 Vse je bilo razlišo č moistlj. Zelo sem seči oči, da so jaz bomo vse prijetiti, ki so kkega slike... atoms, tudi mikrwater. Tako, zelo se! Zelo si, da jaz bojtenja, da se je, si tega... jaz so vse izgleda, tako da bojtenja. Tako, to sem... tako, oči si seči objoru, na numer 14ga minulija, s kaj modeli. da pa srča je vsega zelo je vsega poslednja. Zelo jaz sem vsega zelo vsega poslednja. Zelo je poslednja vsega. Tama taj se počak naredilo, zato se pačečnega poslednja z rada Landa. Načal, je poslednja iz mnogu stave. Tama stave PSI pripoča na poslednji srča. Zato pradljamo LA vziv, kjer tih ostaj v tem, eno bananas, renovačujem mi v drženju. Kaj je z Vojt a, kaj je namaljil? Zato this is a. So Always is a Gaussian imposition. Sentrji drugi a, ki je vziv vziv, This is the stochastic part. Rc is the width of my Gaussian. This is just normalized. Then the nonlinear part comes in the renormalization of my wave function. I have both my features of collapsed models here. In tudi probabilite, da je tukaj tukaj, in tukaj in tukaj in tukaj in tukaj in tukaj in tukaj in tukaj. Ok. Danes, kako je zelo zelo zelo zelo, imam zelo, da je moja standa zelo zelo. V timeline imam zelo zelo. Zelo zelo. Zelo zelo. Zelo zelo. Zelo zelo. Zelo zelo. Zelo zelo. Okay. Which are the effect of this kind of model? Consider, for example, a superposition, the model squared of the psi is x, a superposition is half in minus a, half in a, two gautions. So the effect of this kind of dynamics for example in a is to have, see, the application of gautions, this is the GRW gaution. Then I re-multiply everything, and what happens is that at the effective level, this part of the superposition becomes smaller, this part of the superposition becomes larger. There are more probabilities of finding my system in a that in minus a. Is it simple? Then there is also something also important here which also solves our first question, why we don't have macroscopic systems in superposition. Okay. So for example, if I take two particles centered both in a superposition, minus a and a, then what happens to the center of mass? So one can show that if I apply the same GRW gaution, in that case my system will collapse twice as strongly in a, that before. So the center of mass now will be going like here and twice as huge. Okay. So bigger is the system, stronger will be the collapse of my weight function. Then this naturally comes out that macroscopic systems are actually localized faster than microscopic ones. Okay. And so this is the so-called amplification mechanism. And this amplification mechanism, again solves the problem of having macroscopic systems of not having macroscopic system in a superposition. Okay. So in some way, the GRW, the GRW model is some sort of toy model of a more complex one. And this is the so-called CSL model. The continuous, spontaneous localization model. I will give you the equation for the psi of this model. And this reads in this way. So we have the Hamiltonian. So the Schrodinger part. We have the nonlinear part. So this is the expectation value of this operator M performed on the state psi. The stochastic part. And we have here a white noise acting on every point in space and time. Okay. And this noise here is defined in this way. X, T of Y is white. So the expectation value of this noise gives a dt and a direct delta in space. Okay. Then now the definition of this guy here M is a Gaussian at the end of the day of this form here. So we have this operator is a smeared mass density. I'm counting how many particles I have in Y, then I smeared with this smearing centered in X and just waiting with the mass of the particle. Okay. Nice. So this equation here has the same structure as those we saw before but has also another nice feature which is written in terms of creation and creation operators in the second quantized formalism. So what happens here is that we still we have also the preservation of the symmetries of identical particles which is also something even more happy here. Okay. And now that's the nice part also my work. This model here is described in terms of two parameters lambda and rc. Okay. So these are two phenomenological parameters and one has to guess the numbers for these two values to actually solve all the problems. So actually say my experiment I cannot have a superposition that large or with that big mass. So there are some proposed values. For example, there are the remaining Weber proposed lambda 10 to minus 17 seconds minus one and rc to minus seven meters. And this is actually consistent with saying okay I cannot have this lambda too small because otherwise my chair will be can be in a superposition. So macroscopic systems need to be killed. And with how do I describe this dimension? So how fast and how big should be my system? So how fast should be my collapse? How big should be my system? So this is the idea behind that numbers. And then one can also think okay how do I test this kind of models because still this is all theory but then we have we need to go to the lab and to see with our experiments with what we see every day in the lab. So let me start here. So the first kind of experiments one can think about are the most natural one superpositions. I make a superposition and I just see for how long this remains a superposition. Interferometric test. So for example I do my Mach sender interferometer and at the end of it I just measure the interferes franges. Okay. So what quantum theory says is that I would see something of this sort. Okay. But then what happens is that I have noises I have disturbances my lausers is not perfect my measurement is not perfect my I don't know what actually I see is something more of this sort. Okay. So at the end I have a contrast of this interferes franges which is reduced. So now could be this just the effect of my collapse models because actually this is what my collapse models predict are reduction of the interference franges. So my system is not in a superposition is just here or there. Okay. And so this is actually so this is this is quantum mechanics and this instead is quantum mechanics plus collapse models. Okay. And actually the effect of the collapse on my system can be described in terms in these terms. So the effect is that the distance between x and y x and x prime is bigger than r c then this is zero this is one and so the effect will be strong. If instead my distance is smaller then this is almost one so one minus one is zero and so I don't have any effect. Then I obviously need to add here the Schrodinger part. Okay. So the action of of my collapse is to kill the superpositions which are which are