Claude Cohen-Tannoudji (Collège de France), "Atom-Photon Interactions” - MIT Physics Department Special Seminar

[MUSIC PLAYING]

INTERVIEWER: It gives me exceptional pleasure to introduce Professor Claude Cohen-Tannoudji of the College de France for the special seminar that we're having today in honor of his having won the Lilienfeld Prize at the American Physical Society. The Lillienfeld Prize is awarded to a scientist of exceptional distinction in science and also in the ability to communicate science to others, and I won't labor the appropriateness of that prize since I think that will become self-evident during the course of his talk. I would like to talk a little bit about Claude's background, because we've been friends for many years. And I have known and appreciated his work for many years, and admired it, and admire the tradition from which it comes. It is the tradition of Alfred Kastler, who essentially played the pivotal role in reconstructing science in France after World War II, who pursued a life of interest in the interactions of atoms with radiation, and set a style for teaching and for lecturing which has influenced his school, the Ecole Normale Superieure, and who established a tradition of which Claude Cohen-Tannoudji is the inheritor.

He has passed on this tradition-- many of you have heard his students speak-- Serge Haroche, for one-- Alain Aspect who technically was not a student but worked very closely with Claude in recent years is another exemplar of that tradition. Claude's work has been concerned with the interactions of atoms with radiation from his graduate student days, when he attacked the problem of light shifts-- which were then a very mysterious effect, which were confusing people involved with precise optical pumping experiments. And out of those studies, eventually came his theory of the dressed-atom theory. It really sounds so much nicer in French-- [FRENCH]-- but in English it's the dressed-atom theory. And this is a picture which is now widely used to discuss phenomena, of atoms, and radiation and many other phenomena too. So this whole new way of looking at quantum systems was developed by Claude.

In addition, many of you are probably familiar with his name through his books-- his famous textbook on quantum mechanics and more recently two monographs on quantum optics and interactions of atoms, the radiation field. The latest one, of which, had just been published in English, I am sure you're going to be the standard works in these areas in the future. Claude is at the Ecole Normale Superieure, but actually he is a professor at the College de France. And this is all too complicated to explain, but I do want to point out the uniqueness of being a professor at the College de France, because of it is the most demanding institution in the world for a teacher. It is a demanding institution because there are no examinations that it gives, no grades that it gives, no degrees that it gives, and no students that it has.

Now you might think this makes it all very easy, but it isn't. On the contrary, it makes it very demanding, because the lecturers at the College de France are open to the public, and people come simply because they want to learn and not because they have to. And if you are not a good teacher, you lose your audience, and this puts a tremendous strain on the teacher, I think. At least, I couldn't conceive of trying to really attract my audience willingly week after week and, in fact, year after year, because there's another rule there that very few of us would put up with. Namely, you do not repeat your lectures. Each year's sequence is a new series of lectures, so it is for these reasons the most demanding institution the world for teaching.

And Claude is one of the great teachers at the College de France. His lectures there are packed. People come from wide areas around to hear him, and without saying more I think you will see why. Claude.

TANNOUDJI: Thank you very much for your introduction. Before starting, I would like to tell you I'm honored and happy to be here again in MIT, because I was here in '76. I spent some sabbatical stay in MIT and Harvard, and it's, for me, a very great pleasure to be here for a few days. And I would like to express my warmest thanks for Dan, who, immediately after the announcement of the Lillienfeld Prize, invited me to give one of these lectures here at MIT. Thank you very much, Dan.

My lecture will deal with photon-atom induction, which as you know play an essential role in atomic physics, and that can be considered from two different points of view. First, they provide information of atoms. They can be considered as a source of information, because by looking at the light which is absorbed or emitted by atoms, you can get information on the structure on energy levels-- that the main-- the whole field of spectroscopy. There is a second point of view, which is that this interaction can be used for acting on atom and for controlling their degrees of freedom. In the '50s, '60s, as Dan mentioned to you, in the group of Alfred Kastler and [INAUDIBLE] there have been a lot of experiments where exchange of angular momentum between atoms and photons were used to control the internal degrees of freedom of an atom as a field of optical pumping.

