 Thank you very much. It's a pleasure to be here. I was wondering whether my invitation was due to the fact that Nokia subsidizes my chair. I have an Augustin Fresnel chair at the Institut d'Optique and Nokia is a sponsor of that chair, so I'm grateful. So when Nokia asks me something, I say, yes, I'm benevolent. I'm also glad being here among people who are mostly mathematicians or physicists on the side of mathematics to say that I am also supported by the Simons Foundation on a program which is a joint program between mathematicians who have invented a new method for treating a celebrated physics problem which is Anderson localization, which is a quantum phenomena. And there are mathematicians in the United States who have invented a new method for treating this problem and they have embarked me into a network which is well subsidized by the Simons Foundation. So it's and it's for me an opportunity to interact more than I ever did in my life with a mathematician. Okay, and it's very interesting. Okay, now what is my talk about today? It's about the discoveries which established quantum optics starting in the 50s, 1950s, not 18, okay, 18 is more classical optics, but starting in the 1950s there were a famous paper called the, or effect discovered, the Humbury Brown twist effect, which is considered the founding phenomena which started quantum optics. Then there is something fascinating, strongly related to entanglement, Hong-Wen Mandel and of course Bell, and I have a pretext and I love explaining this story and it turns out this is why I have this subtype here. It turns out that quite often people come and say, oh, all my life I heard about HBT, but I did not know what it was. So I'm so happy that you explained that. And the pretext for explaining it is that all these effects which have been discovered and studied with photon, we are revisiting them with atoms in place of photons. And so you will see in the talk from a certain point of view after all it's a quantum particle, but it's special quantum particle. It's a particle which has a mass, okay, and a photon has no mass, and in some cases it makes a serious difference. And there is another difference that I will emphasize during the talk. So let us start. While this is to say that we have a group with plenty of people working on many different subjects. So we work on quantum simulation with ultra-cold atoms, Anderson localization, and other subjects. But the topic of today is quantum atom optics. That is to say quantum optics, not with photons, quantum optics with atoms. Okay. And these are the people who have performed the experiments on metastable helium that I will describe today. Okay. And I like to start with the result of my thinking about quantum physics. I really am still absolutely amazed by quantum physics. I think quantum physics is a big mystery. But I think when thinking deeply about it, there are two different kinds of mysteries. And because I would not dare to present it if I was alone, I have a strong support by citation from Richard Feynman. So let us start with two quantum mysteries according to Feynman. In his book, the famous book, Lectures on Physics, published in the early 60s, Feynman emphasizes wave particle duality and says, this is the only mystery of quantum mechanics. Okay. And wave particle duality is about a single particle. If you have a single particle, a single electron, a single atom, or a single photon. Okay. But let's say a single atom, a single electron. We know that it can also behave as a wave. And vice versa, something that we think is a wave, that is light, for instance, can in some circumstances behave as a particle. Okay. And this is wave particle duality. Twenty years later, after here claiming all the mystery of quantum mechanics is here. Okay. After 20 years in the early 80s, it was a time when I was completing my experiments on the test of Bell's inequality and it about that, Feynman recognizes that there is something, a second mystery. And the second mystery is around entanglement. And the first obvious difference between wave particle duality and entanglement is that wave particle duality is about a single particle, while entanglement is about two particles. Okay. And so there is a series of wonderful effects, Hamburg-Brahme-Tweez effect, Hongumandel effect, and test of Bell's inequality, Bell's inequality violation. And I really think that these two mysteries, or call it whatever you want, are different in nature for the following reason. And again, this is a result of thinking hard about it for decades, unfortunately. In the first case, although it's strange, it's bizarre, you can anyway always come with models describing things in our real ordinary space, three-dimensional space where we live. Okay. Plus time. Okay. Space plus time. Because when I have a particle, I know what is a trajectory of a particle. And if I have a wave, I know exactly how to describe the propagation of a wave in ordinary space. So I can describe everything in our ordinary space time. When it comes to entanglement, the good way of describing it is to describe in a Hilbert space, which is a tensor product of the space describing the first particle and the space describing the second particle. And in that abstract space, there is no problem. But as a great theorist of quantum optics, Asher Peres once said, quantum phenomena do not happen in a Hilbert space. They happen in a true laboratory, in a real laboratory. And a real laboratory is an ordinary space. And as soon as you try to represent things in our ordinary space, then you have bizarre things like non-locality, etc., etc. Okay. And so the fact that the only good model for describing it is in a Hilbert space and the fact that there is no reasonable classical model in ordinary space time makes a major difference between these two. So today, I will really concentrate on this, that is to say, on the experiments which suggested that entanglement is really something important and fascinating. So to finish with this global presentation, what we call the quantum revolution, but let us call it the first quantum revolution, was first a revolutionary concept about wave-particle duality. And it allowed to understand the structure of matter, its properties, its interaction with light and then transistor, etc., etc. It has given us the information of communication and communication society. Okay. And of course, no need to emphasize that it is us revolutionary, for instance, as the invention of the heat engine. It has totally changed society. So it was first conceptual and then technological and it changed society. And the second quantum revolution that I will focus on that today, I will focus on the fact that there is entanglement, which is really a very strange property. And there is another thing which will be implicit, but I like to emphasize it. A major ingredient of the second quantum revolution is the fact that starting in the 1970s, physicists, experimentalists have learned to control, observe, manipulate single microscopic single quantum objects. You have to know that in the 50s, people as smart as Schrodinger brought strong sentences saying it's clear that quantum mechanics is only to describe large ensembles of microscopic objects. Schrodinger continues, as soon as we pretend to describe a single quantum object, we are led to ridiculous consequences. And claiming that is just as fantastic as if our colleague paleontologist would tell us that he has an alive dinosaur in his lab. Okay. So there the feeling that quantum mechanics is only for a large ensemble of microscopic objects. And then we can understand why in contrast to Einstein, why most of the people were not annoyed by the fact that quantum mechanics was making probabilistic prediction. Because if you have a large ensemble of objects, the theory makes probabilistic prediction. Of course, there is no problem to accept it. Okay. So the fact that starting in the 1970s, the physicists showed in the laboratory that they can observe and manipulate single quantum objects really totally changed the situation. In particular, because there is no question of observing entanglement if you cannot isolate single pairs of entangled objects. Okay. So, yes. Well, the point that I want to make here is that single quantum objects are first natural objects, atoms, photons, etc. Photon of Serge Arrache, etc. And then we also now have artifact objects made in laboratories, which are quantum objects, for instance, Joseph John John, etc., etc. Okay. Let us try with the Humbury-Brown and Twizz effect. This is Mr. Humbury-Brown. If you look on the Google image, you have dozens of images of Humbury-Brown. If you look for Mr. Twizz, you will find no image because, in fact, he became a member of the intelligence service or some secret service. Okay. And people in secret service do not publish their photos. But somebody privately gave me this photo of Humbury-Brown and Twizz. So it's a personal photo. It's not on the web, unless if people copy my transparencies and put it on the web. So this is the famous Humbury-Brown and Twizz effect that it was published in 1956. The experiment is the following. You have a lamp, a classical lamp. Laser was invented in 1960. So it's a classical lamp. There is no doubt. There is a filter to select more or less monochromatic light, a single line, in a mercury lamp. And you put it on a small hole or rectangular aperture. So filters to isolate a single line. And then this light beam is