 Greetings and this will be the last class of this course and I will provide a very rudimentary introduction to some further applications of atomic physics and in particular we have been talking about atoms which are in the presence of external fields. We considered the electric fields and the magnetic fields and there are other kinds of fields that one can talk about. So, there will be a very rudimentary introduction to things like laser cooling, Bose Einstein condensation and we will see how it is how it leads to very accurate measurements of time and we will talk a little bit about auto second metrology and so on. Essentially, we will discover that we need additional tools to study these topics in details because we will need further base in quantum collision physics and also in relativistic effects and in studying electron correlations and so on. So, that really becomes the subject of a whole additional course. So, let us go back to what we studied in the Zeeman effect and we studied the sodium atom in a magnetic field and we found that the d 1 d 2 2 lines split into 10 lines 4 lines from the n p 1 half and 6 lines from the n p 3 half. So, in total of 10 lines is what you get, but then there is more to atomic structure than what we have considered so far and that is the hyperfine structure because we did in the context of perturbation theory we always said that all perturbations which are of the same order of importance must be considered together and the more important ones must be considered first and the less important later and the perturbations can be because of internal structure which we have not considered at that until that point like earlier when we ignored the relativistic effects or when we ignored the spin orbit effects, spin orbit interaction. Now, these are internal to the atomic structure the atom exists along with these properties you cannot add these properties when you put it in an external magnetic field on an electric field you have some control on that you can switch on that field or switch it off, but you cannot switch off a spin orbit interaction in the atom it is there. So, likewise there is an additional internal structure which is the hyperfine structure and this comes from the nuclear spin and the nuclear spin angular momentum which we have represented here as this I. So, this is the nuclear spin angular momentum and this would couple to the net angular momentum which is coming from the L plus S coupling and you get the hyperfine structure coming from this I dot J interaction and this has some very exciting you know applications in atomic physics and this comes together in the quest for measuring time and here is a quote from an article by William Phillips who got the Nobel prize for laser cooling and this paper is in the reviews of modern physics which we have uploaded at the course web page and Phillips points out in this article that the desire to reduce emotional effects in spectroscopy and atomic clocks was and remains a major motivation for the cooling of both neutral atoms and ions and this is where the hyperfine structure plays a big role in enabling us to go for things like laser cooling both science and condensation and measurement of time in a very precise manner. So, you need to slow down the atoms to be able to see their structure and properties and then measure the frequency of transitions between two different levels and accurate measurement of frequency is what will give you a standard for measuring time because frequency is just the inverse of time. So, how would you slow down an atom and one knows just by looking into the sky that you can slow down the atom by shining light on it because you always know that the comets have their tails which are directed away from the sun no matter where they are on the orbit it is not that the tail trails the atom on the trajectory, but it is always directed away from the sun and that is because of the radiation pressure. So, here is the NASA picture of the Halle's comet and one knows that if you shine light on an atom you can actually slow it down. So, this is called as a scattering force and this is all again a figure from this wonderful article by Phillips which I very strongly recommend and I will only provide a very brief introduction to this article that when you have an atom which is moving from left to right let us say and a photon which is moving from right to left and the collide and the atom absorbs this light then it is going to slow down its velocity will come down by the linear momentum of the photon divided by the mass of the atom and eventually the atom will get excited because it is absorbed that energy and when it is excited it will also radiate that excess energy after it has lived its lifetime in the excited state and then it will cool down. So, I will show you this cooling cycle how this cooling actually takes place because what is happening is the translational kinetic energy gets gradually converted into the energy which is radiated away from the atom. So, there is an average on an average the energy which is radiated away would be radiated in any arbitrary direction it not necessarily in a given direction because that is coming out of spontaneous radiation of energy. So, this is the wonderful piece of work for which Stephen Chou, Turnout G and William Phillips shared a Nobel prize in 1997 for laser cooling and to understand this process of cooling we consider a two level atom. So, let us consider a two level atom in a state J equal to 0 and absorption of a photon could raise it to J equal to 1 excited state and let us say that light falls on it resonant light which is appropriate for this transition from the lower state to the excited state and the atom gets excited. Now, what is going to happen is that the angular momentum of the photon is absorbed and the internal atomic quantum number the angular momentum quantum numbers they change and the internal changes in the atomic structure raises it to through a delta J equal to 1. So, that is where the angular momentum is being taken