Now more recently during the last ten years, a new field is exploding, which is called laser cooling and trapping. And one uses it is laser light to control the external degrees of freedom of atoms-- the velocity and the position. And what I would like to focus in on in my lecture is this second point of view. I would like to show you how it's possible to manipulate atoms with photons. And I would like also to show you that there is a certain continuity between these two lines, and how optical pumping turns out to be a very important aspect of laser cooling and trapping.

My lecture will be the following. I will first give you a few examples of processes where-- one uses photon-atom interaction for controlling the polarizations, the energies, the velocity of the position of an atom. All these effects use the basic conservation laws of energy, and angular momentum, linear momentum, and are just a simple illustration of quantum mechanics. Then in the second part, I will describe a more recent development in this field, which allowed us to cool atoms to the micro Kelvin range. And to get what we call now cold atoms. And I will describe, mainly, two effects-- two cooling mechanisms-- the so-called sisyphus cooling mechanism and the velocity selective coherent population trapping which results from an interplay between optical pumping, light shift, and radiative forces. And finally in the last part of my lecture, I would like to discuss a few possible applications of these cold atoms, and I've chosen two applications.

First are caesium atomic clocks using Zacharias atomic fountain. And I am particularly happy to see because of that here because Zacharias was trying to do such a scheme here some time ago. And the second example will be a gravitational cavity for neutral atoms. So let me start with a few processes.

And the first one will be optical pumping, which is a transfer of angular momentum from polarized photon to atoms. As you know, the angular momentum, which is carried by a photon, depend on the polarization of light. If you take a sigma plus light propagating along the z-direction, the photon associated with this light beam carry an angular momentum plus each bar with respect to this axis. Suppose that this beam is resonant for an example of atom, which is contained in a cell. And suppose, to simplify and correspond to actual experiment, the atomic transition connects a ground state to be the angular momentum 1/2 to an excited state with an angular momentum 1/2. So we have two Zeeman sublevels in the lower state and two Zeeman sublevels in the upper state.

The sigma plus light can excite only the transition between g minus 1/2 and to e plus 1/2, because when the atom goes from this state to this state it gains an angular momentum delta m equals plus 1. It goes from m equals to minus 1/2 to m equals plus 1/2, and these angular momentum is provided by the absorbed photon-- sigma plus photon-- so only this transition is excited by the atom. Then when the item is an excited state after a short time, which is a lifetime of the excited state, it falls back to the ground state either here, in which case it returns to the initial state, or by emitting a pi photon, delta m equals zero in which case it falls back in g plus 1/2 where it remains trapped, because there is no sigma plus transitions starting from g plus 1/2. So at the end of such a cycle, atoms have been pumped from here to here via e plus 1/2, and they have gain on the average an angular momentum delta m equals plus 1. And if the absorption time due to the collision of atoms on the walls is long enough, it's very easy to put nearly 100% of the atom in this state. So that optical pumping mechanism is a very efficient method for polarizing atoms with 100% efficiency.

Now I'd like to emphasize another important point that such a scheme is also a very sensitive detection method, because as you can see on this picture only atoms in g minus 1/2 can absorb sigma plus photon. So by measuring the absorption of the light beam of the amount of remitted light, you have a signal which is proportional to the population of this level. Now if all atoms are accumulated here, the vapor is transparent. Now you can-- if you apply a static magnetic field along the axis, introduce the z match splitting between these two states, and apply an resonant radio frequency field which induces radio frequency between the two Zeeman sublevels. Usually, magnetic resonance is detected by the absorption of the RF photons, but you see here that any RF transition bringing the atom from g plus 1/2 to g minus 1/2 can be detected not on the RF photon but on the subsequent absorption of an optical photon from g minus 1/2 to e plus 1/2. So we have an enormous amplification factor, because it's much easier to detect an optical photon than an ideal frequency photon. This is why such methods have provided the possibility to study a very dilute substance.

