us a benchmark for saying that we can do very well in terms of describing these line shapes particularly in the case of large lattice relaxations large one grease factors we can also do it in the case of intermediate values of the one grease factor and that's what we've tested in the case of DNV center in diamond where you see this structured phone on sideband is a configuration coordinate diagram for DNV center in diamond here's the electronic structure once again and based on that configuration coordinate diagram again we can put all the vibrations into it but we can't do it anymore in a one dimensional model we would not be able to match the details of the experimental line shape we actually are forced to take three dimensional picture digital phone on's into account both the lattice phone on's as well as more localized phone on's in the vicinity of the defect in order to be able to reproduce the experimentally observed structure of this phone on sideband this was the status of these calculations in the meantime has actually been able to optimize the description even further such that the overlap between the theoretical curve and the experimental curve is essentially perfect so this gives us a very important tool for helping identify single photon centers in materials and also for designing single photon centers the NV center has gotten a lot of publicity but it's really not an ideal center the zero phone online is the line that you're going to use in the quantum applications but as you can already see just by visual inspection of this spectrum the zero phone online has quite a low overall intensity it's only like less than 3% of the overall intensity of the emission here so you're basically working with a very inefficient emitter from the point of view of using it for quantum information so the search is currently on for centers where most of the intensity is concentrated in the zero phone online so the ability to calculate these line shapes is really very valuable so let me switch gears and talk about non-radiative recombination we're going to be using the same ingredients the same principles but now not for optical transitions but for transitions that are mediated by phonons so the context in which non-radiative recombination is often discussed is once again for defects in materials where the defects are playing a detrimental role and are trapping carriers either by trapping electrons from the conduction band or trapping holes from the valence band this was a problem that was recognized shockly and read and whole in the early 50s and flagged as an important obstacle for making highly efficient electronic devices so while it was flagged as an obstacle in terms of doing quantitative calculations of these rates of capture that hadn't really been possible until recently with the advent of highly accurate electronic structure calculations but also calculations of phonons as well as electron-phonon interactions so we want to calculate these rates for instance how an electron gets trapped on a defect level like this it's a product of these three factors obviously the number of electrons that are available the number of defects that are present in the particular charge state that will capture an electron and most of the information is in this capture coefficient which is what we are actually going to to be calculated and again we do that with the aid of these configuration coordinate diagrams so there are three curves here because ultimately you first want to capture an electron and then go to a state where you can capture a hole that has been completed so the transitions between these various curves here are now not transitions that are mediated by photons but what we are going to do is for instance if we start in the upper curve make a transition to the curve below it by using energy provided by lattice vibrations to cross over between these two curves so if this were a classical picture we'd immediately know what the barrier for that process is it would be a thermally activated process whereby you really have to go up this amount of energy to be able to cross over but this is quantum mechanics and we don't need to go all the way to that crossing point we can have a quantum mechanical process to make these transitions and that's what we can explicitly calculate the formula we use for that is essentially for a means golden rule where you recognize a matrix element squared there should be a density of states which is included in this energy conserving delta function which when sum or integrated over all of the possible electronic states that reflects the density of states the electron phonon coupling is what enters into this matrix element and calculate that very accurately formula is actually on the right hand side here this is the matrix element that we need to calculate and with some simple manipulations you can actually reformulate that in terms of an overlap between a wavefunction and a wavefunction at a different set of atomic positions and we have extracted these matrix elements based on different possible codes with the help of Georg Kresser we were recently able to also obtain this in FASP so we combine this information with the information about the configuration coordinate diagrams that we already have and again as our benchmark because it has been so well studied I'm plotting here the capture coefficients for holes onto a carbon level in gallium nitride plotting it as a function of temperature the red curve is what we calculate the blue diamonds are experimental values and you see the agreement is really very good so if I have a few more minutes no okay I want to make one more point which is about conventional wisdom in terms of the position of defect levels in the bandgap to give you maximum non-radiative recombination going back as far as Shockley and Reid and Hall when people have thought about centers that are going to be highly efficient non-radiative centers they've usually thought about levels that need to be in the middle of the bandgap the reason for that is it's kind of intuitive the rate with which you can capture for instance an electron from the conduction band becomes lower and lower and further away this defect level is from the conduction band and conversely the whole capture rate becomes lower if the defect level is further away from the valence band so combining those two principles you immediately say well if you want to have a center that can capture both electrons and holes it needs to be somewhere in the middle of the bandgap we recently found an important exception to that very relevant for identifying centers for quantum information we looked at transitional impurities in gallium nitride looking at iron for instance the iron level is very close to the conduction band it's within half an EV of the conduction band so you would say very strong non-radiative capture of electrons that's obvious you would think the non-radiative rate for capturing holes would be very low or even non-observable experimentally however it was found the rate for capturing holes was as high as the rate for capturing electrons I don't have time to go through the complete story but we've been able to explain this based on taking into account that a transitional impurity such as iron has excited states very easy to see to see within the D state manifold you can put electrons in excited states that correspond to higher line D states and if you take those excited states into account you can actually calculate rates that are very close to the experimentally observed rates so that's the context in which we came across this but we find this to be a very general issue for instance this is an example for a gallium vacancy complex in gallium nitride that careful consideration of excited states and you may notice that this sort of picture is actually very similar to what we calculate for the NV center in diamond these excited states can play an extremely important role in non-radiative recombination and going beyond quantum information in our opinion also explains why defects can still be important non-radiative centers in extremely wide bandgap materials or even insulators where the traditional picture based on energy difference between defect level and bandage would tell you non-radiative recombination would not be important so I'll just put up my conclusion slides here and I'll actually leave you with a slide of references in case you're interested in more details and I'll stop here and be happy to answer any questions Thank you