 Perhaps this is my mobile phone, so I better not create any interference. I have a... Can you pass me the... Okay, so I thank a lot for the invitation to Fabrizio and Salo Fazio and all the organizers. I have been challenged by the chairman who happened to be Fabrizio to make it one minute before the 30 minutes to have the chance to have five minutes of questions. But after the break I had now, I have only 28. Okay, so I will start my talk before I have 27. What I offer is this title of Analog and Digital Analog Quantum Simulation of the Quantum Rabbi Model. I have to say that we are in times of quantum supremacy or quantum advantage depending on your political view on the names. Which I don't care at all by the way, for me it's quantum supremacy. And I just have prepared a pedagogical example of the Analog Quantum Simulation of the Quantum Rabbi Model and an introduction, pedagogical introduction to the concept of Digital Analog Quantum Simulation that is more or less what we are boosting in my group in Bilbao as a paradigm to replace the correct obsession towards scalability in digital quantum computers by merging these two concepts of Digital Analog that are not incompatible. In fact, as usual, especially in this 21st century each time we feel too original, one PhD student sent me a paper of the 50s or the 60s where there was a concept of Digital Analog classical computers and classical simulators. So the merge, especially after the work of the Greek philosophers and before and all the great physics through the history is very hard to coin original things but I will try to convince you that in the quantum there are some interesting new things. So this is a photograph, of course, out of date. It's impossible to make in our case up to date photographs. The topics of researching my group are quite diverse. We wake up every day and what comes to our head sounds interesting and original. We work on this. If it has a queue in front, much better. But I cannot describe what we are doing in the group. So given that I missed the opportunity to lecture last week, I'm very, very sorry. I have mixed my slides in such a way that it is half-lectured because there are a lot of young people and also some advanced results. So concerning my view, depending on the day, I have a definition on quantum simulations. As you can see here, there is important to mention the intentionality for reproducing and mimicking models in quantum devices. In this sense, graphene should not be called a quantum simulator of the Dirac equation but just you are forced to do it. You don't have any other chance. Of course, if you build on it, you will have a quantum simulator. Another feature of this definition of today is that I mentioned that sometimes we make the mistake to say that quantum simulation is a physical system onto a physical system. No, it is always a physical model onto partial information of a physical system and this difference in the wording is important. And also given that it is models that are abstract concepts, of course taken from observations, you can also map unphysical things. I mean, you can see the work of Dimitris and others, Alexander Samite, and many experiments where we coined and created the field of quantum simulation of unphysical operations, which is not absurd. If you doubt that you can map onto physics and physical things, I can go to the blackboard and write 1 plus 1 equals 3 and you will see that it's a classical simulation of a violation to mathematical logic. Of course, in the quantum, you can do the same. The origin of this is Richard Feynman and I was very inspired and I work on this topic based on some aesthetic views more than scientific but this is just the motivation of each of us. Since I visited Athens, I was very impressed to see that five centuries before Christ we had already definitions of what is an imitation and a mimesis. So it's not a honor to Dimitris but you can take it also as that because I typically present this. So for me, quantum simulation is a quantum theater. The motivations sometimes also are misled by the grants. You see, the goal is not to produce companies and devices but it is also, of course, a component. For me, it is a communicating vessel between unconnected fields and I just give this example of black hole physics and BSEs. A second motivation is to the possibility of studying phenomena that are difficult to access or even absentee nature. I did some work in the past about Dirac equations, and I was impressed at some experts in quantum field theories not coming from my CTP, but I'm here after 26 years by the way. I'm very impressed how long. When I was interested in high energy physics and many experts told me that the Dirac equation in first quantization is unphysical or it should be erased from textbooks. So I was happy to realize that I was working in quantum simulation of unphysical stuff, no problem. So you should be open to these issues. Of course, the third is about really solving important problems that cannot be handled by classical computers and the typical examples are condensed matter, quantum chemistry, quantum field theory, machine learning, artificial intelligence, artificial life and many other things. And we don't run from our responsibility of creating quantum technologies. We are in the