Now you could see now-- ask two questions. I have taken here a resonant light beam. What happens if the light beam is slightly off resonance? I've taken the light beam along the magnetic field. What happened is the magnetic field is no longer parallel, but perpendicular, to the magnetic field? I would like to study, now, these two cases and to show you that interesting things have happened. First let me consider what happens if the light beam is perpendicular to the magnetic field. It's called transverse optical pumping.

You see that if the light beam was alone, atoms would be optically pumped in the plus x direction, and they will be prepared in a state which is the eigenvector of gx-- the component of the angular momentum along the x-axis, which is, itself, this linear superposition of g plus 1/2 along z and g minus 1/2 along z. But that's not the whole story, because you have now a competition between optical pumping, which prepares atoms in this state, and the larmor precision around the magnetic field, which makes this beam-- this magnetic moment-- persisting around the [INAUDIBLE] with the Larmor frequency proportional to b0, omega l, Larmor frequency. So you have a competition between two processors with different symmetry.

So if you want for a given value of the magnetic field determining what is your steady state orientation in the ground state, you have to sum the contributions of all atoms which have been optically pumped in the path of t. That is to see for t minus tau with tau postive, and which a time t has not yet been disoriented by collision or by absorption-- which have survived all relaxation processes. So what you have to do in the x or y plane is to consider atoms which have been just pumped at time t equals-- at tau equals zero times t, and which are oriented along x with a magnitude 1-- let us take 1-- and then atoms which had been pumped at time tau before which are rotated by an angle omega l tau-- and which have survived relaxation process, so their contribution has been decreased by e to the minus tau over tau g is the relaxation time of the ground state. And then you have to integrate over tau from zero to infinity.

And you see that in this case, it is clear that you have two limiting cases-- the case where during the relaxation time tau g the larmor rotation is very small in which case you have to make the vector here sum of vectors which are nearly parallel. So you get the large value of gx and the small value of gy which is roughly omega l tau g-- and the second case where omega l tau g is much larger than 1 in which case all the vector you have to sum are nearly isotopically distributed-- in which case gx and gy is equal to zero. And you see that your critical value of the field is field set that omega l tau g is of the order of one radian. So if you plot the steady state value of gx and gy, this is omega l. You see that you have an absorption shape for gx, a dispersion shape for gy, and the width of these two curves is delta omega l-- where delta omega is said that the corresponding delta d0-- delta omega l is such that delta omega l tau is of the order of 1.

If you take a cell with a very good relaxation time, let us see one second. And if you take a paramagnetic atom in the ground state 1 megahertz per gauss, this width a magnetic field correspond only to 10 to the minus 6 gauss, so you see that this is nothing but the Hanle effect. But the Hanle effect in ground state, where because the lifetime is much longer than in excited state, the [INAUDIBLE] resonance squared are extremely narrow. And actually, we observed such effects a long time ago on 87 rubidium with Jacques Dupont-Roc and Serge Haroche.

And let me give you, here, an example of Hanle curve in the ground state of 87 rubidium. You see that the width is 1 micro gauss. The signal to noise is about 10 to the 4. So if you put here the-- if you put yourself here and you vary the film, you see that you can detect, very easily, a fraction of nano Gauss, so that's a very sensitive magnetometer. I think its sensitivity, which is comparable to the screen. And we choose this at that time that was using just ordinary lamps. And actually, this magnetometer was sensitive enough to detect the static field produced by a gas of oriented nuclei. One takes a cell of 10-centimeter diameter filled with optically-oriented helium 3 nuclei with the density of 5.10 to the 15 centimeter cubed in 100-centimeter-cubed volume.

And we put the rubidium cell nearby, and we make the Hanle resonances on this cell. And we detect the field-- the macroscopic field produced by this oriented nuclei-- nucleus, a very small magnetic moment-- at the macroscopic descent of 10 centimeters. And to make things with more fun, we apply a very small magnetic field to the helium nuclei in order to have them processing with a Larmor period of a few minutes. And that the field we detect on the rubidium cell, and you see a very good Larmor-- detection of the Larmor frequency three hours later and 11 hours later, and we still see the nuclei rotating. Actually, this experiment is done in a magnetic shield, and we tried to apply it to-- to see the magnetic field produced by [INAUDIBLE], but the magnetic shield was too small for introducing even a baby in the sheaf. But now it's great. It's currently used to measure the magnetic field produced by the brain or by the [INAUDIBLE].