analyzed with two photomultipliers. It's after World War II. And during World War II, they have developed this technology, like photomultipliers, et cetera. So you have two photomultipliers, which are very sensitive photodetectors. And there is a beam splitter here. And you know what is a beam splitter? It splits a beam in two. And they look at the correlation between the photo currents. And nowadays, well, people laugh when you see that, because nowadays you have electronic circuits and you just look at the correlation. At that time, to make the correlation, you need to have a multiplying system. And it's just mechanical with a motor. You have a current coming on one side, a current coming on the other side, and the rotation of the motor is proportional to the integral of what you have. It's really amusing. And the correlation, of course, what is it? The correlation should be written somewhere. It's not written? Okay, so I will write. I will do like a mathematician. I will write what is a correlation, piece of chalk somewhere. Voila, merci. So the correlation, the current correlation is nothing else than the average of the product of the photo currents with a certain delay divided by the product of the averages. Okay, so this is a correlation function. It means that if there is no correlation, the average of the product is equal to the product average, and G is equal to 1. And when they measure it, okay, what they find is that, oh well, what I should say. Excellent question. Both. It's, and I missed that step. It's I1 and I2, but you can replace the problem. You can think that what you are doing is going in the image space. So you can think that you have put this photo multiplier on the other side and so you are in the same space. So it's a good question, and the answer is both. Technically, it's this and that, but when you are on the other side, okay, it's I of R1 and I of R2, okay, which are in the same space. Okay, and so conceptually, you can think that you put these there. So what you find first, let me, okay, I'm here, what you find first, if you put the first photo multiplier exactly on top of the other one, exactly the image of the other one, so R1 equal R2, and you look at the correlation versus the delay. When the delay is null, the correlation is two, which means that the two photo currents are, there is a correlation, are correlated, okay, the fluctuations are correlated. This is not obvious because if the fluctuations were due to each detector individually, there would be no correlation. It means that what they measure is correlation in light at the end of the day, and not correlation due to noise in the photo multiplier, okay. But, and when they separate the two images of the two photo multiplier, then it drops to one, here there is no correlation. No, sorry, when they change the delay, and to change the delay, you just add meters and meters of cable. One meter of cable is 20, is five nanoseconds, one meter of cable is five nanoseconds, so you can change the delay by adding meters and meters of cable. I was doing that during my thesis. And if we were writing the formula, I would be quadratic in the photo operator in A. Yes, exactly. So it's square times square, it's a classic correlation. Yes, we'll see that in a while. Now, if rather than changing the time, they work at delay, which is null, and so of course when the two, this is the same as this one, now they separate the two photo multiplier, and it drops from two to one, and there is a characteristic distance here, just like there is a characteristic time here. We call it the correlation time, and we call it the correlation length, okay, or the correlation length. Okay, good. What they find is that this time, this correlation time is nothing that's what we call in optics the time coherence, which is just the reciprocal of the bandwidth of the line, okay? So a typical line will be one gigahertz, so it is one nanosecond, something like that. And then here, and this important thing for them, the coherence length is what is called the spatial coherence, and the spatial coherence is nothing else that the wavelength divided by the angular diameter that you have here. And this is the reason why they were doing the experiment, because by measuring this coherence length, they could determine the angular diameter of stars. This was the goal of their experiment. I will show you that, okay? So the idea was the following, okay? You have two detectors. When the detector is close, you watch a star. When the detector is close to each other, there is a factor of two, and at a certain distance it drops to one. This gives you the coherence length Lc, and from Lc you can determine the angular diameter of star. And you know that measuring the angular diameter of star is very difficult. It's very difficult for the following reason. There was a system used by Mr. Michelson, which is an interferometer, in which you just look at ordinary interference between the light coming from the same star on two points. And what you observe here is fringes, and a calculation which is well known in optics tells you that, again, what happens is that when you increase the distance, the visibility, the contrast of the fringes will decay, and the typical size will be again Lc, the length of coherence, okay? And so just by observing how the visibility of the fringes decay, you can measure the angular diameter of star. But there is a problem, and you have all heard of this problem. It is the fluctuation of the atmosphere. And because of this fluctuation of the atmosphere, there is a difference in the path here and a path there, and there is fluctuation in the relative phase, which means that the fringes move like crazy. Michelson was a fantastic experimentality. This is a true story. He was able to observe the fringes, and although they move very fast, he can say, hmm, good visibility. Let us decide this 90 percent. They would put a point 90 percent. Then he would increase the distance between the two detectors, and he said, hmm, 50 percent. And he really made measurements of angular diameter of stars with that. But honestly, it was not very convenient. And of course, in modern times, we want to have objective measurements, okay? And this is why Amberie Brown and Twiss invented this method, which is different, because this method is insensitive to the atmospheric fluctuation. I have not planned to explain it, but if you ask me the question, but after I finish my explanations, if you ask me the question, I will tell you why it is insensitive to atmospheric fluctuation. But this was the goal. Michelson is the same, wasn't he? Absolutely. That's a great Michelson, American physicist of the end of the 19th century. And we can note that it's not the same thing which is measured. Now, Thibault, you are going to be happy. In fact, what we are measuring here is what is called G2, which is second order in intensity. And intensity is a square of the electric field, okay? And so here, we measure a fourth order correlation function for the field. While here with Michelson, we measure a second order correlation for the field. So obviously, it's not the same correlation function which is measured. But, and so Amberie Brown and Twiss, this is insensitive to atmospheric fluctuation. But although it's different function, in fact, they are strongly related for a reason which is a mathematical reason. Well, mathematical from my point of view, probably for you is not. The idea is the following. Let us describe the light arriving here. It's an electromagnetic field, fully classical at this stage, okay? I only think of classical waves. In fact, I have an incoherent source, which means many independent emitters. So the wave that I receive here is the sum of many contributions, which is a sum of waves with phases that I take as uncorrelated random variables, because the various sources are independent. The result here is at a given time and a given point is the sum of independent random variables with the same statistical properties. And we know that then we can apply the central limit theorem, which tells us that we have Gaussian statistics. And when we have a Gaussian statistics, we have the Vick theorem, which allows us to express a fourth order correlation function as a function of a second order. And at the end of the day, we have this formula. The correlation function measured by Amberie Brown and Twist is equal to 1 plus the square of the correlation function measured by Michelson. So within a factor of root 2 or something like that, the width of this function is the same as the width of this function. Maybe there is a factor of root 2. We don't care. And so this is a good understanding of what happens here, at least an understanding from a mathematical point of view. Well, it's not in my formalism, but a hand-waving explanation. I could do full calculation, but a hand-waving explanation is the following. In the Michelson experiment, you look at an interference between something passing here and something passing there. I will get extra. And so what is important is what is the difference between the path and the fluctuation is of the order of a few micrometers, which is several wavelengths. This is why the fringe is moved. Now the Amberie Brown and Twist effect at the end of the day amounts to looking at a bit-note here and a bit-note there and compare the two bit-notes. But the bit-note will be in the gigahertz because, you know, when you look at this fluctuation, it is in