times of the flagships and this is not the main topic of my talk today, but we are heavily working in extensions of quantum annealing, coherent quantum annealing and other paradigms of quantum computing and soon you will see papers with the title digital analog quantum computation as also as an effort to merge concepts. And this is my true, my honesty behind all what I do. You see some people we wake up and we are unhappy with the reality. So when you are unhappy with reality you have two choices or you are a scientist to create things that do not exist or you are an artist. For me it is exactly the same. Okay, this does not need the introduction, but I just mentioned there are huge efforts in decades in Traptions while I think that Traptions are suffering to upgrade to several dozens of qubits with one and two qubit gates and local control, but it is coming with a huge efforts essentially in Innsbruck and Maryland and other groups. Sorry, these are Traptions. In optical lattices you have heard already from Sebastian before and other people. Superconducting circuits has the highest gradient in my opinion towards scalability. There are offers of great companies like Google IBM and also a consortium in Europe that will try to put 50 qubits on a chip with local control and near-neighbour coupling to be able to scale up towards quantum computing and quantum supremacy. And of course quantum photonics and many other issues. So let us start more or less with the technicalities. The first thing that I wanted to share especially with the young people is that from a historical point of view I like to think what was the first quantum simulations. So for me it was the work of Dave Weinland in the 80s with the quantum simulation of the James Cummings model where you replace the quantized electromagnetic mode of the cavity by a vibrational mode. Dave Weinland has accepted kindly last year during my visit so there is no problem. A second example that I tend to give concerning quantum simulations is what happened in this nature paper of the Yale Group in 2004 which is considered the origin of circuit QED. There is nothing else than the implementation of the James Cummings model that I think it has been described probably during the previous talks but you just have a two-level system, a single mode, a quantized harmonic oscillator and a dipolar coupling after rotating wave approximation. So this was reproduced in this paper and of course the authors in this case are not so optimistic about my denomination of quantum simulation but here happened the opposite that in trapped ions. In trapped ions you replace the electromagnetic mode by the motion and here you replace the two-level atom by an artificial atom but the electromagnetic mode is still there. If you can pass from optical regime to microwave for the sake of Maxwell equations it's irrelevant for the sake of matter it's not irrelevant we have the universe we have with the chemistry we have but in my opinion and I think I said this since 2004 and I was massacred by a lot of quantum opticians but nobody doubts that nowadays there are plenty of experiments that show that you can do much better and much more interesting quantum optics than when the original quantum optics set up. If you disagree we can use a coffee break and I will convince you in five minutes. In the case of trapped ions as I said you had this James Cummings model inspired from cavity QED and it was implemented in trapped ions and has been the origin of great physics and in fact of many key applications. I wanted to repeat pedagogically a sentence that probably most of you have heard from me already there is also a misleading concept when you make quantum simulations and people tell you if you make me the Dirac equation in a single trapped ion what is new I can do this with my laptop right I think these people have not got the point about what is physics in theory and experiments and the pedagogical sentence is the following the same quantum model implemented in a different quantum platform always brings novel physics it's not redundant if you don't find the novel physics it is the problem of your imagination it's not the problem of the implementation right and this is what we have improved through our careers and through these fields and the example I like to give is that in trapped ions the implementation in the language of lambda parameters and amplitudes of electric field in lasers and exactly the structure of the James Cummings coupling was a paradigmatic example of a quantum simulation in the 80s at that time there were no quantum computers and people said nobody stood up and said irrelevant what is if we know everything about cavity QED why you do it in a single trapped ion and my example goes ahead because the experimentalists that are also clever people were able immediately to use an acousto-optical modulator to change the frequency in some megahertz so to produce for the first time in history a weird coupling in the lab that is the anti-James Cummings so this Hamiltonian as you know does not violate energy conservation as it's written in many books of really standard textbooks and the imbalance of energy is not coming from the laser at all you see any time independent Hamiltonian conserves energy therefore it's a completely physical and