Before leaving this experiment, I have described it simply with magnetic moment, larmor position. We can give a more sophisticated description, not for complicating things but because I think it provides a basis for understanding a more modern experiment. We have cooling atoms below the recoil, in which is called coherent population trapping. Because suppose that you have zero field. If you have zero field, you can take any quantisation axis. Now if you have zero field, you know that the atoms are optically pumped in the direction of the light beam, and they no longer absorb light. How can you understand that?

If you take the beam along the x-axis, and if you take, as a quantisation axis, the z-axis, a sigma plus beam along the x-axis as polarization which is a linear superposition of the sigma plus, sigma minus, and by polarization along the z-axis. So sigma plus along x is within a constant factor. Pi over z along z plus 1 over square root sigma plus along z, plus 1 over square root of 2 sigma minus along z. Now optical pumping prepares atoms in the g plus 1/2 x state, the eigen state of x with eigenvalue plus [INAUDIBLE] 2, which is a linear superposition of g minus 1/2 z and g plus 1/2 z. So atoms are in this state, and from these two states you have pi, sigma plus, sigma minus transitions. But what is nice here is you have a linear superposition of these two states, and the polarization of the field is a linear superposition of sigma plus pi and sigma minus.

And if you take this superposition, the amplitude for absorbing a sigma plus photon is the amplitude to be in g minus 1/2, which is 1 over square root of 2, times the amplitude for the beam to be sigma plus which is 1 over square root of 2. And the amplitude of absorbing a pi photon is the amplitude to be in g plus 1/2 which is 1 over square root of 2-- and the amplitude for the photon to be pi, which is pi. And if you multiply these two amplitudes by the Clebsch-Gordan coefficient of this transition-- of this transition-- you easily find that the absorption for sigma plus, and the absorption for pi, interfere exactly destructively. So the atoms are in linear superposition of two states, the photon is a linear superposition of two polarizations, and the two absorption amplitudes from here and from here to here interfere destructively. And they interfere also destructively from here and from here to this state e to the minus 1/2. What?

STUDENT: What is pi?

TANNOUDJI: A pi is the polarization which is linear and parallel to the z-axis. The pi polarization with respect to the z-axis is the linear polarization parallel to the z-axis. There are three basic polarizations for photons which are sigma plus like that, sigma minus [INAUDIBLE] if I take the z-axis, and pi like that-- corresponding to the three possible values of the angular momentum of the photon which is g equal 1.

STUDENT: I thought there were only two levels.

TANNOUDJI: There are two-- if you take at the quantisation axis, the momentum of the photon. But a pi photon is a photon which propagates in the x or y plane and which has a polarization which is linear and parallel to the z-axis.

STUDENT: But are there still only two states?

TANNOUDJI: A pi polarization is a linear superposition of sigma plus and sigma minus, but it's called, usually, atomic physics. It's a transition delta m because it's called a pi polarization. Now another important thing here is that in the old magnetic field, the two states g plus 1/2 and g minus 1/2 have the same energy. So if the atoms is in such a state, it remains there indefinitely, because this state does not evolve. Now if you put a small magnetic field-- so if you have the optical pumping along x, and if you put a small magnetic field along z, the state g plus 1/2 along the x-axis is still a non-absorbing state-- an uncouple state. But once the atom has been put here, this state is no longer stationary, because the phase factor of applying the time evolution for g minus 1/2 and g plus 1/2 are not the same. This means that the state is evolved. The state g minus 1/2-- g plus 1/2 is coupled to the state g minus 1/2, which can absorb light. So I think the general idea we can keep from this experiment is that there are situations where atoms can be prepared in states which are linear superposition, and for which there exists several absorption amplitudes which interfere destructively. That's a very interesting quantum interference effect. Now two situations then can appear.