care of, but what happens to the linear momentum of the photon the linear momentum of the photon cannot change the internal structure of the atom. So, the linear momentum of the photon ends up changing the velocity of the atom in the laboratory frame of reference because it cannot change the internal structure the angular momentum changes the internal structure, but not the linear momentum and if you look at the expression for energy which is equal to p c which is which are my initials and there is absolutely no coincidence about this the momentum is h cross k. So, this momentum which is absorbed by the photon must change the momentum of the atom in the laboratory frame. So, this is where the change in velocity of the atom will result. So, this is let us see how this process actually takes place. So, you have got this energy which is absorbed by the atom the atom then goes to an excited state. So, this is the cooling cycle which I am depicting in these in this picture. So, the atom is raised to an excited state and then it loses energy. Now, there are so many different possibilities of course, when there is one transition from the excited state to the lower state only one photon is going to be emitted and it is going to be one of these. So, it is not that there are so many different photons which are being emitted in all the directions that will not even conserve energy. So, only the energy difference between that is going to be radiated away and it will go in one of these directions, but it could be any one of these it does not have to be any chosen one of these because this is happening through the process of spontaneous emission and the radiation can take place in any arbitrary direction. So, what happens after the radiation is emitted through spontaneous emission the atom would go down to the lower state as it started out with. So, it comes back to the lower state and now it is ready to absorb another photon and this is what is called as the cooling cycle because now it can it is it now becomes ready to absorb the next photon and the next photon raises it to an excited state and now once again it is going to emit light through spontaneous emission, but it will not be necessarily in the earlier step if it lost light in this direction in this type around it might lose it in this direction or it could lose a light in some other direction and when it goes through a number of such cycles every time it is going to lose light in a different direction. Now, this is where all of these different directions come into play and what is the net average of all of this when it goes through a number of cycles whatever you know kick it gets because of the recoil coming from this emission spontaneous emission of a photon it gets averaged to 0 and the result is that it ends up getting a net extra momentum in the direction of the laser light which is the original direction. So, that is how it gains momentum in a particular direction because the recoil kick it gets by emitting the photon in different directions in different cycles it gets averaged out. So, this is the principle of laser cooling because so it does not happen in a single step, but when it happens again and again and again and again over a number of cycles and you can estimate how many times this will have to take place because you know that the rate at which it will lose energy will depend on what is the level width of the excited state. The excited state is of course, not a sharp one if it is sharp the atom would never decay because then it would have infinite lifetime and it the excited state has got a certain width and therefore, a finite lifetime which goes as the inverse of the width through the uncertainty principle and that is what results in the recoil which is the momentum divided by the mass of the atom and this will happen at half life of the excited state. So, if gamma h cross gamma is the energy width then this will happen at a rate which is gamma inverse by 2. So, this will result in an acceleration which is the velocity multiplied by this rate because acceleration is just the rate of change of velocity. So, this is the rate at which you know the atom will be accelerated and it will be accelerated it will be losing its kinetic energy it will be losing its velocity in the direction in which it is approaching the laser and that is what results in cooling. So, this is how cooling results what is happening is that the atoms translational kinetic energy gets converted into optical energy through spontaneous emission of photons. So, there is an energy transfer through this process. So, this is like taking an additional degree of freedom into account which was not there originally which is losing light to the surrounding. So, you can estimate how long it will take because if you plug in these numbers now you if you plug in these numbers for the sodium atom for example, if you consider its lifetime and mass and so on the resulting accelerations are as large as 10 to the 4 or 10 to the 5 times the acceleration due to gravity. So, you the atom goes through very high you know accelerations because of this and you can work out these numbers if you have the sodium atom of photon can be radiated at about every 30 to 32 nanoseconds on an average and the atom can be brought to rest in about a millisecond. So, this is you need all together a number of something like 10 to the 4 cycles for this to happen because every time it goes through this cycle it is losing a little bit of kinetic energy it is losing a little bit of velocity and that is what results in cooling because thermodynamically we associate temperature with velocity of motion. So, this is the cooling cycle as we have seen and the other thing you must remember is that the atom and the laser of course, approach each other in different directions. So, they are coming opposite to each other. So, now there are some complicating factors and you have to worry about them because what you really need is not a laser of a frequency which is exactly appropriate corresponding to the energy difference