at the distances larger than r c. So r c is describing how much I can how big superposition can stay in my experiment. Larger. I just kill it. So this kind of experiments actually interesting and I will try to to give you an idea of this parameter space here. Let's see. So this is log lambda. This will be log r c. Just put some some numbers. So ten minus twenty. Ten minus two. Here we start from ten minus ten. So this is and this is in meters. This is in seconds minus one. So for example the GRW values are somewhere here. Ten minus seven. Ten minus ten minus seventeen. So because of the same idea two big particles should collapse really strongly there is a theoretical lower bound here which is more or less going in this direction. Ten minus ten. Ten. So everything which is below this line cannot actually describe the fact that at the macroscopic level I don't have superpositions. Then I do this kind of experiments and I see that at ten minus four there is this experiment much more of making this a few years ago they made with single atoms a superposition of half a meter. So it's more or less half a meter and this is the bound. So all these values of lambda RC are actually excluded by this experiment. Then another experiment is done with something really similar to this one. So again superposition of macroscopic of macromolecules of masses ten to the fourth mu. And the bound more or less as of this shape here ten minus six. Well there's still quite a big region that should be explored here. And what actually we did was to consider not just this direct effect of of collapse models but also undirect ones. So what do we have here is at the end of the day a stochastic modification. Yes it's linear but it's also stochastic. So the new linearity makes the collapse. The stochasticity makes some noise. So what happens is that I can actually describe my my old dynamics in terms of stochastic potential which is of this form here. So this is the same operator as before and then I have a white noise here or if you prefer respect to the one before this was it. So I just starting a stochastic potential but this means that I can also retain a stochastic force from it. I V CSL from the Schildinger from the Eisenberg equations. Ok. So I have a noise and this noise is kicking my system my particle and so can actually the first thing is ok look I start with a particle of momentum kI ok then my my noise is kicking in so this is CSL noise and so my particle at the end my particle will have some other momentum ok ok but I can describe this kind of system I know how to do that so for example I think that I can see is how the energy of my system changes and ok then I just sum over all possible initial states over all the possible final states and I have the statistics of my kind of particles fermions arbosons then I have my transition energy so this is the difference between the final and the initial energy of my system and then I just compute the probability of having this transition and this probability I know I can compute it in terms of the standard perturbation theory so I start from the initial state I make it evolve as for the Schrodinger equation I kick my particle then I have still some evolution and I consider which is the projector on my final state I just integrate in time and so this is the amplitude this is the amplitude of this transition and so I get my evolution of the energy and what happens is that you do all this computation and you get that this actually gets a really nice and simple equation but what is important is that it doesn't depend on the statistics of my system on all the form of my system depends just on the mass on this straight lambda on the value so far c squared but what is really nice is that it is growing linearly in time so if I just leave there a particle a free particle then the energy of my particle the total energy of my particle would just grow due to this CSL mechanism so this is an actual measure for example there are dissipations there are other kind of emissions but consider for example a star and let's assume that this star here is actually heated only by this kind of process so everything that the energy that is flowing in is due to CSL instead whatever is flowing out is due to radiation so this is something that I can measure this is the black body radiation of a of a star for example so I measure this so from this I say ok look the system is actually in equilibrium so I have that all the energy coming out due to radiation is actually coming in due to CSL so I can put around on the values of lambda energy and surprisingly I get around which is really strong something of this sort it's kind of surprising that I did a lot of tests I make really big superposition or with quite big particles then with just a single observation I just rule out part of this parameter space wow so maybe I should consider more carefully this kind of systems this kind of tests so maybe this noise is really can be really strong goal we are actually really good at measuring it ok so I have still 20 minutes so I will go with a last example and this is the optomechanics but yesterday yesterday we we already saw something what is optomechanics what is which is the idea behind that so briefly what we have here is a fixed mirror movable mirror ok there is some light here the system is is a little bit more complex this is an open system we have the laser also so this is the so this is the spring this is the laser which is pumping my cavity my system here will oscillate but then we have also that we have dissipations on my system we have some noises effectively noise then we have that the cavity is also leaking energy but now what we do is just to add an external noise ok we just add here CSL this is the idea we are really good at measuring all this object here or we can try to push some of the parameters in such a way that we do that and then why shouldn't we just test CSL also in this way and in particular I will write it here so I can don't erase the graph in particular ok, yesterday we saw the mass equation for malisma now we will give you the quantum lunging equations for describing the system it's the mass and then we have the equation for the field in the cavity ok so the idea is that the one of yesterday we have our our harmonic oscillator here which is coupled through some light we have dissipation we have noises due to the environment and then we just add the stochastic force due to CSL and the field is described as an harmonic oscillator is coupling with the position of our mechanical resonator clicking because the cavity is not perfect and then we