If the state is stationary-- if the states which are linearly combined have all the same energy, an atom which is here remains there indefinitely, so all atoms will be optically pumped in such a state which no longer absorb light, and the fluorescence stops. But if you add a perturbation such that the states which are linearly superposed are no longer the same energy, then the state that's not stationary is coupled to other states which absorb light, and the fluorescence reappears. That's the main scheme of what is called coherent population trapping, and we will see another application of this effect later on.

Let me now give you a second example. What happens if the light beam, which makes the optical pumping-- which is now I come back to longitudinal pumping-- what happens when it is non resonant? So I would like to show you now that it's possible to displace atomic energy levels by quasi resonant photons. That's light shift or still called AC stack shift. And I would like to give you an interpretation of this effect in terms of dressed states. Let me first consider the uncoupled state of the atom plus photon system.

For example, this state is g n-- atom in the constant g in presence of n photons. e n minus 1, this state is the atom in excited state e in presence of n minus 1 photon. If the laser frequency omega l is equal to the atomic frequency omega a, these two sets would be degenerate. If we are slightly off resonance, the two states are not degenerate, and the splitting between them is by delta, where delta is equal to omega l minus omega a. Now the atom in g can absorb one photon and jump into e. That means that the laser atom interaction Hamiltonian VAL has a non-zero matrix element between g N and e N minus 1, and this matrix element is nothing but each bell may go one of two well omega 1 is the so-called happy frequency, which is well known in magnetic resonance. So these two states are coupled. And it's well known in quantum mechanics that when two states are coupled they repel each other. This state is shifted down. This state is shifted up.

And if we are in the perturbative regime where the coupling, h bar omega 1 over 2, is small compared to the splitting, h bar delta, we get the shift which is omega 1 squared divided by 4 delta. And that is just the light shift. So you see that the light shift is proportional to the light intensity, because the square of the high beam frequency is proportional to the light intensity. Its sine dependent on the tuning, because it's 1 over delta. If omega l is larger than omega a, delta prime is positive, its omega l is equal to omega a delta prime is negative. You could ask what happens if omega l is equal to omega a. Now we have two degenerate states. When you introduce the coupling, you get a doublet, an anti-symmetric linear combination, and that also is well known in atomic physics. It's the Autler-Townes splitting of spectral line. And finally, of course, it depends on the light polarization.

Actually, a light shift were observed about 30 years ago. This was the subject of my thesis. This light shift were called lamp-shift by Alfred Kastler, because they were produced by the light coming from the lamp. That was a play word for comparing them with lamp-shift. And actually, the experiment was done on the 1/2 to 1/2 transition on mercury like that, and the experiment was done in two steps. In the first step, optical pumping, when measured very accurately-- the transition frequency between g minus 1/2 and g plus 1/2-- that's your green curve. And then in the second step, one adds a second beam of light-- second light beam-- consisting of light coming from a different isotope of mercury, so it's shifted in frequency. And this light beam can be polarized either sigma minus or sigma plus, and you see immediately on this picture that if it is sigma plus, only g minus 1/2 is shifted. If it is sigma minus, only g plus 1/2 is shifted.

So you see that in one case, you increase the splitting between the two levels-- in those occasions, decrease the splitting between the two levels. And this is what is observed experimentally. The magnetic resonance gap is shifted in the opposite direction, depending if the polarization of the light is sigma plus or sigma minus. Now I come to a third example, which is the transfer of linear momentum from photon to atoms. Suppose that we have a laser beam, a laser plane wave, and an atom in a laser plane wave. When the atom absorbs the photon of the beam it gains the momentum h bar k, because each photon has a momentum h bar k. When it falls back to the ground state by spontaneous emission, giving rise to fluorescence light, it loses some momentum. But because spontaneous emission can't occur with equal probabilities into opposite directions, the mean momentum loss is equal to 0 on the average.

