Thank you very much. Thank you for giving me the opportunity to give a talk So this is a collaboration So there are we are half theories half experimentalist So in theory part is Julian Barre is here, you know him a alum of Olivetti Which is a former student of Julien and on the experimental part is David Wilkowski, which Who was before in in East now is in Singapore and Marie-Vonne Charlonie who did the experiment I will explain Just now which which what which who was the former student of of David So the My talk it's divided in two parts first I will explain an experiment which has already been done in in which It involves one-dimensional pseudo gravity So cold in experiment of cold atoms which mimics one-dimensional gravity and then I will explain a Project or an experimental proposal for a For an experiment cold atoms which mimics two-dimensional gravity, so So the motivation of doing that so I think we're here everybody in in a in a long range Conferences, I think we know which is the which are main motivations So there are many systems as for example Julian's plane which presents long range interactions and cold atoms are experimental systems which permits to Manipulate such long-range system doing experiments and playing for example with the parameters and also Looking at the dynamics for example when you look at galaxy the galaxy the time scales Which are involved in a in a galaxy are very very long, so you don't see the dynamics So here you can see the dynamics and also there is a new ingredient here in the second part of the talk We will see that there are forces which do not derive from a potential, so that's interesting from the for the statistical mechanics of such system as They're not well understood and there are other fields in which they have this kind of of Forces for example in self-propelled particles. They have the very simple when you look at the equations They are very see equations are extremely similar so I Will start to explain the this experiment which has already been done and then we explain to you the basics of Of how works This cold atom experiments, so what we use is what we it's called a far of resonance minute of the trap so a there is The trapping of the here is The atoms are here and they are trapped with a single focused a laser Which is here this laser and this this laser induced a depot in the atoms and because of the depots they induce a Potential which which is proportional to the intensity of this of this of the laser beam and This this potential is more or less harmonic and in the experiment that has been done It has been measured in this in this longitudinal direction this frequency of the of the harmonic potential and this frequency in the other Direction so this this this the cloud of the of the of the atoms are more or less one-dimensional It's more or less one-dimensional so Okay, so this is called atom experiments So there is a mechanism in order to cool of the atoms and it's done by doctor effect so The force of an atom is the sum of this F plus of F minus Which are which is this expression This is semi classical formula to compute interaction between the lasers and the and the atoms And what is important in this formula here for us are the terms which are in red So this term is the intensity of the laser which decreases when when the when the photons enters in the of course in the in the in the cloud Here is the velocity of the atoms and Delta is the frequency detuning between the the levels of the of the of the energy levels of the atom at rest and the lasers and So playing with with this Delta if you if you if we plot here in the next slide I will plot F as a function as a function of the velocity for detuning which is negative You see that There's the force okay here the units. I don't I'm not using Physical units the force is a has this fish this shape So you see that for particles we have Positive velocity the force is negative. So it means that they are cool So there is a force in the opposite direction of the velocity And when the velocity is when the particle is going in which has negative velocity is going in another direction The force is positive. So It's also against the it's in the in the opposite direction of the velocity of the particles So this is a mechanism of the other of coding in this kind of experiments So Now, okay, we have the experiment is very complicated and now what the physics quite complicated and now the idea is to Take the main physical to understand what the means the main physical forces which are which act over the atoms so I already spoke about the confining force of the traps this this this With this with this force The atoms remain confined and there's another another force which is called the shadow effect and what The idea is the following so imagine that you have here one laser Okay, I didn't tell you about I just tell you about the The laser which are trapping but there are two other laser which are This one which is going this direction and this one with exactly the same With the same frequency which is going in the other direction. So there are contra propagating lasers And so here we have one laser. We have another laser So what happens when so imagine here we have two atoms to imagine that there is a photon which This which is captured by this by this atom. So this So what it will happen? So after a while this this is the photon will be remitted So if it's a spontaneous emission, it will be in some random direction. So it is only one Okay, and probably it will be interact with another another Another atom so the this this spontaneous emission cause an effective force, which is Coulomb, it's repulsive. This one is repulsive is an effective potential view of R 1 over R as Julien explained in his talk So this is the second force I have