 Hello. Well, pleasure to be here. Thanks, Andrea and David. So what I would like to discuss today is a fairly unusual kind of interactions, which can be realized in optics. And they are much longer range than what the previous speaker was talking about. I will make the connection with that. And it has other merits, except for the length of the range. There is a very rich variety of physics which can be associated with it. So oops, what did I do wrong? Yeah, yeah, this. What I'll be talking about is mostly the PhD thesis of F.E.F. Ryan Shachmoul at the Weitzman Institute. He's now at Harvard, still continuing this kind of work. And then another collaborator of ours is Igor Mazitz, a former postdoc and now a scientist in Vienna. Now, what I would like to do is first present the outline, which consists of three parts. The first has to do with the unusual long range vacuum forces, which we can encounter in wave guides, in transmission lines, as I will explain. Then I will move on to long range dipolar interactions, which we find are extremely advantageous in fiber gratings. And there they allow to transfer entanglement in a deterministic fashion, without losses or dissipation over very long distances. And finally, and this is a view as a highlight, I will discuss the possibility of realizing highly known local nonlinear optics, proper nonlinear optics, nonlinear Schrodinger equation by a combination of electromagnetically induced transparency and laser induced dipolar forces. OK. Now, let's move on to vacuum forces. Everyone knows that vacuum forces in the quasi-static limit scale as 1 over r to the sixth. And when they are retarded, they scale as 1 over r to the seventh. But the question is what determines this scaling dependence? And the usual answer is that it is the geometry of the objects. OK. So for example, if you have two planes, their van der Waals interaction scales as 1 over dq. However, there can be another approach. And that is to consider what is the essential mechanism of these vacuum forces. And that is the exchange of virtual photons via, or if you like, vacuum fluctuations. And therefore, to affect the space dependence of these interactions by shaping the geometry of the environment in which the objects are placed. So it's not the geometry of the objects that we're looking at. It is the environment in which they are placed. And in particular, we find that you get very unusual properties when you enclose polarizable objects in a one-dimensional environment in a wavegate. OK. There, you can get giant van der Waals and Casimir forces. So the quite amazing discovery which Effie made was that the very particular geometry that one should look at is the coaxial cable, which everyone knows, everyone has at home. OK. There is the open version of a coaxial cable. That's a Coplaner waveguide. But it's the same thing. You see this thin stripe in between two large metal plates? It's the same. So the claim is that if you place two polarizable objects within such a waveguide, and only such a waveguide, you get a huge enhancement of the vacuum forces. OK. Why is that? That's because they have an amazing mode. OK. That's their fundamental mode. It's the TEM mode. No other structure has this mode. It's a mode which is diffraction less. OK. It has the same dispersion as a plane wave in vacuum. It's normalized by a constant factor. And otherwise, it just propagates as e to the ikz. OK. Nothing else. And you see this is the dispersion. Free space dispersion. OK. If you look at different kinds of waveguides, because they have a cut off frequency, OK, you won't get this kind of dispersionless propagation. And therefore, you will have inevitably diffraction effects. And diffraction effects are the killer for the enhancement or the long range of vacuum forces. OK. It's far from being obvious. OK. So this is the key property, TEM mode, free space in one dimension. Now, you can evaluate vacuum forces between two polarizable objects in such a way by a diagrammatic technique. OK. It's fourth order perturbation theory. If you did that, it's cumbersome but doable. But you don't get much insight from it. But there is an alternative approach, due to Miloni, where you actually pretend as if you have real dipoles and real propagating fields, although these are all vacuum fluctuations. So you say there is a vacuum field which induces a dipole, OK, number one. Then this becomes a source for further propagation of a vacuum field to dipole two, which in turn is induced by these vacuum fluctuations and affects back dipole one, OK. And you do the calculation just using Maxwell equations, nothing else, OK. Once you do that, you realize that what really determines the distance dependence of these forces is the ability to propagate virtual photons between these objects without diffraction, OK. Because then every virtual photon that emanates from one will end up in two and vice versa, OK. So it's extremely simple, but it has taken us some time to realize. Now, OK, so once you do that and you get the same result from both approaches so they check, you can in fact analytically solve for the distance dependence of the vacuum force between two such polarizable objects. You take some characteristic transition frequency in these objects, but of course it doesn't really matter what transition you take because you have to sum eventually overall possible transitions. But there is one say that contributes more than others. And what you find that really the crucial quantity is this small distance which determines the core thickness of the coaxial cable or transition or waveguide, all right? So it's this A. Once you have A and lambda as characteristic wavelengths, you can write the interaction energy between two dipoles within this environment as a function of z normalized by this wavelength. And lo and behold, you obtain completely different scale dependencies than you might expect. In the van der Waals regime for z smaller than this characteristic wavelength, it scales as a constant plus z log z, as opposed to 1 over z to the 6 in free space. That's why I was saying, much longer range. Furthermore, if you go to the