valid Hamiltonian but however you don't find it in fundamental systems from first principles and I like to give this example because you start by just trying to copy a banal system, a banal model that you know from cavity QED to create novel interactions and immediately after if you see the celebrated PLL 95 of Sirac Solar that created how to implement quantum computers in physical setups you realize immediately that you require both interactions to shelf the quantum information to create the first control phase and the phase control node gate into trapped ions and that was the origin of many many great things in the history of quantum computing so that was more or less the introduction and now I will tell you briefly I'm really challenged by Fabrizio with the 29 minutes but I will manage so you tell me how many we have now oh great, fine I will arrive, I have prepared I think exact material to make it so let's go for the analog quantum simulation and as I said this is meant to be a pedagogical talk when I did my master's thesis I had a heavy frustration because it was the James Cummings model beyond rotating wave approximation I suffered a lot because I did plenty of analytics and numerical simulations that I have never reproduced again in my life, that's why you have PhD students and postdocs and computers and the frustration was that when you make beautiful models and calculations and then people tell you which are the experiments, you go to Paris you go to Garching, you put the parameters the coupling divided by the frequency of the oscillator in the microwave was 10 to the minus 6 30 years ago, nowadays is still 10 to the minus 6 because these are atoms and cavities nothing has changed in the optical domain is 10 to the minus 9 more or less and then every effort that you do to go beyond rotating wave is lost and destroyed except by some block-seeger sheaves and some special effects but tiny and not so amazing in terms of producing some scalable effects then I was interested in this James Cummings beyond rotating wave and in the year 2010 I cured my frustration because we should always try to cure our childhood frustrations and I started to work heavily in this what nowadays we call it quantum rabbi model to make the difference with the semi-classical approach the 30s of rabbi that is a semi-classical model and by the way this beautiful figure was done by one of my students one of these people that believe that aesthetics is crucial for science at least for the motivation and in 2010 my group collaborated with these two independent experiments the first one was done in the group of Rudolf Gross, Fran Depe, Akin Marx in Walter Meister Institute in Garching with Flux Cubits and independently Paul Forndias in the group of Case Harman and the people at Delft Hans Moy and others produce also in Flux Cubits all these in superconducting circuits and in a single shot experiment let's say in the first effort to go beyond the rotating wave the ratio G over omega passed from 10 to the minus 6 in the microwave to 10 to the minus 1 that means 5 orders of magnitude increased the coupling or the ratio in a single effort and in this way we started to cure this frustration to see effects beyond rotating wave and independently at that time in 2011 I had to write a review of a paper by Daniel Braque who is now a great friend and collaborator where he was able to present historically what some colleagues considered the first let's say analytical solutions of the quantum rabbi model which was considered for 80 years an unsolvable model it is of course debatable what you call analytical or not but this is a work that has led to plenty of developments and was also developed independently from these previous experiments these experiments were done of course in circuit QED and in this case with Flux Cubits it is crucial for these Flux Cubits to have galvanic coupling not only just and just some junctions on the line not enough just to to design Nobel Cubits the reasons why this field of the quantum rabbi model beyond rotating wave let's say is relevant is because the ultra strong coupling that people consider between 0.1 and 1 more or less let's say you have at the beginning a perturbative regime and later a non-perturbative regime was leading to some applications block-seeker shifts as I said but with some colleagues we developed even possibilities of making faster quantum computing and faster quantum operations given that you have a coupling that is larger so we developed also some ideas that were at the end not very original because they are our previous works of how to make faster and stronger operations and right after we realized because we are theorists the ultra strong coupling in the perturbative and non-perturbative regime was studied and developing in mathematical physics, in mathematics in experiments, in theoretical physics we immediately speculated about the possibility to have the ratio G over omega larger than 1 and we named it deep strong coupling regime just to make the difference with the ultra strong coupling not because we wanted to coin a new name but because there was new physics and that's what I will comment briefly here we studied this deep strong coupling regime of the quantum Rabi model or they come to kick me out already Fabrizio, you have