here This is the multiple diffusion from spontaneous emission. This is a repulsive column like Because to multiply diffusion, okay So what's this this force is very important in three dimensions But it's not important in one and two dimension because when you have This geometry the photons can escape very easily. So the probability to interact with another atom is very low And the another and the other force is what he calls the shadow effect So if imagine I have one atom here and another atom exactly aligned with the laser if this atom has Interact with this photon, of course, this atom will not interact with this photon Okay, so because of the transfer momentum this atom is kicked. So there's a kick in this direction and Okay, that's what's happened physically. So now imagine that I forget that there is The lasers and I see what happens So if I look what what is happening here, you will say, okay What's happening is this in an attractive force between these two atoms Okay, this is an effective attractive force between the atoms and it is what is called the shadow effect It's just this screening of This atom is screening the lasers and then it's really this atom will be interact with these lasers With this laser and then there will be this This is effective attractive force Okay So I will not enter too much in mathematical details and how we we model this This system but it's it seems it's reasonable to to to to work with the because the system is long-range with lots of equation and With with some noise because of the lasers, okay the lasers Give a put energy in the system and there is also a here hidden There is also a friction because the force here depends on the velocity So in order to to talk in the experiments So when the the atoms are loaded in the trap and after a while they will arrive to a to a to a stationary distribution so one way to by the way to compute the station distribution is to Assume that the distribution of the distribution of function. It's separable So it depends on a function with depends of the position multiply by the function with depends on the velocity so if you do that and You assume also that the optical depth is small the optical depth is measure the here you have the your cloth is How depth the photons can enter in the in the in the system? So how transparent it is so if the system is very transparent you can expand in in The exponential here we have an exponential in this formula So you can expand here for small b and you get Like that So in the experiment B was around zero five But here this is the density so the density is normalized to one. So this this at most zero five so it's zero five zero six, so it is more of a reasonable approximation and When you do okay, I don't I will not Detail too much the mathematics here, but the details But you have two character with scales one which is related with a with a trap The other on the other one with the characteristic interactions, then we are able to compute with with different Scales in the system their value and then you obtain this This equation which is exactly the same equation for the Stationary solution of a cell great I think gas in one dimension so this This is the this is a function sign because in one dimension the force between in one dimensional gravity the force between Two two particles it just the sign because the potential is For 1d gravity the potential v of x is proportional to And so this this equation has this very one solution Which is a solution of the stationaries solution for 1d gravity with this this sec square function Okay So this is the theoretical part so that's relatively easy now the experiment has been done and now the point is to see if really what we have in the system is if this model is valid and We have really in the system in effective interaction, which is Equivalent to 1d gravity So how can we look at to to convince to convince our server that that is the case So we can look at the scaling of the size of the system with a number of atoms So if there is if the interaction is attractive and when you increase more and more the number of Atoms the system size will be we get smaller and smaller We can see also the launch the density profile density profile coincides with the one that they just brought before And we also look can look at the breathing oscillation. So you have the system, which is stationary So you change a little bit the intensity of the laser So you perturb it and then it would try it will start to breath and this frequency of the oscillations is related with with for a for a pair force which is Which is a poor law. It's related with with this exponent We expect to to see a alpha equal to zero if it's 1d gravity So this the experiment it is This is the Longitude of size in some function of time So here in this this blue this blue crew. It's with the intensity of the lasers equal to zero So there is only the trap So what happens so it starts to oscillates and then so the oscillation stops And I think that the main mechanism here is just that the trap is not totally harmonic So it's relaxed because of the unharmonicity and here there are two other cases here the intensity of the laser is different from zero and this one is larger So we see that of course it oscillates, but this is getting The size is much smaller than of course in the case in which there is no There is no interaction and where the intensity is larger and is larger even larger. So I remember you with that Interaction depends on the intensity of the lasers Is here This is the intensity of the laser. So when the intensity is larger the force is larger. It's expected to be larger Here the system is smaller even smaller, okay so this is a measurement of the size of the system The inverse size of the system as a function of the