the gigahertz range. So in fact, you look at bit-notes. The bit-notes are in the gigahertz range and in the gigahertz range, the wavelengths is of the order of several centimeters and the fluctuation of the atmosphere is very small compared to that scale of centimeters. I can't show you full calculation, but the real idea is what I explained you. Okay, so let us continue because we have not yet entangled man and things like that. They move to Australia and I will tell you in a while why they move to Australia and they built an observatory. So you see you have like a railway here, diameter is 200 meters, so you have here one detector, another detector, and they look as a function of the distance. And for instance, for this star of the southern cross, beta of the southern cross, they observe G2 decaying after 60 meters and from this, they get the angular diameter of the star. So it works, okay. So, fine. Why annoy you with that? Because after all, Hamburg-Brahme-Twist correlation were predicted, observed, and used to measure star angular diameter more than 50 years ago. Why bother? Because the question of the interpretation provoked a debate that prompted the emergence of modern quantum optics. And the question was, can we describe it in a classical way? I have shown you that. Yes, it's easy. Can we describe it in a quantum way? And then it turns out to be very subtle. So let us come again to the classical interpretation. The classical interpretation amounts in a sense, if you look at the fluctuation here, it amounts to saying that the average of the square is bigger than the square of the average. And if we have a Gaussian process, there is exactly a factor of 2 in the ratio. This characteristic of a Gaussian process. So this is what we have seen, okay. There is another image that I love, and I would like to show it to you, or I forget. You all know what is a laser speckle? No, yes, no, you don't know. Ask anybody a small piece of plastic of anything through which I can pass my laser beam. A piece, a little bit of plastic, whatever. Yeah, at least transparent, even if it's a little diffusing, it's okay. Yeah, yeah, something, perfect. Thank you. And then I'll take an after. If I go further. Apart from the cross, you see that you have black and dark, dark and white spots. Do you see them? A granularity, okay. It's a random granularity. This is a laser speckle. Okay. The idea is the following. Let's suppose that I freeze what I get here at the scale of 10 to the minus 15 second, okay. So nothing evolves. What I get is this interference between these many independent random variables. This is exactly what I get here. That's a speckle. Now if I let time go on, what will happen is that this speckle will change, okay, all the time. If now I put two detectors separated by a large distance, it's totally incoherent. But if I put two detectors at a distance which is less than a typical size, that is to say less than the correlation length here, the fluctuation will be correlated. This is a very nice classical representation of the calculation I have shown. So we understand perfectly well from a classical point of view. Now when people try to understand the Ambribrand and Twizz effect taking into account the quantum nature of light and thinking of photons, this is a description that you give. The G2 correlation function is the ratio between the probability of joint detection of two photons divided by the product of the probability of single detection. Okay? And then if you find a factor of two, there is a problem because if you have independent detection event, the probability of joint detection is just equal to the product of probability of single detection. So if you find G2 different from one, it means that the photon tend to come by pairs, tend to come together. The factor of two says we detect the photon by pair close to each other. But how can it be? You have a star. The photons are emitted by different points of the star, maybe separated by thousands of kilometers. Obviously they're independent particles. So how can it be? So you know what happened to Ambribrand and Twizz? They applied for grants. They were in UK, okay, in England. They applied for grants to build an observatory and they did not get their money because the referee says, hey, it's stupid, because of course photons are independent. They are not going to have this bunching factor. And so how might independent particles be bunched? No, they are not. The referee says, okay, we don't fund you. This is why they immigrated to Australia. And there are two interpretations to the fact that they were funded in Australia. Are there Australian people in the room? No. There are two interpretations. The first one, okay. The first interpretation is that quantum mechanics was not yet arrived in Australia. So they did not know about photon and the second interpretation is that they were very smart. So you choose. Anyway, the Australian funded them and they built the observatory and they fund, they use it, okay. Of course, before immigrating to Australia, they tried to argue with the referee, as we all do, okay, when referees are not fair to us, we try to argue. So the first thing they did was a small tabletop experiment I have described. They say, hey, look, there is, okay. And then they made basically the reasoning that I have made here. But they added the fact, you know, light is both a wave and a particle, so in a sense, et cetera, et cetera. And uncorrelated detection can be, well, they gave a very nice interpretation of this formula, which is the following. They said the first factor here is the one that you would get if you had really independent particles. This is what we call the shot noise. The shot noise is what you have when you have many independent particles. And they say the second term is the fluctuation that you can represent by the following model. Let's suppose that you have many independent waves. Each pair of waves give a big note. And so you have an addition of many incorrect big notes. And when you do the correct calculation, you find it did this second term, okay, big notes of random waves. And the reason why I like to cite this formula is because my hero, which is your hero also, wrote this formula in 1909 in the famous Salzburg conference, 1909, okay. Einstein was a young physicist. It was his first appearance in a real serious conference, okay. He was probably still an employee of the patent bureau in the bourbon. And in 1909, Einstein, who had invented the concept of photon in 1905, was obviously obsessed by this question, okay, we are photons, but we also have interference, okay. He could not dismiss a young Fresnel and all these people. And he was obsessed by that. And he came to this conference in Salzburg and presented several reasoning saying that light is both a wave and a particle. So, he really invented the concept of wave particle duality in 1909. And he explicitly wrote this formula for fluctuations in the black body radiation, in the thermal radiation, okay, Planck formula, et cetera. But he could write this. And he correctly interpreted this formula. In fact, okay, well, I should depart her to Australia, but I wanted to say, no, voilà, no, no, say, Einstein. What he did, what he did was calculating the fluctuation of the pressure on a wall due to black body radiation, okay. And he found the formula, which is his formula. And he correctly interpreted the result in saying the one is a fluctuation that you have if you have many particles arriving on that. So, you have a randomness, okay, a Poisson process, et cetera, and you have some randomness. And the other term is a term that you have if you consider as many waves with relative phase, which are random variables, big notes, et cetera, this is absolutely fantastic. I mean, we never recognize enough the genius of Einstein. We always discover new things in all the papers of Einstein. Okay. So, I recommend this and there is a book which explains that very well. Okay. Now, good as it was, the defense of Amber Ibrahim Twizz was not fully convincing from a fully quantum point of view. And the first really convincing description of the phenomenon in terms of full quantum mechanics and full quantum optics is due to Glauber. Well, Fanot had the preliminary of that, but the clear explanation is due to Glauber. And Glauber presented a toy model. You all know what is a toy model, okay? It's a small model which certainly does not describe reality in all his details, but the basic idea is in it. So, Glauber came with the toy model, which is a following. We have two emitters which are excited, ready to deliver a photon. And we have two detectors which are in the grand state, ready to detect the photon. And then the two photons are emitted and they are detected. And so, at the end, what do you have? You have the two emitters which are in the grand state and the two detectors are excited and when they are excited, the electron is free and then you can amplify it with an electron multiplier and you observe a pulse. Okay. And Glauber said, okay, we all know from the basic principle of quantum mechanics that in order to calculate the probability to go from an initial state to a final state, we must consider all the possible paths and add the amplitudes associated to this path. And in fact, he says there are two possibilities to go from here to there. The first possibility is that the photon emitted here is detected there and the photon emitted here is detected there. So, he