Casimir-Polder retarded regime, you find the dependence on 1 over z cubed instead of 1 over z to the 7. And when you evaluate the overall magnitude of this interaction, you find that in the van der Waals regime, at distances that are 1% of this characteristic wavelength, you have an enhancement of the order of 10 to the 7 compared to free space. In the Casimir-Polder regime, the enhancement is even larger. You see, up to 10 to the 23rd, if you have distances on the order of 50 wavelengths. However, the absolute magnitude of the effect becomes very small at such distances, and this is probably not detectable. But this limit we contend is something you can test in the lab. So this is what we have. And what we think is that at present, the best setup for testing this prediction is using two superconducting transmount qubits placed in such a superconducting waveguide because they have giant dipole moments. And this, of course, is what you're looking for. So if you take typical parameters, what you find is that at 1% of their characteristic wavelength, which is this, you get an energy shift of 28 megahertz compared to what you would expect for free space. And this shift is much larger than the line width of the transition. So this is clearly detectable. Actually, John Martinis even promised us to do the experiment before he was drawn by quantum computing. And we never heard from him before. But it's definitely doable. The Casimir-Polder regime is much more challenging because the shift is then smaller than the line width. So unless we are able, technologically, to come up with transmounts that have much smaller line widths, this probably is going to be hard. Not impossible, but hard to measure. So much for transmission of vacuum forces over large distances. Now, let's turn to the interaction of dipoles via what we call resonant dipole-dipole interactions. We've been interested in that for a long time because we always felt that this should be a good mechanism entangling atoms or dipoles over large distances. But there is a problem there. And the problem is the following. That when you have a resonant dipole-dipole interaction, that means that two atoms share an excitation. One atom, say, is excited, the other is not. There is always the exchange of a real photon to worry about because the exchange of a real photon is a dissipative process. You exchange a spontaneously emitted photon, which is then absorbed and re-emitted by the partner. This kind of processes, indeterministic, dissipative, diffusive, what have you, and this is not what you want. This occurs at the resonant frequency of the atoms. But then, OK, so let's look at how this force scales. At long ranges in 3D, it scales as sine kR over kR. But in 1D, it scales as cosine kZ. If you are able to confine your field to 1D only. By contrast, there is the dispersive or coherent part of the interaction, which is mediated by virtual photons. They are virtual in the sense that they are off-resonant with the atoms. Nevertheless, they contribute to the energy shift of the atoms. So all frequencies that are off-resonant with the atoms will contribute to that. They are associated with the real part of the self-energy as opposed to the dissipative part, which is associated with the imaginary part of the self-energy. And in 1D, they scale as sine kZ. Now, unfortunately, both in 3D and in 1D, it turns out that you cannot really get rid of the dissipative part of the exchange of spontaneous emitted photons. With Duncan O'Dell, who was an excellent postdoc some time ago, we did a very interesting work concerning long-range interactions in dipolar gases, which were mediated by these cosine kr over kr terms. However, what was the killer then was this sine kr over kr spontaneous emission term. There is no way to ignore it because it scales in the same way. And even if you switch to 1D, you still face the same problem, whereas the coherent part, which is the proper resonant dipole-dipole interaction, scales as sine kZ, the spontaneous emission part scales as cosine kZ. And there is no way to have one without the other. So it looks as a no-go theorem for a long time. But then we found a solution, again, thanks to Effie's work. And the solution was to have not just 1D, but a grating, which is embedded in such a fiber. Now, what difference does a grating make? The difference is that now you have a band gap in the dispersion. And this will turn out to be crucial. What you then want to do is have atoms whose frequencies are within this band gap created by the grating. If this is true, then you block spontaneous emission along the fiber where the grating is placed. That's the effect of the band gap. Yes, frequencies which are within the band gap are forbidden along the fiber. And this still leaves us with the spontaneous emission of photons sideways into transverse modes outside the fiber. So let us say that the strength of emission or the rate of emission into such modes is gamma-free. It's the same as what you have in free space. You obviously need the interaction, the dipole-dipole interaction that you are interested in. You need it to be much stronger than this gamma-free. And this is what the band gap does for you. Because when you are very close to the band edge, you have this huge upsurge in the mode density, which physically means that a photon which is approaching such an edge will be slowed down at the vicinity of this band edge. So it will spend a long time interacting with the atom and, effectively, enhance the interaction by a factor which is one over the square root of the frequency minus the band edge frequency. So as you approach, it becomes huge. Now, this is the enhancement factor compared to free space spontaneous emission. It's this eta. But it is also the enhancement factor for the length scale of the interaction. Because as you see, within the band gap, the interaction decays exponentially with an exponent psi that is, again, inversely proportional to the frequency distance between the atomic resonance and the band edge. So as a result, you are able, very close to a band edge, to ignore spontaneous emission both inside and outside the