number of particles So and we expect that this this function will Increase with N as N. We cannot We don't know the pre-factor, but we know that if it's long range it will increase with N And we see that this is for the two intensities I of the laser actually before we see that this is the case Another measurement It's possible to do an experiment is a equilibrium profile So this is a measure of the equilibrium profile in the experiment and this is a fit for For the equilibrium distribution Corresponding to different This is a one to different interactions Poorly interactions, okay And for a gravity, of course, it should alpha is zero So we see it here that the okay the minus one is clearly excluded and between alpha equal to zero which is So gravity and alpha equal to one half are more or less What alpha equal to three salutes slightly better, but a half I equal to one half. It's also compatible And there are the measurement for the breathing oscillations. So here so okay. This is a formula. So I show you before alpha is a Is the exponent of the parent interaction and P is the amplitude of the perturbation So this is a plot of P as a function of the frequency of the breathing oscillation and so there are the The fits which corresponds to Alpha equal to zero this one correspond to alpha equal to one this one correspond to alpha equal to two and we see that the best fit is alpha equal to one and Okay, alpha equal to zero. Sorry and alpha equal to one is also possible But it's a worse fit so the conclusion for us we have quite strong Strong indication that what the effective integration there is in the system is is Say do one dimensional gravity So this is for This is the first part of the talk Which is the experiment with has already been done and then this a Think that in two dimensions so The interest of that is that in one default gravity in one dimension there is no Face transition for gravity in two dimensions. There are a face transition So if you go to a temperature with is sufficiently small in a When you have the system in in a confining potential there is a collapse Okay, and in 1d. There is no collapse. There is no face transition So this interesting to see face transitions in in the in experiments Also like in 1d the the photons the scattering photons of Escape on the traffic direction. So there is a negligible repulsive force Also, we expect that the authority force will be the most important and in addition here the effective Force between the atoms does not derive from a potential. So it means that the difference of the force is is still follows the Poisson equation, but there is a Irrotational term which is different from zero. So this here You will see that there are theoretical quite a lot of theoretical difficulties and This has been published in in this work in purel So the idea is so the experimental setup is more is the same than the previous one So instead of having only two lasers, we have four lasers and This is a traffic beam which makes the Which makes the distribution of atoms quasi two dimensional and The force of the system so the idea is the same kind of calculations and before you have the force in the so now we have we have the system is two dimensional So imagine that here we have the x direction the y direction so we have One laser here. We have another here Another here another here So the shadow force how it works So imagine that you have one atom here and you have one another atom aligned with the first one So this this atom will not see this laser. So they will be an effective force between the two Like that so this true in this direction, but also if there is another atom here, which is screening This one is screening this one to this laser and there will be an effective Gravitational pseudo-gravitational force like that But it is not it does not correspond to gravity to true gravity in two-dimension it's it's two times gravity in one dimension because the this force is like that is also with a sign but the force of Gravity in two dimensions the potential is v of r is proportional to log of r So if you want to do this this kind of to achieve this kind of interactions with In a with this kind of experiment you can do it. It's you have to put in infinite number of lasers like that We can see quite easily that's in the limit of an infinite number of laser this the if the potential is the effective potential is like that and And so it derives from a potential the force there are five potential, but I think it's quite expensive So this is there are this more is the same equation is that the same equation than before but with another dimension and We expect that the Dynamics to be overdone. So instead of using velocity equation, you can use the Smoluchowski's question, which is Which is this one. So here these are fourths. This is the mean field force. This is the temperature this any question for the density So this part corresponds to the to the pseudo two-dimensional force gravitational force. This one is the The trap and this one is a diffusion So what's nice here is we have only one important parameter that so if you rescale here time and Space you see that this equation depends only on one parameter that we call here theta it's kind of temperature So, okay, what's the difficulty now? So imagine that you have a force with the pond with depends from which device Which come from with which is a gradient of some potential? So the question of the Smoluchowski equation the question wrote before you can write it in this way you can write it like The division of the flux and the flux is is this This expression and here okay for simplicity. I I neglect the trap So, how do you how can we compute the Caribbean solution of this equation? So that's simple If the force they are for the for a potential