 Okay so first I'd like to thank the organizers to invite me to give this talk but also to put this conference together because I think it's a very nice thing to do for people working with long range interactions. So I will present collective effects in light scattering by cold atoms so it's about long range interactions and I will try to motivate why it is why we have long range interactions when you scatter light on cold atoms and where exactly do you see these long range effects. It's a bit my motivation first I will try to explain you why we have a long range interaction a long range problem and so in this community we don't speak about long range we speak about collective effects and I would say our cooperative actually and I would say people do not differentiate yet what would be long range, cooperative, collective. Cooperative or collective just means many many body effects but you can have many body effects that are local and I will try to discuss a bit what is really long range and what would be more local and in particular the big difference I will discuss is between dynamical effects and the space effects. Okay so first thing why do we have a long range many body problems so we send light on large cloud of particles here I will consider atoms so microscopic particles much smaller than the wavelengths I will consider they are motionless so no dynamics but they are dipoles interact why because each time a particle receives light it will scatter it will re-radiate a spherical wave a spherical wave decays as 1 over r and the other particles will see the spherical wave this means that all the particles interact together through the spherical waves and this way you have sorry a many body long range problem that is classical or quantum here I will treat everything classically and I will discuss just at the last slide what would be the quantum aspect but what I will do is a classical so I will consider two level atoms with a fixed position in space and the dynamical variable will be the atomic dipole beta square so beta is the atomic dipole and beta square is a probability of my atom to be excited and in a linear optics it's beta is proportional to the electric field and if you look at the dynamics of a single atom the dynamic of the dipole you have a decay term gamma that gives a lifetime of your atomic transition you have the detuning between the atomic transition and your incident laser is just a rotating term and then you have an open system of course because your atoms are emitting light so you need to inject light and so this is one big difference with many long-range system that has been studied here I will have an open system because we traced over the degrees of freedom associated to the light okay so this is the dynamics of a single atom if you look at the dynamic of my cloud with many atoms coupled the term you need to add is this one here you are seeing my dipole J also receives the spherical waves emitted by all the atoms M and this spherical wave needed to travel from atom J to atom M and this way you get the long-range coupling it's many body because typically in a cold atomic cloud you get 10 to the 6 to 10 to the 10 particles it's quite flexible system because you can experimentally it's quite easy to tune the detuning to change simply the frequency of the incident laser you can also by changing your trap the characteristic of your trap you can change the density and the size of your system so it's actually quite a flexible system okay and I mean the linear optics regime so the atoms have a linear response to the light so this means that I just have linear problem here I just have N by N my matrix that describes my problem so a lot of the physics will be in the eigenvalues and in the eigenvectors of this matrix and so I will discuss especially with the eigenvalues and here I should say it's a microscopic approach I consider my medium as a made of point scatters and I will present another approach a bit after I would say the hallmark of a long-range effect in the system is super radians the original context in which super radians was introduced was with two level atoms in the small volume limit so you take many atoms and you put them in a volume much smaller than a wavelength and you start with all atoms in the excited states so the atom will start to decay but the decay so typically one atom will lose will go to the ground state but the point is that you don't know which atom went to the ground state so you quickly go into super positions of state so you go to states that are symmetric that look like this so here I have just three atoms of course you get many possible symmetric states like this if you have many atoms and when you go from this state to this state or from this state to this state you have actually a much smaller much faster sorry emission rate particularly in the original work by a dicker in a 54 he showed that in this small volume limit you expect an emission of light not at rate gamma like a single atom but n times gamma means that n atoms strongly coupled emit light much faster than a single atom so and the point is that in this small volume limit you actually have kind of you can treat the system like in a mint like defining a microscopic mode because you don't see any more the effect of the distance because the atoms are in a volume smaller than a wavelength so they just interact with a unique light mode then if you have a large system much larger than the wavelength you can still have a super radiant effect you don't need to have the atoms very close and in this case it decays as n over r square here r is okay is missing and this is already a long range effect because if you take a fixed density and increase the system size this super radiant state will rate will go to infinity okay so I started with a quantum system before because they had done all strongly all my atoms excited and I decayed to the states with many photons but actually you just need to you can just study this problem with one photon in your system and it's called super radians with a single photon and already if you send a laser on your cloud imagine you can