fiber and have pure dispersive long range effect which extends as long as allowed by this exponential constant. Yes? Now, first, when we saw this, we thought this was a way of entangling atoms because that's what such an interaction can do for you. They will exchange an interaction in a non-dissipative coherent way. And since you can presumably make this psi as large as you will, you will be able to entangle atoms over enormous distances. But this is not the case. The reason is that the treatment breaks down when you are extremely close to a band edge. Then you will have to resort to a much more sophisticated theory, which is a non-Marcovian theory. It is then the case that the band edge actually binds the photon. All right? This is an effect which we discovered already in the 90s for the binding of a photon to an atom if the atom is placed within a band gap very close to the band edge. The photon then stays bound to the atom and does not propagate. So you will have the same effect here with the result that if the atoms are very distant, they won't be able to exchange a photon. The photon will remain bound to the atom. But still you can have enhancement of the interaction by orders of magnitude. And we have checked that you can entangle atoms with very high fidelity over distances of 100 wavelengths or even more. So let me turn to the final part which concerns the possibility of having nonlinear optics, which is mediated by such long range interactions. Now, here the starting point is the result by flash hour and looking that in a three-level atom with a very strong field operating between these two levels, a probe field will first not be absorbed by the upper level because there will be destructive interference between this probe field and this coupling, yes. But furthermore, there is the possibility of mapping the excitation of an atom or conversely, mapping the field, this probe field, onto an excitation of this D state. So you can create a coherent superposition of G and D at will by tuning the probe field and the strong coupling field accordingly. So this is the first observation. Then when this entity propagates, you get sort of a spin wave because this excitation then will wander from one atom to another. And you will have these superpositions of G and D propagates along the sample. But this is still, at this point, a local effect as far as the atom is concerned. And very early on, we have noted, and this was the work of in Balfriller. She was an excellent PhD student and David Petrosian, who was also an amazing postdoc. We noted that there is the possibility of converting this effect into a long range effect. For that, one would have to resort to Rydberg atom interactions of the kind that we heard about. So the idea is that you first populate this D level by means of the probe field. And then this D level is a Rydberg level with very large polarizability. So if there is a Rydberg atom somewhere in the vicinity sufficiently close to it, there will be a dipolar interaction, dipole-dipole interaction between this Rydberg atom and another one, which, again, has been prepared in the same way. And this will create cross coupling between the probe fields that induce the excitations of the corresponding D levels in the two atoms. This is essentially the care effect but extended to long ranges. So instead of having a delta function here, which is what you have in a care effect, and these are the populations of two adjacent D level atoms, you have here the dipole-dipole interaction, which is mediated by two Rydberg atoms, which scales as one over RQ. Now this actually gives rise to the most effective mechanism to date of entangling two photons. Photons usually are extremely weakly interacting, of course. But in this way, they can interact because they first are converted into an atomic excitation. And this atomic excitation, in turns, interacts with another atom. And then you can convert it back to a photon. So if you have two beams, each associated with an ensemble of atoms of this kind, there will be cross correlation between these fields. And we are pleased to say that this prediction of ours from 2005 was realized in Lukin's group by Ofer Fistemberg, who is now at Weizmann and is continuing this kind of work. So in fact, it has been proven that this is the most effective way of entangling photons because they interact over the longest distance possible. But we are still not completely happy because we are then limited by the interaction range of dipole interacting atoms. And this is number one. Number two is that this is really very effective as far as cross phase modulation is concerned. Yes, you take two beams, and you correlate them in phase. But what about self-phase modulation, which is really the heart of nonlinear optics? The analog of a nonlinear Schrodinger wave equation. And this is what we finally achieved, again, mainly thanks to Effie. So here is the setup we have in mind. It is a combination of all the elements I've been talking about. It's a fiber in which you have a grating embedded. And the atoms populate this grating. They may, but need not be Rydberg atoms. It works also for regular atoms. There is this strong coupling field which is needed to induce this electromagnetically induced transparency in the atoms. And this is the weak probe field. So together, the weak probe field and this omega give rise to electromagnetically induced transparency in the sample. But then in addition, you illuminate the sample by this laser. And as you see, it is slanted compared to the axis. And there is a good reason for that. So basically, as far as the dipole-dipole interaction is concerned, you have the dipole-dipole interaction which is mediated by this laser. The laser excites an atom of resonance, in fact, because you want to avoid dissipation. So you tune it off resonance, but this laser is strong enough. And the length scale of the interaction, this dipolar interaction, is, again, what we find in a fiber, which is essentially limited by this exponent that diverges as you approach the band edge of the grating. So it can be much, much longer range than your usual dipole-dipole