represents it by this diagram. But there is another diagram where the photon emitted here will be detected there and this one will be detected there. It's the second diagram. And what you must do to calculate the probability is first add the two diagrams, okay, and then take the square of the modulus of the complex amplitudes, okay. And so, you have an interference between these two diagrams because there is a phase which depends on the distance of the emitter. So, when I am going to have many pairs here, these various interferences will produce a fluctuation and it's not very difficult to find that this fluctuation is a Gaussian fluctuation and you get the factor of two. Why is it important? It's important because to my knowledge it is probably the first time when somebody explicitly invokes interference of two photons. I don't know an example in the literature of somebody explaining an effect by saying what I have is an interference between amplitudes associated to two photons. It's one two-photon amplitude, another two-photon amplitude, they are complex numbers. I add this complex number and then I take the square modulus, okay. So, this is what we call this two-photon interference effect, quantum weirdness of the second kind that is to say the one that Feynman discovered, well discovered, realized that it is important in the 1980s, okay, and is related to entanglement. Now, again, I will cite Einstein. We have to understand, because this is a referee who says, look, we have independent photons, et cetera. They are right, the photons are independent. So, we have to understand that this effect, which is called the bunching effect, it's not, it's a lack of statistical independence, but there is no real interaction. We should not think of it as the two-photon having some kind of attractive interaction between each other. It's just like both Einstein condensation. This is why I cite the letter from Einstein to Schrodinger about what is called both Einstein condensation. Both Einstein condensation is a fact that when you have bosons who like to be in the same quantum state, the statistics is different. And when Einstein published his paper in 1924, Schrodinger sent him a message in saying, you have made a mistake in your calculation. And Einstein said, no, no, no, I don't have made a mistake. It is true that everything happens as if the bosons would like to be on top of each other, but don't think of it as a real interaction attracting them. It's the way we describe the problem with the different statistics, or here with complex amplitudes, which at the end makes it look like if they were attracted towards each other. It's extraordinary subtle, and it was understood again here. So it's the first time, the Ambulance with Effect, when somebody invokes interference between two photons. And more than that, Glauber develops the full formalism of quantum optics, which will be available for future discoveries. So it's kind of ironical. So there was terrible and hot discussion. Claude Coentanogee was a young researcher, and he went to conference and told me, Glauber would shout, and then Mandel and Wolf would shout. There were terrible fights in conferences about that. So it was a really hot discussion, and when laser was invented, there was a terrible discussion of which you can find the trace in physical letters. I mean, when you look at the introduction of the physical regulators about these subjects, they literally insult each other. Literally, you read the introduction and say, they have not understood anything. Okay, this is a real explanation. So what was the debate? Some people thought that the laser was invented in 1960, and some people thought that with the laser, it will be easy to observe the Ambulance with Effect. And Glauber, who had made the correct theory of the Wolf Effect, immediately realized that it was not the case, and that with a laser, you would not have any Ambulance with Effect. That is to say, in a laser beam, the photons are really independent of each other. So there was an experiment which showed this. This is a single-mode laser. This is a chaotic light, and you see the difference. Single-mode laser, the G2 function, is one all the time, while with the other, with a chaotic light, then you have a factor of two. And again, this is extremely easy to understand in a classical vision. And in the classical vision, the idea is the following. Laser light is a light of which the amplitude is extremely stable. There is an extremely tiny fluctuation, which is very small. The only significant fluctuation in an ideal single-mode laser is a fluctuation of the phase. And of course, this fluctuation of the phase is enough to provoke a line width, because if the phase changes, the instantaneous frequency, which is a time derivative of this whole factor, changes. So you have a laser line width, okay? But this laser line width is due to phase fluctuation and not to any amplitude fluctuation. This is negligible. Well, negligible. Well, don't look at that. This is technical. So if really you can neglect this term, then when you take the square modulus of that, you have a constant intensity. Although the phase fluctuates, the intensity is constant. There is no big note, because there is only one wave. And if the intensity is constant, the probability of photo-detection is constant, if the probability of photo-detection is constant, G2 equals 1. And we say the photons in a laser beam are really independent, okay? The quantum description, basically, is a technical way of rephrasing this classical explanation, okay? So what is the conclusion? The Ambrébrantou's effect is a landmark in quantum optics, because although it is easy to understand if light is described by random electromagnetic waves, it becomes a subtle quantum effect if light is described as made of photon. It's an interference of two photon amplitudes. And Glab had to develop the full formalism to describe it. I would not say he started from a standard formalism of quantum physics, but he really adapted it specifically to the case of light, these quantum optics. And of course, from this, we understand that it is an intriguing quantum effect for particles, okay? And there is a first example of that, which is described by Gordon Bain. You may know he is a famous theorist. You may know him or may not. It depends on the communities, where he explains that you can use the Ambrébrantou's effect for knowing the size of the cross-section of collision between nuclear particles. Rather than looking at the size of a star, look at the size of the cross-section of collision of particles. It works, but anyway. And in our lab, we have a question, can we observe the Ambrébrantou's effect with atoms? Because with particles, it's a really interesting effect. So of course, I will not talk about all I have planned, but I prefer to go slowly, unless you tell me that I am too slow, but I prefer to explain slowly one thing, that not explain well many things, okay? So if you agree, I continue with this speed, which is not very fast, but I am using ultra cold atoms, which are very slow, so I'm contaminated. Okay. Oh, let us skip that. This is technical. So how are we working with atoms and doing quantum optics experiment with atoms? The key ingredient is metastable helium, and I will explain you one. Let us come back to quantum optics. The development of quantum optics is totally linked to the invention of single photon detection, the photodeplier, the fact that you can detect one photon, then the next, then the next, and look where and when was this photon detected. And then you look at the correlation between two detection events, et cetera, et cetera. What we have here is metastable helium, is an atom that we can detect individually for the following reason. Helium is an atom with two electrons, and I'm pretty sure you know that the electron has a spin. So you have states where the two spins are parallel and states where the two spins are anti-parallel. These states are called triplet states because the spin is equal to one, and then you have two possible projection one zero minus one. While when the two spins are anti-parallel, there is only one state, so it's called singlet. Now the important thing is the following. You cannot go from a singlet state to a triplet state by absorbing or emitting a photon. It's called the selection rule, and the reason if that's a strong transition with light are due to electric dipole interaction, and when you flip a spin, which is a magnetic moment, the electric dipole plays no role. So the probability to go from a singlet to a triplet state by either absorbing or emitting a photon is extremely weak. It's not exactly zero, but it's extremely weak, which means that in fact you have two series of levels for helium. You have the levels which are singlet, which are called like that, and there is a full series of levels that I have not represented here, and then you have a full series of levels which are triplet, and they don't all to each other, and it turns out that the lowest level which is a triplet is 20 electron volts above the real ground state. So now this is a story. If I start with an atom in any of these triplet levels, by spontaneous emission, it will arrive here, and then I can excite it with a laser, etc. And for all practical purposes, this atom is just like an atom with a ground state here, and I can use all the technique of manipulating the atom with laser to decelerate the atom, stop it, trap it, etc., playing onto these series of levels. So this is called a metastable level. It's metastable because it's almost stable, but the lifetime is, I don't know, hours. Okay. But the real ground state is here. Now let's suppose that this atom comes close to a piece of matter. What happens? When you are close to a piece of matter, you have an electric field which is responsible for the Van der Waals force. You have some kind of electric field. And the atom, sorry, the electron is moving rapidly in this electric field. And when an electric charge moves rapidly in the electric field, in the frame of reference of the charge, there is a motion magnetic field. And so because of this motion, there is an effective magnetic field which can flip one of the spin. And if you flip one of the spin, then you get in a singlet state. And now there is 20 electron volts available, and 20, well, 19.8, almost 20. 