of the atom when it is at rest, but the energy difference which is appropriate when it is in motion and there will be a Doppler correction which is required right and if that Doppler correction is not done the atom will not be able to absorb that energy. So, you need the atom to absorb this energy radiated, reabsorb, re-radiate, reabsorb and go through these cooling cycles. So, the Doppler effect is going to play a big role in this. So, you really need a laser which is red detuned you know with respect to the resonant atomic transition. So, there must be a red detuning, but then when you have a red detuning you have done it for the initial velocity, but then the atom now gets slowed. So, this red detuning has to be continuously changed and you can do this by changing the frequency of the laser that you are using and this is what is called as chirping. So, you need to do some chirping, so that you can achieve this cyclic you know loss of translational energy into the energy which is radiated away. So, there are other things that one has to worry about that if you of course, need a two level atom and you might think of taking something like the sodium atom which has got the 3 s ground state and the 3 p excited state, but then of course, there are other candidates in the alkali atom you know group one you can work with potassium, rubidium, cesium and so on. All of these are candidates for two level quantum systems, but then these are not strictly two level atoms, because we know that there is a fine structure. So, the excited state 3 p is already a doublet this is the spin orbit doublet and then there is a hyper fine structure there is a nucleus spin which for the sodium atom is three half and there is this hyper fine structure interaction which I mentioned at the beginning of this class. So, you have to take this interaction also into account, now when you do that the resultant angular momentum will be given by angular momentum coupling of j with i and the f value will go from j plus i to modulus of j minus i. According to the laws of angular momentum coupling and this means that if you take this value of j which is three half and this value of i which is three half the resultant value of f will go from 3, 2, 1 and 0. So, you will get a quarter. So, you get four levels from the excited 2 p three half state likewise for j equal to half and i equal to three half you will get f equal to 2 and 1 for the 2 p one half that also spreads into a doublet because of the hyper fine structure and the lower level is also not unique that will also split into f equal to 2 and f equal to 1 levels. So, there is a lot of detail that one really has to be concerned with. So, these are the this is how the splitting takes place and now you have a large number of transitions which are possible. So, you have got a quartet coming out of this 2 p three half state and then you can have a large number of transitions which are possible and all of them are not conducive for the cycling process. So, this creates some difficulty, but it also enables some solutions and the difficulty it poses is coming because of this additional hyper fine structure and there is this considerable splitting between these energies and this is given here for the sodium atom and units of frequency. So, this multiplied by h would be the energy differences. So, let us see what how we deal with this additional splitting and the additional lines which result from this. So, the ground state we have seen is not a unique level it is split by the hyper fine structure into 2 levels f equal to 2 and f equal to 1 and what it means is that if you have an excitation from f equal to 2 to this excitation and then in the cooling cycle when the de-excitation takes place through spontaneous emission it is quite possible that the atom would lose energy and come down to this state rather than to f equal to 2. Now, that is not good for cooling. So, this is a difficulty which William Phillips observed in his experiment and actually if it were to come down to the same level you could repump and then you would have an appropriate cooling cycle, but in the absence of that since it decays or it has the possibility of decay to a different level then you do not have the appropriate cooling cycle available. So, the line widths of the transitions of course are much smaller than the differences between these energies otherwise that would have taken care of it. So, that is not the case over here and what Phillips observed is that the absorption of light would get shut off. So, when he was carrying out his experiments he found that you know the cycle would not continue. So, I quote from his article here in the reviews of modern physics that this optical pumping made the atoms dark to my laser after they traveled only a short distance from the source. So, this is the difficulty that he faced and for reasons that we understand and the solution therefore, involved using a repumping laser. So, that it raises it from this level back to this puts it back over here and then you know regenerates the excited state which will then decay into the desired level. So, these are some of the tricks which are used there is a lot of detail that one has to really work with and this is certainly not an easy task which is why it ends up fetching a Nobel Prize. So, I strongly recommend that you read this article by Phillips. So, the using the repumper he was able to get a good amount of laser cooling achieved what essentially is happening is that atoms from a very narrow velocity range are transferred into a narrower range and you do it gradually step by step till you really achieve a lot of cooling. So, this is to be done using chirping is one way of doing it. So, adjusting the frequency of the cooling laser. So, that you know the cycling process is enabled is one strategy which is the chirping technique. Then as we are alerted by a comment from Metcalfe's article, one has to be careful in using the thermodynamic idea of a temperature because you are really dealing with quantum systems and