you you write f equal to minus the grade in the view and you look for a solution in which this the Caribbean solution for the density is exponential minus beta The the mean field potential and went you you have to compute but just technical problem and you had just Technically simple you should just to compute the self-consistent leader relationship between the you and the Density and then directly that's just because you put this answers in the equation you obtain that theta is equal to beta equal to minus one Equal a beta to me a one over beta and you see that the flux is zero So in order to have here a stationary solution What's What's important is that derivative row with t to be zero? But here and so it's that it should be sufficient that the difference of the flux to be zero But in this case when the force they are from a potential we see that the flux itself is zero So what happens now with a force which does not derive from a potential? So if you try to do the same thing a It's impossible. You will not achieve to do that If you write because now the force is the gradient of some potential the me feel for plus one one one part with just Divergence Which is not divergent less and then you for example you could try to put this in some way in this equation You can to look for it and that's all that's it does not work And this and you will see why it does not work one We will come I will show you the numerical solutions of this kind of of system and you will see that's Not strange and not to have simple analytical Computation of it so What can we do theoretically? So theoretically we cannot do too much, but we can try to estimate at least a Bound of the phase transition So if you compute this count the quantity s of t which is the integral of row log of row and you compute the derivative of s and you use the Smoluchowski equation you get this equation And then you can you can using this This inequality this functional inequality you can relate this term with this one and so you obtain this equation So in order to have a collapse See S point S as to go to minus infinity And we see that this only possible if Here this term is negative and then it's only possible with theta smaller than 0.17 It's not saying this this inequality is not saying that there is a Collapse for this effect for this reduced temperature, but that it could be a collapse for this reduced So let's see. Okay. First of all, I can see you a Simulation so the simulation we have done We have discretized the Smoluchowski equation and you have used n-body simulation molecular dynamics simulation so we start from some initial condition and What the system look like it looks like this like that this can it forms this kind of structure with a cross Yeah Okay, this we will see that there is a A phase transition what we think there is a phase transition and here we are quite close to the to the phase to this phase transition and show you Two different snapshots this snapshot correspond to the simulation as to show you and this one is that higher Temperature and if you go to higher air temperature, the system gets more and more isotropic. Okay, so The idea here is to try to identify if there is a phase transition or not So here we have measured that the density at the gene so the density here in this at the center of the system Or is here around the center of the system And we see that for temp quite large temperature So the there is there is it's more or less a constant So here this in in log linear scale here is in linear scale So here it's more or less a constant and it arrives some point which is between 0.0 point 12 more or less or 0.15 in which this it's increased a lot and here so We estimated this the transition here in this region And here there's two kind of points the black points are pointing which With the simulations we trust in the simulations and the white points We don't trust in the simulation because when you have a collapse to something which with a Density which goes to infinity. That's not possible to simulate. So in this region if you if you if you do conversion states Decreasing more and more the the time step in your simulations you get different result That is what you expect. It's also a sign of the of a phase transition So what about the currents the flux I I tell you before so we can do the same kind of things so this is the The the flux which are in the system. So here we see that they are non-zero So we see that the particles which are here They they there's a flux of particles in this towards the center here and they escape Along the diagonal here in this way This way here and here they do the same and here they do the same and also if you measure the The average value of the flux as a function of theta We see also the signature of the of the phase transition So here it increase quite a lot around the same value of theta so this So what this summary of the talk what I I show you in an experiment with had already been done of things that mimics gravity in one Dimensions we have also a a project and of experiment in two dimension, which is Schedulates, so there is a PhD student working on it a So there are many many theoretical things still we To do which is a natural phase transition res really a phase transition What happens with if we using under dumping dynamics, so here the The phase space has only one dimension But if you if the dynamics is not on the dump and we are we have the friction and the dissipation for what happens in this in this case more general case and Just to say that I hope the experiment will be done soon and estimating the the regime of attainable with the with With with experiment we are in the same regime that We have a model here in the in the theory Thank you


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