to protect me they are not going to shoot right, okay, fine only photons but not x-ray the story of the deep strong coupling regime as I said we studied G larger than omega when we did that work it appeared in 2011 many especially the old guard of quantum optics attacked us intellectually that this was all wrong and physical impossible, only theory that cannot be done in the lab that you break two level approximation single mode approximation not only rotating wave multi level system impossible and each time I hear this from people that are more clever than me I work heavily on this to show that they are wrong and this is a pleasure that I have in my group when every time you hear that somebody that is more clever that you tell you something is wrong go back home and work and prove them wrong it's really very nice so that's what we did but for the moment at that time we rediscovered but in this paper we discovered that the essence beyond the quantum rabbi model beyond the rotating wave is this parity operator that we trace back in some unknown papers of the past you see this sigma set minus 1 to the n and then we realize that instead of James Cummings dabblets and all this wording of the James Cummings model what appeared naturally is that the whole Hilbert space was just divided in two slices and these two slices were just infinite dimensional states with parity plus 1 or minus 1 following this you start in G0 which is of course not anymore the ground state of the James Cummings and it's not a dark state and then you couple anti-James Cummings James Cummings, anti-James Cummings James Cummings and if you start in G0 you have a similar sequence of couplings and of course the James Cummings dabblets happened, evidently happened as an effective model as a phase transition that nobody has been able to study if you are experts in phase transitions I invite you to work on that because I have not figured out how these two parity chains melt down and create James Cummings dabblets you see because obviously this is a natural limit you see that these parity chains just break into dabblets to go back to the James Cummings when you apply the rotating wave it's not a discrete transition it's a continuous transition and that's what makes this problem interesting among the features that we discovered why we needed a new name is because you discover this is just photon statistics as usual in quantum optics you see that you can start with the population of the vacuum of population 1 and you start as a function of time to create something that in this paper we named after photon number wave packets that propagate to the right suddenly this photon number wave packets stack and bounce back and recover to recover the population 1 you see is similar to the generation of Schrodinger cut states when you do spin dependent forces only that the spin dependent forces as you know from trapped ions and from cavity QED is just has analytical solutions when you have the quantum rabbi model and you have all terms different from zero the solutions are not analytical and the other feature that we discover is a collapse revival phenomenon that cannot be traced back to the original Everly Sanchez-Mondragon works where you start even with G0 and then you create sudden collapse of populations in the qubit that are repeated as a pattern and the interesting thing as you can imagine just to finish the publicity of this hip strong coupling regime is that if you start in superpositions of states of the two chains there is absolutely no other feature than just collapse revivals but if you start in superpositions of two or more states of the same parity suddenly you see interference patterns because this photon number which packets start to cross you see is a very very interesting feature and this is what you see here like interference patterns of the collapse revivals because you are superposing states of the same parity from initial state and then people have discovered plenty of other properties I just wanted to tell you this another interesting feature of this result is that we don't have analytical solutions so all this physical intuition that I am telling you we just recovered from the numerical results and that's interesting the genes coming is very intuitive also because it is analytically solvable but this deep strong coupling regime has no analytical solutions but you can think in it and you can predict you give me any initial state and I can tell you what will happen just because of the features we have started in the last years so what are the chances to observe this if you cannot go to atoms and cavities and increase the coupling we were discussing with plenty of labs about these issues and some of them told us that the only hope to trace back these properties was to develop ultrafast switching techniques where you can couple the couple and so on but the second chance when nature betrays you and tells you no is to be an artist and to use quantum simulations or your quantum theater so that's what we did and we have been working heavily in the quantum simulation of these devices and recently very recently in IQC and other labs in Waterloo in other places in Garching people have been able to produce experiments not only for simulating but also for producing true USC and DSC regimes