there are as Metcalfe and Stratton point out in this article and also in their book that there are very many different distributions which have the same energy, but they are very different from each other. So, the idea of temperature is not completely appropriate as such, but nevertheless it is used in the sense in which slowing down of the atoms is considered. So, it is in that spirit that the idea of temperature is used and the term cooling is used and not quite in the manner in which it is used in classical thermodynamics. So, there are other ways of enabling the cooling cycle, chirping is what we mentioned earlier. The other thing you can do you have studied the Zeeman effect and now if you switch on a magnetic field very gently and you can control this. If you switch on the magnetic field, the again the energy level spacing between the different hyperfine levels can be controlled using the magnetic fields. Now, this is exactly what was coming in the way of the cycling process and this was compensated to certain extent by chirping. The alternative way of doing it would be to use the Zeeman effect and use an external magnetic field and this is done by having a large number of solenoids of different lengths and this is called as Zeeman cooling or the whole process is sometimes called as Zeeman slower and using the Zeeman effect which you know that there is a mu dot b coupling and that is proportional to the magnitude of the applied field and that is something that you can control and that will you know give you control on the spacing between the energy levels which is what you want to optimize to enable the cooling cycle. So, the Zeeman effect will fan out the hyperfine structure into a large number of different levels and then you can pick the transitions which are of interest to you and adjust the magnetic fields because your ultimate goal in this process is to enable a large number of these cycles of electromagnetic radiation absorption and subsequent emission through spontaneous decay in arbitrary different directions. So, that over a number of cycles the recoil from the spontaneous emission is averaged out to 0 and the atom slows down. So, this is enabled further you have to worry about the fact that if you just slow down an atom in one direction it can still escape because it has got velocity in different directions and if it has got a very high velocity in some other direction it is going to escape. So, somehow you have to make sure that it remains there and you have to inhibit its escape in other directions. So, one way of inhibiting its escape in other directions is to put it in a molasses and what the molasses this is an optical molasses that you have 3 orthogonal pairs of lasers which are traveling opposite to each other and they generate a molasses kind of atmosphere for the atom which prevents the escape of the atom and a molasses does something like that a molasses is actually it is a very thick you know kind of liquid kind of thing like honey and so on and it is a lovely picture that mouth watering perhaps, but then as you can see if something is stuck in this it is not going to escape. So, that is the idea over here that this is an optical molasses in which you inhibit the escape of the atom out of this zone. So, that is the reason this is called an optical molasses then you do use additional you know techniques like magneto optical traps and then you also exploit what is called as evaporative cooling you know what evaporative cooling is that if you have a hot cup of tea over here it eventually cools down because you know the hotter molecules they are jumping out of the surface and they just escape. So, what is left behind is cooler than what was along with the faster molecules are the ones which run away from the cup of tea. So, that is evaporative cooling and all of this really is exploited in the laser cooling process and what is going to happen as the atom cools as the atom cools its momentum would go down because the whole idea or it is rather the other way round it is because the momentum is lowered that the atom cools. So, the momentum goes down through this cycling process and correspondingly the temperature gets lowered and as the momentum goes down the de Broglie wavelength is going to increase because the de Broglie wavelength goes as inverse momentum. So, as the atom is slowed down the de Broglie wavelength increases and if it increases significantly to overlap with the wavelength of the neighbor you will start losing the distinction between the first atom and the other atom. So, that depends on how many atoms are packed together in a certain region. So, it has to do with the density of atoms and you can work out these calculations in details that if you have if rho is the number of particles per unit volume then if this product of rho times lambda cube where lambda is the de Broglie wavelength corresponding to a certain temperature T then it turns out that if this number is greater than 2.612 you can expect to see that you begin to lose the distinction between different atoms and all the atoms then undergo a phase transition into a condensate which is the Bose condense matter. So, this is a completely new kind of phase and you can achieve Bose Einstein condensation which was predicted by Satrin Dronath Bose in his very famous article and you can then get a Bose Einstein condensate of atoms. But then of course, the atom has to be a Bose on for that and an atom is a Bose on if the number of fermions in the atom is an even number. So, that is the criterion to get a Bose Einstein condensate. So, for any neutral atom the number of protons is equal to the number of electrons. So, that gives you an even number when you add them up. So, whether or not an atom is a Bose on depends on the number of neutrons in the atom. So, certain isotopes will be Bose on and some other isotopes may not be Bose on of the same atom. So, the statistics is determined essentially by the number of neutrons in the nucleus you recognize the instrument Bose is playing. What is it? Loudly, what