20 electron volts is much more than enough to ionize anything. To extract an electron from a piece of metal, you need five electron volts. To ionize an atom, a few electron volts. To ionize a molecule, a few electron volts. So as soon as an atom in a triplet state comes close to a piece of matter, there is an electron ejected. So we can extract an electron from metal, which will allow us to make single atom detection. Because remember, what is a photoelectric effect? It's the fact that when a photon arrives on a plate of metal, one electron is ejected. We have exactly the same process. So we can use the same trick that is to say accelerating this electron, bumping it on a dinote, and having a multiplier application. And at the end, we have a pulse of 10 to the 10th electron, and this we can observe. So these are how it works. This is an artist's view. We have our metastable helium atom, which are trapped here with coils, and they drop. And each time they arrive on a piece of metal here, boom, there is an electron which is ejected. In fact, this is not a piece of metal. This is what we call a micro-channel plate. It's millions of very small tubes, which pass through the plate. Each of these tubes is an electron multiplier. You have 2 kilovolt voltage difference between the upper part and the lower part. And when an electron is emitted here, it bounces on the walls, and there is a multiplication factor. Each time it bounces on the wall. So if you have a factor of 3, if you bounce 10 times, you have 3 to the power of 10, which is a lot. Okay? It's probably like 2 to the 5th or something like that. Okay. So now what we have on the other side, we have a macroscopic pulse. Each time an atom arrives here. We have done yet better. And this is even better than photomultipliers. Because what we have done is rather than just registering the fact that we have detected one atom, we can tell where the atom arrives. Because here, we have delay lines, and when the charge arrives here, it splits along delay lines. And by looking at the delay at the end of the lines, we can tell where the atom was detected. So now we have a detector, which is allowed to tell us where and when the atom was detected. And if you think of it, in fact, you discover that our two-dimensional detector is, in fact, in our experiment, a three-dimensional detector. Why? Because we have cold atoms here with very small initial velocity. You drop them. When they arrive on the detector, all of them have about the same velocity. So by looking at the time of arrival, you can reconstruct the third dimension. So in fact, this allows us to describe the full distribution, volumic distribution, 3D distribution of the atom just before they fall on the detector. Whoops. So we have repeated the experiment, which is not exactly the amber-ribarine twist, but very similar to the amber-ribarine twist, in the following way. We have a trapped cloud of cold atoms. We drop them. And when they arrive here, we detect all the arrival position and time. And so we have a set of points here. And with this set of points, we build the correlation function. That is to say, the histogram, what is the number of pairs which are separated by this vector in the three-dimensional space. And the result is only noise. Most of the time is zero. From time to time, it's one. So we repeat the experiment and we build the histogram again. We repeat the experiment. We build the histogram again. And at the end, we make the average of the histograms. And when we make the average of the histograms, this is what we get. And again, I repeat, we don't average single events on each pattern. These are fantastic experimental techniques. You have noise. You look for a correlation in that noise. It's still noise. But if you average the correlation, then you get a signal. If you average independently, it doesn't work. Okay, when you think of it, it's trivial, but it's beautiful. Now you see a second curve. And the second curve allows me to say why atoms are even more... Sorry for that. The important difference between photon and atoms is that all photons are bosons. All photons are bosons. Why for atoms are bosons and are fermions. For instance, in the case of helium, helium-4 is a boson, but helium-3 is a fermion. And if I repeat the reasoning of Humbury-Brown to his with fermion, when I add the diagrams, I must add them with a minus sign, which means that the interference, rather than being positive, will be negative, which means that the probability of two fermions to be on top of each other is small, is null, but nothing is perfect. This is well known. It's called the Pauli Principle. But you have it directly as a consequence of the same reasoning that I did. And we have done a fantastic experiment in which we have a sample, in which we have helium-3 and helium-4. They are the same temperature, nano Kelvin temperature, etc., same size of sample. And we can at will decide to drop either helium-4 or helium-3. And so we can compare only the statistics. All the other parameters are the same, and we see the difference. Now, I have a joke I like to make. So I tell you, it will be a joke. Here, the size of the hole is different from the size of the bump is different from the size of the hole, which is here. Why is it so? Because helium-3 and helium-4 do not fall the same way. Yes. Now, objection, objection. Galileo told us everything fall the same. Okay, let us me explain better. They indeed arrive with the same velocity. But because they have different masses, the de Broglie wavelength are different. And the size of the hole can be related to the de Broglie wavelength. So in the quantum sense, they don't fall the same. They have the same velocity, but the de Broglie wavelength are not the same. Well, I like to provoke people. When they are Italian people, usually they defend Galileo, okay? And I defend the boy because I'm French. Good. So the previous one was ratio is 4 to 3. It's not related to helium-4 and helium-3. Yes, the ratio is 4 to 3. And helium-4 and helium-3 is the same as 43, yeah? Yes, because the de Broglie wavelength is inversely proportional to the mass for the assembly. The curve goes up and down because it depends on fermions and bosons. No, okay. Up and down because fermions and bosons. But now the width, look at the width here. The width is bigger. And the width is bigger because the de Broglie wavelength is bigger, by a factor 3 to 4. And I should ask that the contrast is less here, is bigger here, because when you make a convolution, the visibility, the contrast of the convolution depends on the relative width. But this is technical. So, no, I can talk for hours, okay? So at the end of the day, the amber-brown effect with fermion, which can be interpreted by subtracting the diagram which is here, leads to the conclusion that the average of the square is less than the square of the average. And this is not classical at all, okay? This, of course, formula is stupid if it is classical numbers. But in fact, in real quantum physics, here I must use second quantization, so I must use operator of creation and annihilation. And if I replace n by creation, annihilation operator, then because of the commutation relation of fermions, this becomes possible. So from this point of view, if you compare for photons and for fermionic atoms, for photons, of course, I can describe it with quantum physics. But if I insist, I can describe it with classical physics. While here, there is absolutely no classical explanation. So in this sense, fermions are more quantum than bosons. I think that it is true because, in fact, all classical waves, as we know them, get their properties from the pact that you can make many bosons in the same wave, in the same mode, elementary mode of the electromagnetic field. And at this point, you get a macroscopic wave. You cannot do that with fermions. With fermions, when you have put one in elementary mode, that's the end of it. Okay, I will describe the angle-mandole effect, and I think it will be enough. The Ambré-Brain-Touis, the angle-mandole effect was discovered in 1987. And you have a complicated experiment, but you can simplify the skip to understand better. And the idea is the following. Here, you create pairs of photons. And the two photons are well-isolated here with diaphragm, et cetera. And you join them, you combine them on a beam splitter here. So you have a pair of photons, and you recombine them on a beam splitter. And now you have a single photon detector, and you look for correlations, so you look for coincidences. And you look at the