have also some noises due to the input laser which is pumping our system ok, so the idea is that we just construct from here we solve this kind of problem we linearize it as we did yesterday so we can solve it in an easy way and this is done the linearization part exactly as we saw yesterday so in particular we take the cavity operator and say this is just equal to to this where alpha is much bigger than 1 ok and so we can neglect some terms the quadratic terms in a tilde so we solve this and then a trick is that all these operators here depend on time and so we can just describe them in the Fourier space let's call it in this way ok so this will be the operator in the Fourier space but in this way all the derivatives are just i omega times the operator and so we simplify this problem we can solve it and then at the end we define the DNS the density in the spectrum and this is the the auto correlation of the position in the Fourier space omega so I want to see how far two frequencies are correlated so which is the correlation among two frequencies the position at two different frequencies ok so this is all standard treatment of auto mechanics so what I get at the end of the day just to have an idea we got something of this sort so we have three contributions to this signal and this is what something that we can actually measure in the lab the first contribution is due to the laser so this is the noise describes the noise due to the laser ok then we have a contribution due to the environment and finally we have a contribution due to collapse models particular ok here this guy here which appears everywhere is something of this sort it's just a Laurentian ok centered in omega zero almost because there are some modification due to the presence of the laser and there is the gamma so the dissipation so the idea is that I will get something of this sort ok then we have S of CSL which put it here I really don't want to raise the picture here if I do the measurement along the x axis what I get is that CSL is of this form where I have an integral in K in the momentum in the transform of the mass density of my system and there is the smearing that becomes a Gaussian minus K square R C square ok and this actually quantifies which is the action of my system on my system ok and this is something that we can actually measure in the experiment so which is the idea the idea is that let's see ok the simplest way to see which is the the modification of of CSL so take for example the term of due to the environment no so we have something we have this this guy here mega missing there yeah some mega missing here so this guy here in the limit of high temperatures and we already saw yesterday that is always the case in the experiments this becomes so t much bigger so actually it's kBt much bigger than hbar omega this becomes something of of this sort can be approximated to this so which means gamma kBt ok then I just multiply by the mass I divided by the mass and this is the mass which is missing between the two terms ok and so you can see that if I take again the last line of that now I get something of this form if the if gamma is actually fixed the effect of CSL is to change the effective temperature of my system so this where is it this is equal I don't know if you but I revert it here so the effective temperature of my system is equal to that of the environment plus something else so if I can measure this something else or the error of this measurement will actually describe will actually give me which is the bound of my collapse parameters ok and we are so there are a lot of experiment doing this for example I can just cool down cantilever with a ball ball there is a squid on it measuring how this ball jiggles and so we can measure the effective temperature of the center of mass of this system and the bound goes something of this form then there were two two experiments here no which are centered on the dimension of my sphere then another really strong bound was given by LIGO so you all know LIGO so this is the measurement of the experiment measuring the gravitational waves so I don't have any other nice color no you know it yes ok the other as always we have a cavity two mirrors then there are actually other two mirrors here there is a long distance kilometer distance between this then and you do interference between the light going here and going there these two are fixed these two can move ok so if a gravitational wave passes in one of the two directions then you have that the light here is not will the contours will be difference between the one arm but that really doesn't matter for us what we are interested is the noise which is acting on this on this masses which I want to point out these are 40 kilogram masses so they are huge but this system here actually puts a bound which is of this sort so this is all the part of the collapse parameters which is ruled out with LIGO and then there is also another and this is the last one experiment we consider and this is Lisa Pathfinder Lisa Pathfinder is also a really nice experiment so there is this proposal of doing an experiment for measuring gravitational waves in space by using free satellites ok so it's the same ideas there and linked by some light and then you measure the relative distance between these free satellites obviously this is something extremely expensive so before sending up free satellites they sent just one and this is Lisa Pathfinder in the satellite one has two masses and the distance between these two masses as I just mentioned these are one kilogram masses then so you have the two masses and the nice part is that these two masses are in freefall in space this is the nice part because there's actually nothing which is touching them so all the seismic noises all the electric noises everything that you have actually in your lab in ground you don't in this experiments you don't have it and this is why this experiment here is placing above which is more or less here so this is also ruled out and this is more or less the state of art of the collapse model testing up to now you have questions I suppose a lot or none that's ruled out this part here yes these are still unexplored values of this parameters lambda RC so that's the possibility on how to test that one of these values here is actually describes collapse models yes exactly and the challenge now also if you have an experiment or you think that an experiment can be useful to test that is to go as down as possible with these lines with these upper bounds ok no questions then I think I will stop here thank you very much and see you tomorrow morning