is it called? Is it? I do not know, but maybe you have the correct name. I thought it was a Dilruba. Anyway, it is a Bose instrument and for achieving Bose Einstein condensation you the Nobel prize was awarded in 2001. And this was lovely experiment done with alkali atoms and you see that in this picture you have got the number of atoms and the pixels are just color codes and they only tell you that there are cooler atoms with the white color in this particular figure. And there is a small number which really leads to the Bose Einstein condensate and this is what the Nobel prize to Cornel Ketterli and Wiemann. And this has an important consequence as I mentioned at the very beginning on the accuracy with which time is measured because that was and remains as Phillips tells us in his article a major motivation for cooling atoms. Because if you look at the clock uncertainty and this is a picture from this website over here, if you look at the clock uncertainty and this is listed here in units of nanoseconds per day then in 1950 it was about 10,000 nanoseconds per day that was the accuracy of the clock according to the technology which was available at the time. And then slowly this improved and you get more and more accurate clocks and now it is approaching these numbers over here. So, you get extremely accurate clocks which you really need to monitor your global positioning system and many other you know processes. So, here is a table in which some of the alkali atoms are listed and you notice that the number of neutrons over here is even. So, these are good candidates to condense and they have a certain nuclear spin. So, you will have the hyperfine structure and then you can use all the techniques which have been developed to exploit this you know the chirping, the zeeman slowers, the evaporative cooling, the optical molasses and so on. And you can get a Bose Einstein condensate. Now, what is going to happen if the atoms that you are cooling are not bosons because that is going to depend on the number of total number of fermions in the system. You will not expect a condensate because all the bosons can fall into their lowest state, but Fermi particles cannot do that and you know that Fermi statistics is it has this additional property which is the exclusion principle because the total wave function has to be anti symmetric. So, because of the exclusion principle if you have Fermi atoms they will not fall into a condensate, but they will occupy the lowest states, but one by one and you cannot then get a condensate from Fermi atoms. Now, if you look at this isotope of potassium this is a fermion it has got 21 neutrons the number of protons and electrons is equal both equal to 19. So, it is a Fermi atom and this is what has been achieved with this atom this is certainly not a Bose atom. Now, this experiment was done by Cindy Regal group Regal, Reiner and Jinn and this is reported in the physical review letters of 2004 on this particular isotope of potassium this it is a radioactive isotope, but it is it has got a very large lifetime and a condensate of this atom has been achieved by Regal and the way it is done is by having additional controls because they had this magnetic field which influences what are known as Fano-Freschback resonances. And then Fano-Freschback resonance is a very fascinating phenomenon which comes from an interaction between bound to bound transitions and bound to continuum transitions. So, it is a resonant phenomenon and on one side of it you have a Bose science and condensate and so that is what has been achieved through a correlation effect this is essentially like getting a correlation effect and this is what led to the fermionic condensates because the two fermions can pair to give a Bose on like state just like two electrons give you the cooper pair in superconductivity. So, that is the B C S to B E C transition and all of this is very fascinating, but you need some additional techniques additional tools to study these processes you need to understand what is called as a Feschback resonance or a Fano-Freschback resonance and this comes from quantum collision theory in atomic physics. So, you need some additional tools. So, this has led to some very high precision development high precision atomic clocks the cesium atomic clock comes from this transition 6 S to 6 P which has got a fine structure and then a hyper fine structure coming from the nucleus pin which is 7 half. So, the 6 P 3 half state gives you additional levels it gives you this quadrate with F equal to 4 5 4 3 and 2 coming from the combination of these two angular momenta and then the 6 P 1 half state gives you this F equal to 4 and 3 doublet and the lower state is also a doublet with F equal to 4 and 3. So, if you look at the cesium transitions then the particular transition between these two states this transition is between M equal to 0 to M equal to 0 state and that is nice because there will be no first order Zeeman effect on this. Nevertheless, even the second order effects are of some importance and this is what gives you the frequency standard or the standard for measurement of time. So, this transition takes place at a frequency of 9.192631770 megahertz and this is what gives us the definition of a second. The second that we speak about all the time is defined as these number of cycles for this corresponding to this transition. So, this is how the second is defined technically and strictly speaking. So, these are very high precision measurements and this really brings us to other techniques which are required in measuring time because you need to be able to measure time with that level of accuracy. When you develop the theoretical models you need to take into account electron correlations, relativistic many body effects, quantum collision effects and so on. So, there is a whole vast techniques in quantum mechanics and relativistic quantum mechanics that one learns about cause for an independent complete course. The projected accuracy is like 10 to the 18 at this current level, but only few