probability to detect one photon here and one photon there. And the fantastic result, which was observed by Hong and Mandel, is the following. If the beam splitter is at a position such that the two trajectories here are exactly equal, so that the two photons arrive exactly at the same time as the beam splitter, then the probability of joint detection here and there drops to zero. Which means that in this situation, when the two photons arrive here, at exactly the same time, either the two photons will be detected here, or the two photons will be detected there. But you will never observe one photon here and one photon there. This is really mysterious. If there were classical particles, of course, you would have one for probability to have two here, one for probability to have two here, and one half to have one on one side and one on the other side. And if now you displace the beam splitter so that the time is not exactly equal, then the probability becomes large. So it's when the two photons arrive exactly at the same moment, then the probability drops to zero. This is a fantastic quantum effect. And when you read the paper of Mr. Hong and Mandel, they don't even mention the fact that it is a subtle quantum effect. They are extremely proud because they can measure the width of this dip with an accuracy in the femtosecond range. And there is no detector able to see that. And the way they find femtosecond is just that they push or pull the beam splitter with a transducer. And with this transducer, you can control a fraction of micrometer, so you can control a femtosecond. And they're extremely happy because they have a resolution of femtosecond. And one year later, they publish a paper and say, oh, by the way, this phenomenon that we observe is a very nice quantum effect. It's amusing. Okay. So why is it a nice quantum effect? Because the only possible, there is no classical explanation of this effect. The only explanation is again to say, I have two photons and at the end of the day, I want to look for the correlation. And I must look at all the amplitudes for going from an initial state to a final state. And here are the two processes. The first, the photon going up will be detected up, and the photon going down will be detected down. But of course, there is another possibility. The photon starting up will be detected down, and the photon starting down will be detected up. I must add the two amplitudes. Now, it turns out that the two amplitudes have an opposite sign. And the reason why they have an opposite sign is the property of the beam splitter, which is not a technological property. It's the fact that if you have an ideal beam splitter, the, maybe you call it the S matrix, but the matrix expressing the output as a function of the input must be unitary in order to provide energy conservation, or probability conservation, or whatever conservation, the fundamental conservation. And because of this property, which is very fundamental, there is a minus sign between the coefficients that you have here and the coefficients that you have here. So the two amplitudes that you have here have opposite signs, which means that they cancel and lead to zero. And so the probability of detection, which is the sum of these two amplitudes squared, the sum is zero. So there is a spectacular evidence of two photon interference. And again, there is absolutely no classical model for that, because of course, if it was classical particles, they would be split between Let's suppose you have a model when you say, I have a particle which has 50% chance to be reflected and 50% chance to be transmitted. Of course, what you get is one quarter for both here, one quarter for both here, and one half for one on each side. But with classical wave, it's a little more subtle. You can think of a classical wave arriving here and a classical wave arriving here. And then by tuning the phase, you can convince the two waves to go out here, or the two waves to go out here. But the point is the following. You have also a randomness in the effect. So you must add a fluctuation in the phases. And when you add fluctuation between the phases, you find that with classical wave, you can indeed have a deep, but the deep can never go below one half. So there is no classical model for rendering an account of something going like that. I would like to add something which is extremely subtle from a theoretical point of view. This phenomenon can also be observed with independent photon, because in the experiment of Ongo and Mandel, the two photons are produced from the same source. They are produced simultaneously. But if you take two photons coming from two distant galaxies, provided that they arrive at the same time, by chance, but it may happen, provided that they arrive at the same time on the same spatial mode, with the same frequency, etc., then you will have the same effect. So it's really due to a quantum statistical effect, again the fact that it is boson and you have to symmetrize the quantum state, etc. So the experiment has been done, not with distant stars, but with two atoms, by Grangier and Brouès. It has been done with two atoms held in optical tweezers, that is to say you hold one atom here, you hold one atom there, you excite them so that they emit randomly spontaneous photon, then they recombine it on a beam splitter, selecting the mode, etc., and by chance, from time to time, the two photons arrive exactly at the same moment, and then they observe indeed that either both go on one side, or both go on the other side, and never on both side. So this effect is really a purely quantum effect. So can we observe the Hongo-Enlander effect with atoms? The answer is yes. We succeeded a few years ago. I will give you little information about how we do the experiment. First we want to create pairs of atoms in order to test the effect starting from pairs. Why is it better to start from pairs? Because if you rely on random arrival, it takes forever to, by chance, have particles arriving at the same moment. Well, if they are emitted at the same time, it's very easy to recombine them at the same time. So this is why Hongo-Enlander did it with a pair of photons emitted by a source, and we want to do the same with atoms. So we want to create pairs of atoms. So to create pairs of atoms, we collide two Bosan-Schenkonensates, and because of conservation of energy and momentum, if we work well, because of conservation of energy and momentum, when we collide the two Bosan-Schenkonensates, they cross each other, then we create pairs like p3, p4, with p4 equal minus p3, or p prime 3, p prime 4, etc. And in order to observe it, what we do is, again, we use our wonderful detector, which is in fact a three-dimensional detector. So we make the collision, and we observe the ballistically expanding cloud, and because we are far from the source, what we observe here, the position that we observe here, in fact, reflect the distribution of velocities at the starting point. And so this is a system passing through our detector here, and you see that there is a sphere with a shell, and this sphere with a shell is all these atoms which are due to the collision here. I can probably show it, yes, I can show it here. My students have prepared a beautiful image, just like in CERN, okay? So what you observe here is the remaining Bosan-Schenkonensates. This is not an artifact, but a supplementary thing that we have, and I don't want to comment it, and you see we can have everything rotating, so it's clear that we have atoms on a shell, which correspond to atoms produced by this collision. Now this does not prove that they are emitted by pairs. So to showing that they are emitted by pairs, what we do is we look for correlation in antipodes, and when we do that, indeed, we find peaks. So now we know that we produce the atom by pairs, like that. Anyway, it's not ideal for doing the next experiments, because the atom being emitted in all directions in space, we would like to convince them to go in a well-defined direction. In optics we know how to do it, in non-linear optics, by the so-called phase matching condition. So we looked for a condition equivalent. In our collision we would like to have a process forcing the pairs to be emitted along well-defined directions with well-defined velocities. And we succeeded using a process discovered and explained by a nice Danish and a smart Danish theorist, Klaus Malver. The idea is the following. If we force the atom to circulate, so this gives you an idea of the subtlety of the experimental technique we use. First we force the atom to circulate in a double-roll wave guide. That is to say, it's a pompous word, but it has a truth in it, to say that we have a laser beam which attracts atom towards the center of the beam. So transversely it's just like a harmonic oscillator. And in fact the velocity, the transverse velocities are so small that the atoms are in the grand state of the transverse harmonic oscillator, which means that all the transverse motion are frozen. We are in the lowest state of the harmonic oscillator. So the atoms are forced to circulate along this direction. Along this direction we impose with a standing wave a periodic potential. So now we have something like electron circulating in a periodic potential. And this is a well-known problem, block waves, etc. It's one of the most important tools in condensed matter physics. And what you have is, it's a fully quantum problem, and the solution is dispersion