interaction. But this is only one ingredient. The other ingredient is that this D level, which was populated by means of this weak probe field and the strong coupling field, has a shift in frequency which is both long range as far as the dipole-dipole interaction is concerned. But it also is dispersive. And this is very tricky. Let me explain what I mean by this. What I mean by this is that you have now to consider what propagates in the sample in the presence of all these fields. So what propagates in the sample in the presence of this EIT electromagnetically induced transparency is a superposition of the probe field and the excitation of the atom, which is sort of a spin wave. It's a cooperative effect. Yes, there is this square root of n, which is the number of atoms in the sample, which makes it large because otherwise this theta is very small. It's inversely proportional to the intensity of the coupling. But this is the propagating entity. And this propagating entity can obey an equation which is completely analogous to the nonlinear Schrodinger wave equation on one condition. This part is just a nonlinear frequency shift, nonlinear because of the dipole-dipole interaction with the other atoms. But there is the dispersive part, which, as you see here, is associated with the effective mass of the propagation. Now, the effective mass of the propagation is determined by the dispersion in the grating. Yes, whenever you have a grating and you have a band edge, you have a massive character of the propagation. But in order for it to be also proportional to the nonlinear frequency shift, which is induced by the dipole-dipole interaction, you have to do something else. You have to detune the dispersion You have to detune the coupling field from resonance, which is usually not done. Now, why do you have to detune it? Because when you look at the dispersion, which is seen by the propagating probe field, if this coupling field is on resonance, is resonant between E and D, you have no dispersion at all. You have this symmetric profile. This is the so-called transparency window. You see the absorption is 0 here. The width of this window is determined by the intensity of the coupling field, the larger the intensity, the larger the window. But the dispersion has 0 at the probe frequency. But if you introduce a detuning, which is a combination of a linear and a nonlinear frequency shift of the coupling, then the probe starts seeing a quadratic parabolic dispersion as a function of its frequency. And this is what you need in order to have a combination of massive propagation. This is the group velocity at the band edge, which is determined by the grating, and this nonlinear frequency shift, which is determined by the dipolar interaction. So all in all, what we then find is a dispersion relation that is modified compared to the so-called free particle Bogolubov dispersion relation that you find when quasi particles propagate in a grating. You find mode frequencies that deviate from the mean frequency by this UK, which is the Fourier transform of the dipole-dipole interaction potential. And this leads to an amazing prediction that there should be an optical rotom, which is analogous to the optical rotom that Duncan found for a dipolar gas. Here it is for photons. What is an optical rotom? It is a minimum of the dispersion at a certain wavelength, which corresponds to an attractive effect of this interaction. If this interaction is attractive, it lowers the dispersion or the energies. Am I? OK, five minutes. All right. So since the energy is lowered, there is a tendency of any fluctuation or any disturbance to bunch around this wave vector. And this wave vector is the difference between the laser wave vector and the grating wave vector. So you can tune it. And here you can see that you get this rotom at a wavelength that corresponds to 1 millimeter effectively. Now, in the same way that you can have a rotom, you can have an anti-rotom, in other words, you can have, instead of an attractive interaction, you can have a repulsive interaction that would tend to increase the dispersion compared to the undisturbed dispersion. And it occurs at the same wavelength. So these are interesting features. And with this I will conclude. The final feature that we are looking at is dynamical or, if you like, dynamical instability. You see, this dispersion relation can become imaginary. When the interaction is strong enough and has a negative sign, omega k become imaginary, this corresponds to an amplification of any fluctuation which would propagate along the sample. And such amplification would lead to an emergence of self-order. So this will be manifest if you measure intensity-intensity correlations at the detector. Then you will see that the intensity-intensity correlations have oscillations with this period of kr that I was talking about, the difference between the laser wave vector and the grating wave vector. And this will be a strong signature of the nonlocality of this interaction. Because if you have local interaction, you will only have this intensity-intensity correlation if z and z prime are the same. So this points out to a possibility of mapping out everything we know about nonlinear optics, both quantum and classical, into this extremely nonlocal domain which can be engineered in waveguide. So the summary is that there is a simple idea that has drastic consequences. The idea is to confine the dipoles in essentially one-dimensional geometry which can enhance dipolar interactions. In transmission lines, coaxial cables, this can lead to an enormous enhancement of Casimir or van der Waals forces. Whereas in fibers, you can enhance dipole-dipole interaction by taking advantage of the cut-offs and band gaps that exist in the presence of a grating that is embedded in such fibers. And this can give rise to either long-range deterministic entanglement generation or to non-local, nonlinear optics in the sense of a non-local, nonlinear Schrodinger wave equation with an enormous wealth of implications, including thermodynamic implications that I have no time to discuss. Thank you very much for your time.