in the lab so that's more or less what I will tell you now briefly I don't like my style of presenting toxic not to show calculations and to do because I think nobody understands at least I don't and you can check in the papers but to pass you the ideas so our challenge was the following and that's what we did in our group you see give me a James Cummings in any experiment in any lab and tell me I cannot increase the coupling I only have the James Cummings can I produce the physics of the ultra strong coupling regime and the deep strong coupling regime that was a challenge and the answer is yes and the only thing you have to do in this case is a coplanar wave I cheap of superconducting circuits of circuit QED let's say and then we apply what in the slang of the community of microwaves can be called a two tone microband driving or just two laser driving if you want in the optical and frequencies the conditions you put to this orthogonal coupling as you see only acting on the qubit or on the two level atom or in the two level artificial atom the only condition are two conditions the difference in the frequency omega one minus omega two should be equal to two times the coupling omega one and the second condition is that omega one should be a dominant frequency with respect to omega two and to the coupling G if you impose those conditions and of course we have checked carefully that this is feasible in the lab these are conditions that can be met correctly as you can see in this paper you realize that after making standard interaction pictures you can create this effective Hamiltonian out of these chains coming plus the two tone driving and you create the full quantum Ravi model in absolutely old regimes it is so true that it's in absolutely old regimes that you can be puzzled by the following I told you that the key parameter is the ratio G over omega oscillator here the coupling has been reduced from G the usual of the genes coming even is reduced by a factor of two while the effective frequency of the oscillator is just determined by the difference in frequency from one of the driving and the original frequency of the cavity this means that as long as you can put this close to zero the ratio G over two by this zero is almost infinity which shows you that the USC and the G can be matched very easily second if you want that this ratio goes to zero goes back to the genes coming you have to increase this difference so that the ratio G over omega effective is again very small and in this way in this paper we were able to show with some further calculations and numerics that you can produce out of a genes coming the physics of the quantum Ravi model in absolutely old regimes most of them are not accessible or are not accessible by direct experiments but you can fake them that's the art of the artistic view of science when people are unhappy sometimes I give talks in some places where people are unhappy with fake physics I always tell them when you go to the theater or to the cinema you pay to be cheated so this is quantum simulation work too you don't pay because your institution has the access to the journals but you read and you waste your time to be cheated and that's beautiful that's why we art exist and literature and quantum physics so that's more or less the idea and these ideas were developed in the lab as I will tell you now but just a brief comment there is a close relation between the quantum Ravi model and the Dirac equation that was also at least in one plus one dimension that was really weird for us and I invite you to work on that we don't have time but if you take the Ravi model as I am showing here and you choose the frequency omega one equal to omega and this vanishes what remains is exactly the mathematical algebraic structure of a one plus one Dirac equation how can it be that dipolar coupling that is a non-relativistic effect has the structure of a Lorentz covariant Dirac equation that's something that may deserve further studies and I invite you to work on this I will not develop but we prove also that you can observe the Ceter Vevegun like in Trapdion like we did with Reiner-Blatt in Innsbruck in this paper and you can also add an additional third driving that will produce the Klein paradox like we did also with the Innsbruck group time ago in theory and in the lab and you can use all the circuit UED techniques of measurement that we developed also with Walter Meister Institute to observe the Ceter Vevegun, the Klein paradox and so on so it's a very interesting connection between dipolar coupling and relativistic quantum dynamics in first quantization this is publicity of other group you see this is a Karlsruhe group of Alex Ayustinov and very soon I hope you see in nature communications I don't know if I am making any disclosure beyond but that's what I heard it will be published in nature communications and this is a slide of this group where you see that they were able to do some 2 tone driving and they were able to reproduce this is pure math, this is a transformation the Rabi equation and they allowed me to show because this is already in the archive how they were able to observe these collapse revivals and things that happen in the lab how many minutes they have Fabrizio? 