have a be sure that your single photon when you work with very low intensities you will excite one atom but you don't know which atom so you will be in a superposition of state already and because of this you can have super radians with a single photon and because you have a macroscopic superposition of state and there was a recent experiment in recipe fair about this aspect and now I will focus really on linear optics I will forget about the story of super position of state because they are not really relevant when you work with a single photon if you don't look at quantum correlations the reason why it's more or less the same is because if you look at the first order in dipole terms you cannot make the difference between a superposition of state and a separable state so now I will really discuss linear optics the one that you learn much more into undergrad studies and I will go back to my very classical equations where each atomic dipole scatters a spherical wave so here I put back the equations with a single atom and the coupling term and you can see I have a linear equation so I can just look at the eigenvalues and eigen modes of my system the eigenvalues will be complex and I will write them like exponential minus gamma nt minus i omega nt so gamma n will be the decay rate of my mode and omega n its energy and if you look at the spectrum of this system you get this kind of thing so for n atoms you get n eigenvalues and it looks like this so on the vertical axis I have the mode lifetime actually sorry it's a mode decay rate so on the higher part you have super radians you have very fast decay rates which I discussed previously this line corresponds to single atom physics and below you have what is called sub radians which is very long which are very long lifetimes okay so since I have a coupled system I have a new many new lifetime in the system and so I discussed a bit super radians and now it's important that to understand that when you have super radians automatically you have long lifetimes at the same time that appear in your system the reason for that is that when you introduce your coupling between the atoms you do not change the trace of your matrix so if you have eigenvalues that increase because of the coupling you need to have eigenvalues that decrease to compensate because your trace is constant independently of the coupling so as you have super radians that appears with very fast decay rate you also have sub radians that have a low decay rate and these modes couple very little to the way if you send but since they are very long lifetimes you will see them in the end and the group in particular for Robert Kaiser did very nice theoretical and experimental work on this in particular recently they showed experimentally that you have these sub radians rates in atomic clouds with lifetimes that increase with b0 this optical sickness and they will come back soon to this so you have long lifetimes and short lifetimes so back to the long-range aspect you need always to define a thermodynamic limit if you want to discuss long-range effects and of course the question is how do you define your time of dynamic limit so here I will say okay I want to have a fixed density in my system I take an homogenous system with a fixed density and then and I increase a particle number to infinity this will be my thermodynamic limit in this context the optical density scales as the density times the size of the system so it goes to infinity of the in the thermodynamic limit which means that the super radiant rate goes to infinity in the thermodynamic limit and the sub radiant lifetimes also go to infinity in the thermodynamic limit and so you need to understand this optical sickness parameter like the cooperativity parameter in the system that give that tells you how strong long range effects can be okay now you say okay I have I describe my system with n degrees of freedom with my n particles and the super radians sub radians are clearly many body effects but do I really need n degrees of freedom to describe to describe them so we want to look at alternative models to understand if we really need all these degrees of freedom and if some simpler models can describe super radians and sub radians so that to understand the minimal model to capture this effect so what you can do is to do a mean field approach you take your equation and you substitute your microscopic scatterers by a continuous density and so you rewrite your atomic dipoles like a dipole field so you replace the other field and the sum of all the possible dipole is dipoles is simply replaced by an integral with a density times your spherical wave with all the dipole fields so here since we have atoms at fixed positions we don't have any problems of divergence when the atoms would come close or collapse so this is a there are no big technical problems doing this so this will be for the dynamical problem where you look at dynamics dynamical effects and you can also look at the steady state when you put this term to zero and if you do this and you apply this operator nabla square plus k square you will simply obtain that your dipole field obeys l mode equation with a refractive index that is given by this formula with the atomic density and this is well known that atomic clouds at first order in linear optics regime they behave basically as a dielectrics they can have a lensing effects and such kind of effects so we'll now discuss a bit some phenomena in this limit of a mean field approach so first thing the spectrum because we said a lot of information is in the spectrum so the microscopic one looks like this and for the field approach it looks like this and since you cannot see much if you do many realizations you obtain this kind of spectrum you can say okay the spectrum for the microscopic and the field approach are not so different okay you lose a bit this branch of the spectrum here the colors corresponds to the participation ratio so it means red means many atoms participate and you expect that in the super radiant modes many atoms participate and the blue few atoms participate in the mode and overall the