 Okay, so all right next speaker is Jerome Dubai and he's going to tell us about atom atom losses Yes, okay, so thanks a lot to the organizers for this conference. Thanks to the ICTP for posting it and and thank you for the opportunity to talk. So this is going to be about atom losses and it's based on a preprint that we just put on the archive a few days ago with Benjamin Doyon and Isabelle Bouchaud. Can you see my pointer? Yes. Yes, okay, good. All right, so I'm going to start very very slowly. So imagine that we have some model for a for a gas like two-dimensional heart spheres and initially the spheres are at rest and we kick one of the spheres. So then what happens is that very quickly because of collisions the energy and momentum that we have put into the system is shared among all the degrees of freedom and so that after some short relaxation time the state of the system in inside box is entirely characterized by by a small number of quantities here three quantities only so particle density mean velocity and mean energy per particle. And so in this talk we're going to be interested in losses so that means that we have the same system so we imagine we have a again two-dimensional heart spheres but now some of the spheres can just disappear from the system at some rate gamma so you see that some of the spheres get colored in blue and shortly after they just escape the system. So this is what we are interested in in describing and we are going to describe this in the regime where the where the loss rate gamma is much smaller than the inverse relaxation time so that the system always has time to relax. So after each loss event the system relaxes so that and it stays in a relaxed state for some long time before the before the next loss occurs. And so even though in that system strictly speaking the energy momentum and particle number are not conserved anymore we can still describe this as a system as at equilibrium it's just that the state the equilibrium state is going to slowly drift as a function of time here in that simple toy middle according to these equations but we'll see more more complicated equations later. So that's essentially the story of that we would have for a gas with losses in an equilibrium gas that thermalizes but we are going to be interested in an integral gas like all systems in this conference and so what we have in mind is more something like this. So imagine now that we have hard spheres in one dimension and that we do the same thought experiment so we just kick one of the spheres and then what happens is very different from what was happening earlier because now the energy and momentum does not is not shared between all the particles and instead as you can see here at any time there is one and only one sphere that is moving and the reason is that just we have elastic collisions in 1d with identical particles and so the only thing that can happen is that the particles exchange their velocities and as a consequence it's not just the energy and momentum of the particles that is conserved but it's the entire distribution of velocities of particles in the box. So this distribution of velocities rho of v is conserved at any time so if you want to describe the system you need to know this distribution of velocities at time zero and just for the younger people in the audience who might who might not feel comfortable with this expression so this sum of Dirac deltas this means that we have a sum of infinite peaks but when as we take the thermodynamic limit these peaks become more and more dense and this becomes as a smooth distribution. Another way to say this is that you can broaden the peaks a little bit replace the delta function by a Gaussian with sigma for instance and so if you do that then this expression becomes a smooth function like the one sketched in black here and this function gives you the distribution of velocities with some resolution sigma so the smaller sigma the better resolution. Alright now again we are interested in losses and again in this regime where the loss rate is much smaller than the inverse relaxation time so some particles can just escape the system randomly with some rate with the rate gamma and the point I want to make here is that this means that our distribution of velocities now is not conserved anymore but that again it's going to slowly drift as a function of time so we are going to have in this simple toy mood that we have just a simple very simple equation of this form so time derivative of rho is just proportion to rho with a proportionality constant that is directly the rate but more generally in this talk we will have time derivative of rho that is proportional to some function of rho at time t so this is the important equation this is what we are going to see in this what we are going to study in this talk and of course if you if we know that if we know the functional f then we can then calculate the time evolution of the rapidity distribution so typically what will happen is that it starts for instance from this violet distribution at t equals zero and then at t equals with time and ultimately it will just go to a constant zero simply because since we are losing particles in the system at large times the system will just go to the vacuum so the big challenge is really to calculate this functional f in general okay so now that's enough toy models so now now i'm going to talk exclusively about the system that we are really interested in which is the one-dimensional Bose gas so it's described by this Hamiltonian here known as the Liblinigar Hamiltonian so there's a kinetic term and a contact propulsion term so when two atoms are at the same position there's some positive energy and this describes a gas of atoms for instance for instance in a box and so the Hamiltonian describes only the unitary part of the evolution but what we want to do is to take into account losses so experimentally what can happen is that depending on the conditions that you have several atoms can come to the same position and because of collision it can form either some internal excited state or form a bigger molecule that is no longer trapped by the one-dimensional confining potential and if this happens then they can escape the system so experimentally usually the most important processes are three body losses but for the purposes of this talk i can just consider k body losses where k is some arbitrary integers okay it was one of two or three of them so i would just fix k okay some integer that is fixed for the rest of the talk and i would consider k body losses so a good starting point to describe these losses is this equation so the lean blood equation for the for the density matrix of the gas in the box and what's important here is that we have this unitary part of the evolution generated by the libdinger Hamiltonian and the the loss processes that are encoded here by this operator psi to the k which is the operator that annihilates k atoms at a given position and this occurs with some constant g which has dimension of length to the k minus one divided by time and by multiplying this by the density to the k minus one i can form a rate an inverse time and gamma so in the rest of the talk i will use this notation gamma again for the for