2 more? 2, 3 so I will just skip this part of Traptions this is possible also with a 2 tone driving in Traptions to reproduce the quantum Rabi model the experiments have been already finished in Qinghua in Beijing and very soon we will submit a paper where we observe also the way of reproducing all regimes of the quantum Rabi model in Traptions and so on so we proved in this theory paper in 2015 that with a single Traption also you can reproduce all regimes of light matter coupling including the Dirac equation in a certain limit and one of my students transformed this strange plot into a more artistic one for the cover of this special issue about the quantum Rabi model and the last part in the last 2 minutes each minute will have 100 seconds hopefully in a relativistic that's right I wanted to share with you this idea of the digital analog in a very practical manner they are not contradictory or exclusive paradigms and we exemplify with the quantum Rabi model so as I said analog quantum simulators it's an artistic you have to make like pottery you see to build exactly the Hamiltonian the digital quantum simulators are meant to be more universal we have access to error correction but we are proposing this DHUS or digital analog quantum simulator or digital analog quantum computer of course without error correction which is not belonging to our time in my opinion at least from a technological point of view and you can have a look to this paper by the way Antonio Mezzacapo who was an Italian brilliant PHD that I had in Bilbao is now working in IBM making the quantum computer with superconducting circuits we propose this digital analog concept with a disclosure because it's in the website of the group of Leo DiCarlo that the experiment that we proposed will be published in Nature Communications hopefully side to side with the previous work but these are two experimental realizations of our proposals and this will be kind of the first digital analog quantum simulator ever the idea of the digital analog is just to use the natural complexity of the setup produced at James Cummings don't kill it, don't reduce it to qubits that's more or less the moral of the story and we are using the analog blocks that are the natural complex Hamiltonians appearing in any quantum device and we are combining them with digital steps that are local operations to enhance the possibilities you can do also this digital adiabatic quantum computers we are working also in quantum annealing and so on just to use the last minute as the idea again the game is the same give me a James Cummings but you want a quantum Rabi model in all regimes so what can I do with James Cummings I cannot change the atoms, I cannot change the coupling so the idea is to use this analog block of the James Cummings plus local rotations in a trotter-like sequence to transform the James Cummings structure into an anti-James Cummings where you have to change forcefully designed here you see and then you just start to make a combination of short time evolution of the James Cummings and then a local rotation short time evolution of the James Cummings and another local rotation and you can prove using this Suzuki trotter methods that at the end you will be able to compose the James Cummings as a block with the anti-James Cummings as another block into the full quantum Rabi model and we put all experimental parameters to prove that instead of having the usual Rabi oscillations you can engineer in an algorithmic way with this digital analog concept the full quantum Rabi model evolution where the mean number of photons grow as you see which is not usual of course in the James Cummings you have just the n plus 1, n minus 1 structure and then you can produce the ultra strong coupling and the deep strong coupling as I predicted before again I will finish with the publicity and the slides of other groups in this case these are slides provided by Nathan Lamford in the group of Leo DiCarlo where as you see they started with the usual goal of say I want the full quantum Rabi model but I have only the James Cummings you compose the anti-James Cummings by using local rotations and then with the trotter method you make a composition of the composition to make an algorithmic way to build the James Cummings and you will see that the mean number of photons obey the quantum Rabi model and not anymore the James Cummings but they can produce plots that I don't understand but these are the slides the ultra strong coupling here and this is experimental data by the way but they were able to produce absolutely all regimes in a digital analog manner and they were even more clever and they realized of something that I didn't before and it is that they were able to make a bigger state reconstruction and they realized that the coherence of the Schrodinger cuts is a key signature of the deep strong coupling regime and this is data also of the experiment where they proved that the deep strong coupling regime was rich because of the coherence of the Schrodinger cuts ok, so I think this is the last slide if you want more information about this new paradigm of digital analog quantum simulators you can have a look to these papers we developed it for quantum field theories for condensed matter models for quantum chemistry models and so on so thanks and I hope I make it thanks and I hope I make it