So after a cycle, the mean momentum change is h bar k. And the mean force, which is equal to the mean momentum gained per unit time, is just h bar k dN over dt, where dN over dt is the mean number of photons absorb per unit time. That is to say, the mean-- which is also equal to the minimum number of photons spontaneously emitted per unit time. And this is dN over dt can be just written gamma sigma ee, where sigma ee is the probability for the atom to be in the upper state-- population of the upper state. And gamma, the spontaneous emission rate, which is also equal to 1 over tau r, where tau r is the relative lifetime of the other state.

At high intensity and for a two-level atom, the atoms spend half of the time in the upper state, which means that sigma ee is equal to 1/2. So the maximum value of the force is equal to h bar k, gamma over 2, and the maximum value of the acceleration of the deceleration, e max, is just equal to f max divided by m. That is to say, h bar k over m times 1 over 2 tau r. If we put orders of magnitude on this number, h bar k over m is just the recoil velocity of an atom just after the absorption of a single photon. That number is very small-- 3 centimeters per second for sodium, 3 millimeter per second for cesium. Compared to the terminal velocity, which is a few hundred meters per second, that's a very small quantity.

But tau r is only 16 nanoseconds for sodium. That means that you can repeat this process a large number of times. And when you divide h bar k over M by 2 tau r, you get something which is 10 to the 6 meters per second squared. That is to say, 10 to the 5 the gravity acceleration, which is a huge deceleration. So that can be used to stop an atomic beam. You irradiate the laser beam by your counterpropagating laser-- and atomic beam by your counterpropagating laser beam. If you start with an initial velocity of 1 kilometer per second, with such an acceleration you have just to wait 1 millisecond before stopping the atom. And during this time, the travel over distance, which is a 4.5 meters, which is not too big, so we can do the experiment in the lab.

And the only problem you have to solve now is the fact that when the atom is decelerating, the Doppler effect is varying, and you have to adjust the resonant condition to maintain the laser on resonant, and this has been by different methods-- the first one which was developed by Bill Phillips' group was to change the resonance frequency of the atom by applying the space dependent magnetic field, the Zeeman tuning-- and the second one, which is still simpler just by chirping the laser frequency, which is now very easy when one uses laser diodes. Just for illustration, I'll show you some research we obtained in Paris on cesium by Christophe [INAUDIBLE], where he plotted velocity distribution of an atomic beam initially, which is the terminal velocity. And after slowing down one frequency, the scan starting from here to here, from here to here, and leading to a peak of true atoms at 200 meters per second, or 145, 102, 60, 17, minus 8, minus 60. A negative velocity means that the atom had been stopped and go back in the opposite direction. So if you adjust yourself here between these two points, you can have the atom stop with a zero average velocity.

Now that does not mean that these atoms have a zero velocity. They have a zero average velocity, but they have a dispersion of velocity around v equals zero. They have still a temperature. Now I would like to discuss, is it possible to reduce this velocity spread? Is it possible to cool the atom? Because the temperature is related to the velocity spread.

I will show you now-- explain to you the simplest possible cooling mechanism, which is the laser Doppler cooling mechanism, which was introduced by Hamoch and Schawlow for natural atom-- and by Wineland and Dehmelt for free atoms-- and which can be explained very simply. You take an atom moving with velocity v to the right and irradiated by counterpropagating laser beams with the same frequency and the same intensity, with frequency being tuned to the red-- omega l slightly smaller than omega a. If the atom is moving to the right, it gets closer to resonance with this beam because of the Doppler effect, farther from resonance with this beam. So the radiation pressure of this beam will be larger than this one. The two forces would not be balanced. You will get a total force which is opposite to the velocity of the atom, and which for slow velocity is linear in velocity.

So you will get a force minus alpha v, where alpha is your friction coefficient. And the friction coefficient is very strong, and the damping time is very short. And this is why the people from Bell Labs who observed this cooling mechanism in '85, called the scheme optical molasses, because everything happened as if the atom was moving in a pot of honey. Of course, you cannot go to zero by such a scheme, because you remember this friction can occur in a random direction. On the average, it's zero. But from one process to another it varies in a random way so that in momentum space the evolution of the atom is random walk.