the loss rate all right now i need to define the rapidities because i have not defined the rapidities yet so paulato also talked about the rapidities but so essentially these are numbers that appear when you in the mathematical solution of the of the of the of the libdinger model so when you diagonalize the libdinger Hamiltonian you find that the that the eigen states take the form of beta states beta states are superpositions of products of plane waves and the numbers that parameterize those plane waves are homogeneous to velocities and and those velocities we're going to call rapidities so okay so mathematically the rapidities are just the the set of rapidities is just the the label of of the eigen states of the libdinger model but perhaps more physically you can think of of of those rapidities as being asymptotic velocities if you let the system expand in one day so what you can do is imagine that just prepare the gas in a in a box in a small box in a in some eigen state with a label by the rapidities v1 to vn and then you by changing the boundary condition you just let the the atoms expand along a one-dimensional line and so initially the atoms will collide and because of the collisions are complicated processes so the there's some complicated things going on here but after some sufficiently large time the atoms were just the order ordered according to their velocity so the the the one with the smallest velocity will be the leftmost position the one with the the fastest one would be at the rightmost position and when when they are ordered then they don't interact anymore because simply because they will just keep expanding forever without without colliding and so the at this point the velocities do not change anymore so these asymptotic velocities are at large time they turn out to be exactly the the rapidities so exactly the rapidities that were labeling the eigen state at time zero and similarly to to what I said for the for the toy model I was using in the introduction so we can define a rapidity distribution like this okay so just a little bit more about expansion so the fact that the rapidities are also the asymptotic velocities after expansion this is something that has been advertised and used by many theorists but recently it has been it has been measured experimentally so in the in a prep in a sorry in a paper published in science a few months ago by the group of the advice so so you can really measure the rapidities now experimented so what what that what these people did is that they did more or less what I what I explained so they prepared this the gas in in some state here and then let the gas expand in 1d and after several at some time they just measure the distribution of the velocities of the atoms so for instance along this time slice it would correspond to the violet to the first violet curve here and then at the next time slice that would be the blue line at the next time slice would be the green one and so on and so you see that the distribution of velocities evolves with the expansion time but for sufficiently long expansion time it does not evolve anymore and at this point what you are measuring or what they are measuring is really the rapidity distribution in the initial state so really it's something that it's not just a you know it's not just a quantity that shows up in the mathematical solution of the model it's really something that experimentalists can measure now so this is the experimental data that they had and this is comparison with with theory all right now let me come back to losses so again we are interested in describing losses in that gas in the regime where the loss rate is much smaller than any relaxation time in the system and in the quantum system what this means is that we can describe at the level of the Lindblad equation we can just assume that the density matrix of the system at any time t is stationary so it commutes with the Hamiltonian and also it commutes with conserved charges so if we have some conserved charge q then it will also commute with the density matrix under that assumption and then we can just write a simple evolution equation for the slow drift of the conserved charges so which takes this form so it relates the time derivative of the expectation value of q in the system to an operator to the expectation value of an operator that is made out of q and out of psi to the k which is the operator that removes k atoms at the same position all right and then if you apply if you just apply this to the rapidity distribution itself which is a conserved charge in the system then you find this equation which is the one I advertised in the in the introduction where this functional of rho now is given by the expectation value of this operator taken in a state in a macrostate that is parameter or generalized Gibbs ensemble if you want parameterized by the by the rapidity distribution okay so you have to do this for for a finite resolution sigma just to make things more well defined and and ultimately you want to take the limit to sigma close to zero all right so this defines the problem and then and then what we want to do is of course is to evaluate this functional f so okay so analytically it's it's a very hard problem so we were able to do it so far only in in the two easy asymptotic regimes where the gas is maps to a free to a gas of non-interacting particles so first in the in the regime where the where we have essentially free bosons so when the energy per atom is much larger than than the repulsion energy and the scattering energy the the atoms behave like yeah like free bosons and then in that case you can see that it's very easy to see that the functional f is just proportional to rho directly that's simply because you have non-interacting particles so if you if you just remove one particle randomly or a bunch of particles randomly then the probability that you would do that is just proportional to the to the density to the density itself and then there's an interesting combinatorial factor which is k times k factorial which just expresses the fact that there are k atoms lost at each event and k factorial is the is the k but the k but correlation evaluated in for free for free bosons okay so that's very simple now slightly less simple and much more interesting is the hardcore regime so the regime where when the energy per atom is much smaller this time than the scattering energy so the so the particles behave like hardcore bosons which can be mapped to free fermions however the mapping from bosons to fermions is not local and and as a result you find a non-tragical result for the functional f so okay so the formula is what it is but what I want to emphasize is that it is both non-linear non-linear in the rapidity distribution because you see there's some rho squared here and it's also non-local in rapidity space meaning that the functional f at rapidity v depends on the distribution rho of w for any w not just for w close to v but for really for any w so it's a complicated object in general that's the that's the that's what it means all right and away from these asymptotic non-interacting regimes one has to rely on numerics and for this we've implemented a summation of over beta states which is the technique that