relation, that is to say energy as a function of the momentum, which has a shape like that. So which is far from trivial. Now if we have a collision between two atoms, with such dispersion relation, and we impose conservation of energy, and we impose conservation of momentum, for a given initial pair of atoms which collide between each other, there is only one solution. Which means that now if I have a given velocity of my atom in this periodic potential, two atoms can interact with each other. It's dynamical instability and produce two atoms with a given energy which is fixed by this condition. So I know now, this is what is plot here, it's a paper that was published, we know how to choose a parameter to produce two atoms with a well-defined velocity. So now we will do the experiment. We will start with these two atoms produced with a well-defined velocity. And rather than describing it in the usual space with parabola etc., it's annoying. I will put myself in a frame of reference which is following the parabola of the center of gravity, of the barycenter of the two atoms. So the scheme will be this one. At this time I will reflect the two things. And at this time I will recombine the two things. It would be a little long to explain to you how we reflect atoms and how we make a beam splitter. In fact for people who have an idea, we use a light standing wave which imposes a periodic potential. And the name of the game is Bragg diffraction. By Bragg diffraction we can take this and reflect totally. If we apply it for half the time we have a beam splitter. We reflect only partially, we transmit only partially. So here we can totally repeat the scheme of Humbry-Brown of Hongo and Mandel. And so this is a usual explanation. And what we observe is indeed the Hongo and Mandel deep with a visibility better than 50%. So there is no way to explain it by any classical image. And here I'm really exaggerating because what I call a classical image would be two de Broglie waves and deferring. But two de Broglie waves in our ordinary space. It cannot be two ordinary de Broglie waves in our space. It can only be these two atom amplitudes in an abstract Hilbert space. Okay. So maybe it will be time to stop and to let you ask some questions because I have another subject that I'm afraid it's a little too long. So what we have shown here is again a remarkable example of quantum mystery of the... No, I will say a few words. So when you observe a situation of entanglement like that, you can ask yourself the question, if I have an entangled state like that of two particles, could I test Bell's inequality with it? And of course, it is what we want to do. And the answer is we don't have enough here for the following reason. We have two modes only. Our two particles after the beam splitter can only be in this mode and in that mode. And for a test of Bell's inequality, for people who know about Bell's inequality, you need four modes and I will explain you why. If you take the standard experiment of test of Bell's inequality with photons, okay, what do you have? You have a pair of photons, one photon to the left, one photon to the right, but for each direction there are two modes, polarisation, vertical and polarisation horizontal. And same here, polarisation vertical, polarisation horizontal. So this entangled state is an entanglement between four modes. And you need these four modes in order to have something here, which is a polariser, that you can rotate and which explore different combinations of X1 and X2. If you cannot change these, explore these various combinations of X1 and X2, there is no way to test Bell's inequality. Bell's inequality is not just one measurement. You must compare several measurements, okay. And for that, again, you need two modes on one side and two modes on the other side. So in the experiment that we have done, we look for interference between this and between that, it's only two modes. But something fantastic happened in our experiment. It turns out that with our beautiful detector, we record everything. Just like in CERN, you know, in CERN they record everything and then they look for events, okay. We record, well, we are not as big as CERN, we don't cost as much. And we are less people. But anyway, we record everything. And then we are the PhD student, what a very good idea. He looked into the recorded data and said, but in addition to pairs like that, there should also be pairs like this one. So you see, this is position as a function of time. So it's different velocity. And there should also be pairs like that. Couldn't it be that we have simultaneously a mission of this amplitude and that amplitude? And by processing the data, I don't go into the details. We found that, yes, there are certainly pairs, P minus P, with many different values of P. So can't we have a superposition of two things like that? These would be really entanglement in formats because I would have P minus P, P prime minus P prime. So we discovered that full chance we have not anticipated it. In the scheme that we had, if there was a pair P minus P and a pair P prime minus prime respecting some condition, which is here, which is related to the Bragg condition. I don't go into details. It turned out that the experiment which had been done for Hungary for ongoing Mandel, that is to say, applying a first standing wave and a second standing wave, would recombine this and that. And this is exactly the scheme for testing Bell's inequality. But we miss something. The relative phases between the two things are fixed by the experiment. We don't have a knob to change this because we did not plan it. It was there. So anyway, we have looked into the correlation associated between various diagrams like that. And what we have found, whoops, whoops, okay, and what we have found is the correlation coefficient which for a first set is different from zero. From another set is different from zero and for that set is equal to zero. The fact that we have this and that shows that we have entanglement for four modes. But I am a serious experimentalist. I don't claim much when I have something with one standard deviation, okay? So we publish this and we say that the results that we observe are compatible with existence of entanglement, but we must improve the experiment in order to prove it, and this is what we are doing now. But the important thing is the following. I have been working about Bell's inequality for more than 30 years. And for many, many, many years, I was deeply convinced intuitively that violation of Bell's inequality is stronger than the Hong, Wu, and Mandel effect. But I could not exactly know why because we have done this experiment now I know why. Bell's inequality violation is entanglement between two particles in four modes. Hong, Wu, and Mandel is entanglement between particles in two modes. And for the Hong, Wu, and Mandel, I can invent a classical model that with hidden variables, but I can invent a classical model that will reproduce a result. For Bell's inequality violation, there is no classical model to reproduce the result unless you accept nonlocality, which is another story. So what I want to show you is that by doing experiments, people like me are forced to think deeply about impression and tuition that I had. I had the feeling that because we have an experiment, I am forced to think deeply, and so I think it helps to better understand quantum physics. And when you better understand quantum physics, maybe you can develop applications and there is a world of quantum information of which you hear a lot. There is a lot of hype too much, but there is also something in quantum information, and I think that quantum information really benefits a lot by this kind of reflection. So I think I will stop here because you are probably tired, but I am open to any question you want to ask. And by the way, the fact that you have these p minus p, you have these states with correlated things, it's what Einstein, Podolski, Rosen were imagining. They were using special states which were correlated with p. Could you then realize what they wanted to show you could measure positions and noncommutating things, just from the conceptual point of view? You were right, of course, until a certain point, but it's not exactly the EPR wave function anyway. But the second, the most important point is in the EPR situation, you have continuous variables, and it's correlation between continuous variables. But as far as I know, to test Bell's inequality, you must have discrete things. And our trick here is just like with the photons, we select two modes and another two modes, and now we have four modes. And I think that until the moment when you have four and only four modes, you cannot test Bell's inequalities. So it would be amusing to do that. But of course, because of my personal history, I think that Bell's inequality, Bell's inequality violation is really the most quantum thing we can think of. I think that the real value of the EPR reasoning paper, whatever, is to draw the attention on the fact that two particle quantum effects is dramatically different from single. And if we think of the history, we all know of the Solve conference of 1927, when Einstein was coming with an objection to ball and ball would reply, and the replies of ball were fully convincing in that case. And all these attacks of Einstein were based on quantum effect of a single particle. He was dealing with Heisenberg, it was always a single