days ago a paper came out in which an accuracy of 10 to the 19 is projected. So, you can imagine how hard it would be to raise the accuracy by even one order and this is a paper which came out just a few days ago in PRL by Derivianco and his collaborators Zuba and Flambom and they found that if they take highly charged ions like Bismuth ionized 25 times, they predicted an accuracy of 10 to the 19 and what is the age of the universe? Tell me in seconds. That is about 10 to the 17 seconds or so. The age of the universe is about 10 to the 17 seconds and that is the kind of accuracy that these clocks are really aiming for. So, you are not going to go wrong and this really needs very sophisticated techniques in quantum mechanics to study these processes and they clearly go beyond the scope of our introductory course in atomic physics, but the prediction of Derivianco and their collaborators is that highly charged ions will be excellent candidates to build atomic clocks. So, that brings us to techniques which are able to measure time at that accuracy. Well, this is a calculation, this is a first highly charged ion which is predicted to give this accuracy of 10 to the 19. So, you need to be able to measure time and very short time intervals. You want to be able to measure like 10 to the, you know means earlier nanosecond was a big thing and then the femtosecond and now this is like a thousand order, you know quicker than a femtosecond. So, that brings us to what is called as autosecond metrology and autosecond is 10 to the minus 18 seconds and to be able to measure processes of this kind, you need tools which are high precision electronics which can really record these events and if you look at a photo emission process, if light is absorbed by an atom and the electron is knocked out, then it turns out that you can actually measure whether the electron comes out right at the instant at which the photon is absorbed or does it come out a little late and if it comes out a little late, how much is the delay and these are measured using techniques which are called as streaking techniques, electron streaking techniques and using these techniques in this experiment which is which you will find in this article by Schultz, they find that if you have neon atom and subjected to photo ionization, you will get electrons which are coming out of the neon atom from the 2 p shell and also from the 2 s shell. Now, it turns out that there is a little delay in the 2 p electrons compared to the 2 s and that delay is of the order of 20, 21 auto seconds with a certain degree of accuracy, it is not exact 21 seconds with their instrumental accuracy, they are able to measure these within plus or minus 5 auto seconds also, but these are extremely high precision measurements which can be carried out which is really amazing and there is a little bit of delay in the photo emission of the 2 p electrons with respect to the 2 s electron and that delays of the order of some auto seconds. So, that is the current trend in atomic physics in which there is a lot of excitement in cooling atoms and getting both science and condensation, auto second metrology and so on and this obviously needs thorough understanding of electron correlation effects because this when you think of a 2 p electron coming out, you think of a single electron as if it is undergoing a transition from a 2 p bound state to into the continuum, but that is not exactly what is happening because the 2 p electron has got this 1 over r 1 2 interaction with the rest of the electrons. So, the neon has got 10 electrons and the 2 p electron has the 1 over r 1 2 interaction with the remaining 9 electrons and you can average out this in the Hartree-Fock or the Dirac-Fock or the Dirac-Hartree-Fock self consistent field, but that does not take into account the correlations between the electrons and that is something that we studied in an earlier unit. So, you have to consider the electron correlations between these 2 processes. So, the time delay in the 2 p and the 2 s is also determined by these correlation effects and to be able to study these correlation effects, you really need very sophisticated tools, you need relativistic many body formalism and then if you are looking at these B C to B C S transitions affected by these magnetic field controlled phono fresh bag resonances, you need to study quantum collision theory. Actually, it is a parameter called as scattering length in quantum collision theory which is controlled by magnetic fields to bring about this transitions and then you also need to study electron correlation effects, relativistic many body theory and so on. So, that is subject for a different course and we pretty much conclude this course over here. If there are any questions, I will be happy to take, otherwise smiles everybody. Questions, spontaneous emission is what will give you the randomness. So, there will be some stimulated emission, you are not going to be able to choose one or the other, but those atoms which are participating which are getting de excited in spontaneous emissions are the ones which will emit in arbitrary random direction and when they are there in the system, they are the ones which will lose that excess energy and get ready to absorb laser light once again and when they go through the next cycle again and again, may be one of those in one of those cycles they will end up emitting through stimulated emission, but does not matter even if that happens once, twice, ten times or hundred times, you are talking about something like ten to the four, ten to the five cycles of de excitation and through these, there will be a number of spontaneous emissions and those spontaneous emissions will end up providing a recoil which is effectively zero, because spontaneous emission will be in random direction and because that is in random direction, the net transfer of momentum to the atom is what will oppose its initial velocity and reduce its velocity decelerated against the direction of the laser and cool it. So, very well. Thank you all very much.