two spectra look quite the same you get a super radiant part and a sub radiant part so maybe the field approach describes well super radiant sense of radiance actually if you look into more details as a super radiant part you can show that the super radiant modes are well described by both the field approach and the microscopic approach are very similar mode you can also count the number of period super radiant modes and you find that the scale as k r square so like the surface of your system but now if you look at the sub radiant modes present in the field approach you will see that you have actually only surface modes which are something very well known in acoustics especially which are called whispering gallery modes these were first discussed in the context of of dome in a cathedral where you have actually some surface effects like if you go in this dome you can sit actually in this place someone sitting here and someone sitting in the center would not be able to communicate with each other but because of the surface modes that exist the communication is very easy along the corridor along the border the surface of your of your dome and these are the modes with very long lifetimes that we observe in the field model they're very long lifetimes because they're reflected many times before leaking out of the system these modes are actually not relevant for atomic systems because atomic system don't have sharp transition in the refractive index because we usually have traps with clouds with a quadratic profiles or a Gaussian profiles and then these modes disappear completely but this is what we are seeing here because we took an homogeneous density and if you don't take an homogeneous density this mode disappears this means that when we do the field approach we lose a sub radiant that was discussed previously so as a first conclusion we have super radiant which correspond to the very fast time scales that are macroscopic modes well described by your field by a mean field approach and we have sub radiant which corresponds to the very slow time scales it's a disorder effect and which means you need to address a particle-particle correlations and coming from the long range community I think it looks very similar to the situation in long range system such as it was for example discussed in a with the hms model a lot like the blast of equation it's a mean field approach that describes the macroscopic degrees of freedom and it describes the fast time scale and quasi-stationarity corresponds to a slow time scale and it means that you have to address the particle a particle correlation to describe these so there is this really striking analogy although it was not a formalized yet but that tells that maybe what we see as fast time scale and slow time scale in this linear open system is something very similar to what you see in a more generic long-range systems so these effects are probably truly long range now if we forget about the dynamics and look a bit at the steady state as I said before we just get elements equation so your dipole field obeys wave equation in the medium with a given refractive index due to the presence of the other atoms and this is due to the fact that in linear optics your dipole field is simply proportional to your electric field and of course in this case you know that when you look at a wave propagating in a medium the boundary conditions play a very important role the geometry of your system is very important one extreme example is these are these whispering gallery modes that exist only because you have boundaries in your system but we know in general that the diffraction properties of your system depends only on the boundaries so one in the steady state regime one effect that was labeled as a collective that is actually related to this refractive index and I will present because I think it's a very interesting problem is the one of Abraham Minkowski momentum the story is the following so if you take a single atom and you look at the momentum it acquires when it scatters lies in the linear regime in the elastic scattering regime of course the momentum exchanges with light units of h bar k where k is a wave number for the system and there was a very old debate more than one century old to understand whether the momentum of light in a day electric was m h bar k with m's index of refraction and h bar k over m depending I mean there are arguments for both candidates and it was a very long-living debate and it's actually still active with some papers still published on this topic about what is the correct unit of momentum for light in the day electric medium and since atoms exchange units of momentum with light you may be tempted to use cold atoms to study this problem so a very nice sensor was given by a Stefan Barnett a few years ago where basically it's he said be careful to what you momentum you can define different momentum like the kinetic momentum the canonical momentum and this will not give you the same answer so depending on what you observe and how your system is done you will get different answers and this is why there were different experimental results in the frame of cold atoms which is the one I'm focusing on it was shown already experimentally 10 years ago that the atoms acquire recoil of m h bar k here you have a factor 2 because they absorb and then remit the photons so there are two events so this is really typically a classical collective effect that you can understand easily with what I showed you before so what you can show is that the momentum distribution is just a Fourier transform in space of the dipole field and if you accept this this formula that I give you then since you know that the dipole field propagates like a wave in the dielectric with the refractive index the Fourier transform of this will simply be a momentum distribution where the refractive index will appear so it tells you that the atoms will exchange unit of momentum of m h bar k and you can even calculate in more detail the momentum pattern so this is really a nice example of collective effect where due to the presence of the other atoms the atoms each atom