has been developed by Jean-Sebastien Co and the several collaborators over the years and so essentially we are we are computing that double sum so it's a sum over eigenstates before the loss here and eigenstates after the loss here with some probability distribution which is taken which is defined out of the of the form factor of psi k the operator that removes k atom at the same position and yeah okay so for this we are relying on some algebraic beta ansatz formulas by Lorenzo Iroli and with that distribution over pairs of eigenstates we have to evaluate the the difference between the rapidity distribution before and after the loss weighted by the k body correlation and so for this we are also using an algebraic beta ansatz formula uh this time by bullish but uh push good um okay so we did this uh this is what the the results typically look like so for instance if we start here from uh rapidity from the the equilibrium rapidity distribution so the one of thermal equilibrium which is the violet one here for some parameters uh then we can calculate the corresponding functional f which is so the corresponding one is the is the violet one here and then uh since we can calculate the functional f we can evolve the rapidity distribution as a function of time and this is what is shown here after five percent losses 10 percent losses 15 percent losses and so on for k equals one so one body correlations two body correlations three body correlations so it works it works this method works but it's computationally heavy like to to get these curves took it took a few days so it's it's um yeah it's it's heavy and it's not very convenient for practical purposes um okay so i guess my time is up so let me just quickly summarize so for slow losses so for when the loss rate is small compared to relaxation in the system then uh we have we you easily get this equation here and uh to to describe the losses and and the big challenge is to evaluate this functional f for you that's the big thing so in this recent preprint we were able to get it analytically in simple asymptotic regimes um even in these simple asymptotic regimes the result can be highly non-trivial and uh and in general um you can do it numerically but uh we need to we if you really want to do something uh something useful with this unit we need to we need to improve the the numerics because it's it's it's taking too much time at the moment um okay so there are many open questions i think uh on this topic so there have been uh works in the cold atom community in the past years especially on the quasi condensate regime and we have not made the link with between what we did here and and these works so that remains to be done and um also okay so recently there have been several works um by several people in this audience uh on the effect of weak integrability breaking which is essentially the same as as what i explained uh so they they looked at different mechanisms for integrability breaking so not losses but but very similar similar mechanisms and um and they have so analogously to what i said they have some functionals f that they need to calculate and and some people have developed approximations to do that or or or analytical schemes to do that especially these two papers by uh Friedman Coppola-Krishnan-Vasser and and uh Basternello-Denard-Steluca where they used some tricks to truncate uh truncate the sum of over form form factors to evaluate this function so would be i don't think these methods apply to losses but still since we really need some analytical insights it would be really useful to to see if any of these other related works is helpful here so analytical progress is really needed and there's room for for there's room for for more work work there and that's it thank you thanks a lot Shion so yeah we have we have some time for questions yeah me yes okay so just i mean one maybe one thing you could use it's the low density approximation for these form factors but then it would be easier and but actually how do you in the in when you showed the convergence i mean when you did the calculation with uh sum you know all these things how do you check the i mean you have to check that you're including all the relevant excitations right like in uh we are not choosing the excitations it's a mark of chain we are building a mark of chain and we're so we just need to check that it's equilibrated and that we did so it's not it's not uh it's not we are not reorganizing the sum in terms of particle whole excitations okay so we've checked that we've checked that it's equilibrated so that the mark of chain is long enough and we've also checked that it's converged with the system sites okay so we have one question from Dimitri Gengard one moment everybody was uh someone else was trying to Dimitri was trying to ask you a question yes so can i no okay okay so wait until the break yes maybe and maybe Dimitri can you type your question since you have a sound problem okay please have it yeah okay so i said it's just a question i mean you you're just right under in black and so it looks like a natural description but i mean you know how physically set assumption actually the assumption that it's Markovian you know you've done a lot for that particular process right i mean so i mean it's said you know what i mean well okay so we had many discussions with discussions about this with my co-authors so okay i will give you my personal opinion about this so for me Aline Bladian is just the most general equation markovian equation that you can write for a density matrix that preserves you know the trace and the positivity and and that's it so it can't be anything else as long as you as long as you assume that it's Markovian now you can discuss whether or not it should be Markovian but i think at least for losses in cold atom gases it's well accepted that's it's Markovian now the other assumption that the rate is small compared to relaxation that's that's probably less well justified i would say and indeed in cold atoms people have studied a lot like things like the zeno effect which which clearly goes beyond what we are doing here that would be higher order in gamma if you want but i think Markov i think just assuming that it's Markovian i don't think this is really a problem okay so do we have like one very very quick question may i super quick yes and i may ask if there is a simple physical interpretation why the losses in the in the rapidity space in the end the particles are not interacting so i can't really understand can you can you repeat this sorry is there a simple physical explanation why the losses in the tons geodrome regime are not diagonal in the rapidities why they're couple different yes it's because it's because you are it's because of the geodrome beginner stream so it's because you even though the gas is a gas of non-interacting fermions if you want or maps to a gas of non-interacting fermions you are removing a boson so and when you remove a boson you are doing two things you remove a fermion but you also change the boundary conditions or if you want to apply a geodrome beginner stream too so yeah it's like you're you're removing a particle but you're also somehow shaking all the others and and this this results in a in a non-trivial