particle. And he always failed in his attacks with single particle. But when he came in 35 with two particles and with entanglement, then we must say, honestly, the reply of Boer to Einstein is not very convincing. Okay? So the beauty of the EPR paper is to have shown that when you start with two particles, then you have a different problem, totally different problem. But then we must give credit to Bell for coming with a really quantitative distinction between the classical world and the quantum world. And even Feynman in 1982, when he realizes that entanglement is different, he writes something which is like Bell's inequality. And of course Feynman does not cite Bell because Feynman revents everything, so he does not cite anybody. But we can forgive him. So why are you so reluctant to this last experiment? I'm not reluctant. Why am I reluctant? I'm going to tell you, but don't repeat it. In 2015, there were three experiments which made a lot of noise about so-called loophole-free tests of Bell's inequality. Two of them gave a violation of Bell's inequality by 10 standard deviations or something like that. But the third one, which made a lot of noise because it was published as the first one by Nature and Nature made a lot of noise about it, etc., violated Bell's inequality by two standard deviations. As a reasonable experimentalist, I would never claim something seriously if I don't have five standard deviations. In CERN, when they have a bump, when they have two to three standard deviations, they leak the information that maybe they have something. When there is a bump with a signal to noise ratio corresponding to five standard deviations, then they publish. So because I show only one standard deviation, I say my result is compatible with, but I don't claim anything more like that. I'm deeply convinced that in a few months we will have a result with a large enough number of standard deviations, and in addition, we will have a control of the relative phases. So thank you to ask a question. I was asked to write, now it's a trend to have comments about the paper. So I was asked to ask comments about these three experiments in 2015, and I wrote a comment, not in nature, because I was not happy with the way nature processed all that. I cannot say details because I am a referee, an anonymous referee, so I should not give you details, but I was not happy about the way nature processed all that. So I wrote a comment in American Physical Society Journal, which is named Physics, where my position is very clear. I put exactly what are the standard deviations in each experiment, etc., etc. If you read me between the lines, you understand that the three experiments are not at all of the same quality. Okay, on the second hand, I would be really interested to see real applications of all these different... Oh, the application is quantum simulation. For instance, my colleague Antoine Brouès, which by the way is a former PhD student, he has now a network or, I don't know, a lattice of 50 atoms. And for these 50 atoms, he can create entanglement at will. So it is a problem which is a many-body problem for which you cannot do any calculation with a computer, because the size of the Hilbert space, the many-body space describing that is, as you know, there is a exponential explosion, etc. Curiously, I have not looked into details, but I trust Antoine Brouès and the people who work about that. It seems that this kind of thing can not only help to understand problems of condensed matter, but there is much better than that. It seems that there are optimization problems, like the traveling salesman problem, or stuff like that, which could be solved with this kind of quantum simulator. And for instance, I understand that EDF, or the Electric Company, are subsidizing this kind of research because the optimization of the grid is a big problem for them. So I don't say that it works, but these are kind of ideas that people have. Yes, exactly. Yeah, yeah, exactly. Don't ask me details, but serious theorists claim this kind of thing. And, you know, the most obvious thing is to say, I can treat, this is what I do, for instance, I do Anderson localization, okay? Anderson localization is a problem of condensed matter physics. As soon as you pass a certain degree of complexity, three-dimension, etc., there is no exact theory for treating it, and there is no computer big enough with a big enough memory to make a numerical calculation. We do the experiment with ultra-cold atoms replacing electrons. So I simulate a lattice with standing waves of laser. I put ultra-cold atoms, which simulate the behavior of electrons, and I look at what happens. So it's, I don't know, wait, simulator. It's analogous calculation. But it's analogous calculation and incitation where we know that no computer ever will be able to do it, because it would demand the number of bits as large as the number of, I don't know, atoms in the universe or something. I know people write big numbers, etc. So the idea is to try to do something. But the question of traveling salesman of optimization of the grid seems to be serious enough that there is money invested on that. Don't ask me more. There is, of course, quantum cryptography, which is based on, well, the safest method of quantum cryptography is based on entanglement. There are methods which are not based on entanglement, but they are not as safe. This is our changes writers. Yes, this hype. I have already, from the text that they publish, you cannot guess what it is. They speak of subatomic particles. My best guess is that it is just light, and so a photon is a subatomic particle. And then they don't say what they do, but look, in classical optics, well, in usual optics, let's say, we already know how to go to the standard quantum limit, and we even know how to pass this standard quantum limit. So it may be an experiment of that kind. We have an exchange with colleagues, and we have decided that we don't waste our time until they publish a preprint. Okay, when they publish a preprint, it's not a preprint. Okay, we'll see. I think it's part of the hype about quantum information. It's too much. I don't like the way of doing physics, but okay. You know, I've been one of the people who have advocated the emergence of the famous quantum flagship at the European level. I've presented my ideas several times. He knows he was there once in front of the commissioners. I always told them the following. I feel, regarding the universal quantum computer, as I felt 30 years ago, regarding the detection of gravitational waves, okay? No fundamental theory or property or theorem, whatever, property tells you principle. No fundamental principle tells you that it is impossible, but the technological gap looks immense. I feel the same regarding the universal quantum computer. I don't know any fundamental law of physics saying that it is impossible, but I have no idea as we can succeed, okay? I said that several times in front of the commissioners, but some of my colleagues were promising to deliver a universal quantum computer within five years. I can tell you, there will be no universal quantum computer in five years. What we will have is small local applications, microprocessor with a well-defined task, doing a well-defined task here. This, yes, but universal quantum computer, I have no idea. So my personal position is that we have to be very careful about this hype because at one point there will be backlash. So I'm old enough that I will not be hurted by the backlash, but my young colleagues could be hurted. At least they cannot say that I promised too much. I never promised too much. You were there, right? Still China invests 10 billion dollars. I know. And they are good. Sure. No, no, they are good. You know, we'll see. It's a very interesting... IBM claims. Yes, but no, but it's a claim. It's not a universal quantum computer. You know, a good news is that there is a European, a French, but let's say European, company which has taken the challenge. It's ATOS, Thierry Breton. Thierry Breton says there is no reason to live only the Chinese and the American. So there is a quantum computing program with ATOS, okay? And because ATOS is not doing anything in hardware, in fact, it has attacked the problem of software. What he is doing is the following. Because ATOS has now the most powerful European computer, bull computer, not in the world, but in Europe that's the most powerful computer. They can, with an ordinary between quote, but powerful computer, they can mimic something like 30 entangled qubits. And so they say, now, if we add 30 entangled qubits, what could we do with it? And what they come to try to develop is a good interface with a high level language to say what kind of algorithm can we develop if we have a quantum computer? How should we have a convenient language of programming, et cetera? I think it's an interesting, because it is what they know to do. And so I'm happy, I must say, I belong to the scientific advisory board, but I think it's very nice to see a European company taking the challenge to say we are not absent, we are in the game. But of course they are not putting billions, but Europe is putting one billion. Well, Europe is putting half a billion and asking other agencies to put the other half. But it's better than nothing. But it is true that the Chinese put 10 billion. Okay, but Chinese are 2 billion people. Almost. Okay, thank you for your attention, but if you want to chat more, I am open.