exchange a unit of momentum with light a quantity of momentum with light that is not h bar k but modified by the presence of the neighbors okay so this refractive index is collective in a sense that it's a many many body effect but I would say it's not long-range because it doesn't diverge in the thermodynamic limit and it's as soon as you speak about a refractive index you can think of this discussion about additivity if you just take two pieces of the electric of index m and glue them or separate them you still get a dielectric of index m so m is a is an intensive quantity and what changes when you put your pieces together are just a boundary conditions so of course and then you go back to diffraction and you know that diffraction is very sensitive to boundary conditions and geometry for example a cube of dielectric doesn't scatter light like a sphere of dielectric it strongly depends on the shape still I would say this is not really a long-range effect I would say when you speak of this collective effects you must be careful that you're not simply doing diffractions like it was sometimes done in the past both effect long-range systems and diffraction problems share a sensitivity to boundary conditions but overall I would say these are quite different problems so this was for the mean-filled approach and then you may want to go back to disorder to say okay so this is not captured by your mean-filled approach and it corresponds to a disorder effect and if you think of long lifetimes and disorder you should think of Anderson localization because these are two ingredients that can tell you that you have Anderson localization when it comes to light in 3d Anderson localization has been very controversial and a very recent paper published in January of this year summarizes this well by saying Anderson localization of light in 3d are still not being observed yet particularly there were some experimental claims the authors of these papers of these experimental papers themselves recognized that what they observed was not Anderson localization of light any particular recent critical works showed that probably the near-field effects may be responsible for the absence of localization because indeed as you know well light is a vectorial wave is not simply a spherical wave with a rotational symmetry you have polarizations you have electric field and magnetic field and if you look at your coupled equations you need now to consider not only beta j but beta j alpha with alpha equal to x y z and you get a coupling term where you have your spherical wave as before but you have new terms that scale as one of a k r square and one of a k r cube these are still a long range terms because there are one of a k r square and one of a k r cube in 3d so this one is marginally long range but the fact is that the breaking of the symmetry of the coupling leads to a reduction of the dipole-dipole correlation so this actually breaks your collective effects people tend to say okay but if I'm dilute if each atom is far from his neighbor and I still have a very large systems I can neglect these terms but this is not always true and the reason is that they are long still long range terms so if you look at them of a very large size for your system they may still be important because they have a diverging contribution in the thermodynamic limit. One well-known example is that in the case of super radians like he introduced in the beginning since you have in a very small volume these terms actually do become important and they generate what is called van der Waals defacing and the consequence is that you don't observe a rate and gamma like you would expect from super radians but you observe a super radian trade that is much smaller than what you expect just because of his new term but are still long range but they break some symmetry and it makes the things more complicated so back to localization we decided to go for a 2d system and the reasons for this is that first 2d systems allows for study of systems much larger than 3d systems because if you have 10,000 particles you can simulate in your computer in 2d is much larger systems and in 3d you don't have problems of critical density to reach the Anderson localization regime and what is very important for us is that we have a regime of scalar scattering and a regime of vectorial scattering and because we have to decouple subsystems in 2d and this will give rise to a localized Anderson localized regime and a regime without Anderson localization so very briefly in 3d vectorial light couples all sub levels of your atomic transition while in 2d if you really go to a 2d vacuum you will decouple some transition and so you will get vectorial scattering versus scalar scattering. If you put this into equations you get this equation for the scalar scattering where L0's yankl function is basically the spherical wave but in 2d so it decays as one of a square root of r and for the vectorial one you get this spherical waves in 2d plus some near field terms and so the story is that if you look at the spectrum as before so here is single atom physics so you have again super radiance and very strong sub radiance for scalar scattering but if you look at vectorial scattering you get super radiance and a much weaker sub radiance. This sub radiance is actually a very well known effect called radiation trapping it's just the fact that you have a photon in the middle of your cloud you have a very large cloud so before being able to escape from the cloud the photon will be scattered many times but for this you don't need coherences between the atomic dipoles you don't even need disorder so this is I would say not at all the physics of undersigned localization and it's it's something that can be easily understood with a random work of photon models so you can already see a big difference in the spectrum now if you want to really characterize your undersigned localization to say if you have undersigned localization or not you should do two things you should look at your modes and we observe indeed that the scalar light has very well exponentially localized modes in space so this is a typical mode with a logarithmic scale of its spatial profile you see that it's very localized with an exponential shape and if you look at vectorial model you will not see any exponentially localized mode only some extended modes that are a bit sub radiance but nothing localized so apart from this inspection of the mode you should also do a scaling analysis that I will not present here but that was introduced by Abram and with this you can confirm that only scalar light exhibits undersigned localization and vectorial light doesn't means that in 3d I mean it's in agreement with the 3d results vectorial light which is the only physical light doesn't exhibit undersigned localization for point scatterers and if you reach a scalar model then you may have undersigned localization okay so this was a bit for undersigned localization but what we were interested in was a certain understanding the connection between trapping in time which is sub radiance and trapping in space which is a localization so we took again our eigen modes and eigen values and now what I plot is a localization length of the modes on the x-axis and the lifetime of the modes as the inverse lifetime the decay rate and the color corresponds to the distance to the cloud center so actually these modes of these columns are the quite well localized mode with some exponential decay and what is very interesting to see is that okay you have a quite well defined localization length but the lifetime of the mode seems to be quite independent from the localization length while in general you would expect that the more localized you are in space the longer the lifetime of your mode but this is not true actually the lifetime seems to be quite independent from the localization length of your of your mode and another thing that you can see is that when you go in red region means when you go to the boundary of the cloud your localization length increases drastically means you are less and less localized it means that spatial localization is very sensitive to boundary effects while here you don't have this problem meaning you have very long lifetimes very close to the border of the cloud so spatial localization is sensitive to boundary conditions temporal localization is not so this suggests that dynamical effect and space effect are very different and then if you look if you try to understand what is scaling of the localization length and of the longest lifetime in your system you obtain two very two different scaling you can show that the localization length scales as the density of your system means you go to the thermodynamic limit at constant density your localization length is well defined your mode lifetime however exhibit the scaling and two-third over r it means it scales actually as a density times n power one-third it means that in the thermodynamic limit your lifetime diverges to infinity it means again it's a bit the same conclusion as before if you look at space effects you don't have any problem of the thermodynamic limit so I would say it's not really a long range effect because you expect some problems in the thermodynamic limit for long range effect but if you look at dynamics then you get problems so maybe as the signatures of the long range effect are in the dynamics okay so this is what I just said that probably the true long range effects appear to be the dynamical effects which are in particular super radiance and a sub radiance if you look at the steady state at the stationary problem you can show that actually it's much it's mostly about diffraction under some localization shadow effect that was mentioned yesterday and then boundary can play a very important role what I didn't discuss here is that there were other effects that are called collective in the in the community like the collective modification of the radiation pressure force exerted by the light on a cloud the collective frequency shift but actually it goes back to the same thing like if you look at steady state effects it's mostly an effect of density it depends on density and so you don't have a problem in the thermodynamic limit for fixed density okay so the perspectives if you want to go on with long range effects with with these cold atom systems for motionless atoms could say okay one very interesting topics and we had a very nice talk yesterday about this is dynamics of correlations and probably since it's a dynamical effect you expect some nice long range effect okay one problem in cold atoms is that it's for large cloud is very difficult to address the atoms usually what we are good at in cold atoms is collecting the light coming out from the system but probing the atoms is much more of a problem but then the interesting part is that quantum optics naturally addresses several dynamical phenomena especially if you look at optical coherences optical coherences for example the first optical coherence basically corresponds to the fluorescence spectrum you strongly drive your system and you look at the frequencies scattered by your system and then you look actually at the correlations between events events the field emitted at time t and at time t plus tau so ready you will get an interesting signature of the dynamics of your system of the correlation between events at time t and t plus tau and you also have the second order optical coherence which corresponds to photon bunching and anti bunching which gives you a correlation between intensity and time t and time t plus tau so let's say for long range effects this is probably a very promising direction for light scattering in cold atoms I would just like to conclude with an announcement because we are in a ICTP and as was mentioned a few days ago in Sao Paulo we have a cousin of ICTP the ICTP SAIF which organizes schools and we will organize such a school an interaction of light with cold atoms next year but I mean in general it's a structure that organizes a school mostly for South American students plus some students from outside South America and so if you know anyone interested by this school or by organizing schools there there is this structure in Sao Paulo so thank you