And exactly as in Brillouin motion, you have a competition between two processes, laser cooling which introduced the friction coefficient alpha which I've just described, and momentum diffusion heating which is due to the fluctuation of spontaneous emission and which can be described by a momentum deficient coefficient D. You can calculate alpha and D, and calculate your [INAUDIBLE] temperature kB T Doppler, and find it is D over alpha. And you'll find, easily, that this temperature has a lower limit, which is h bar gamma over 2. That is to say, h bar over 2 tau r, which is reached when the tuning is equal to 1/2 a line width, and which is equal to 240 micro Kelvin for sodium and 120 micro Kelvin for cesium. So it's already a very low temperature. But actually, I would like to make the following remark. If you look at the famous Einstein paper of 1917, you'll find that there's already a description of Doppler cooling in this paper, because he was considering how atoms interacting with the black-body radiation, and exchanging linear momentum with this black-body radiation. Because of the Doppler effect, we are able to reach temporal equilibrium, so I think one can consider the first discussion of laser cooling is contained in the 1917 Einstein paper.

Finally, I would like to discuss your last example, which is the control of the position. Can you control the position of the atom? Can you trap the atom? Several traps have been proposed. I would like to discuss one of them, which have been proposed by my young colleague, Jean Dalibard, in '86, which is called the magneto optical trap-- and explain a simple case of where g equals zero to g equals 1 transition. Atoms are put in a magnetic field gradient along the z-axis, so the magnetic field varies linearly with z. The ground state which is your m equal zero state is positioned as an energy which is position independent, where the two states e plus 1 and e minus 1 have an energy which varies linearly with z. And suppose that you excite these atoms with two counterpropagating laser beams, sigma plus and sigma minus, having a frequency omega l slightly smaller than the atomic frequency omega a. What happened?

Look at an atom, which is going like that way. When it arrives here at this point z 1, the frequency of the g0 e plus 1 transition-- that is the sigma plus transition-- is resonant for the laser beam coming from this direction. So these atoms undergoes radiation pressure from this beam, and it turned to the right. It is repelled to the right. When it arrives here, it becomes resonant with your sigma minus wave, and it is repelled to the left. And you see that the atom is bouncing back and forth between z1 and z2. It's motion is limited between two values. It is trapped. And in addition, you have a combination of trapping and cooling, and these traps are very convenient because they work not only at one dimension, as I've explained here, but also at three dimensions. And also, they work in a cell-- not for a beam, but for a cell. And they are deep. They have temperature depths of the order of 1 degree Kelvin. And shortly after the proposal, such a trap had been achieved by a collaboration between Dave Pitcher and Steve Chu.

Let me give you an example of such a trap, which we have now in Paris. That bright spot you see here is a light which is emitted by a cloud of metastable helium atoms which had been slowed down by the magnetic tuning method and which has trapped in a magneto optical trap. And you look here at the fluorescence light emitted by this trapped atom. The scheme also work in a cell, and here you have the [INAUDIBLE] of 10 centimeter diameter-- the atoms are in the cell. And the bright yellow spot you see here at this point is a cloud of cold cesium atoms which are trapped in the cell, so you can have now, in your cell, cold and trapped atoms.

Now I'm going quickly to the second step of my talk, which is advances in laser cooling. Around the '88, more precise measurements were done on the temperature of these cool atoms, which is not very easy to measure the temperature of cold atoms. And these experiments were initiated by the Gaithersburg group, and the big surprise was the temperature was much lower than expected. Usually when you do experiments, you find results which are worse than the theoretical prediction. Here, it was a very good surprise. The experimental research results were much, much better than the theoretical results, and these quite surprising results was confirmed in Paris and Stanford. Let me first explain to you how the temperature is measured. The idea is very simple.

You start with a molasse, which is a bowl of cold atoms like that, and at time equals zero you switch off all the laser beam. So the atoms are no longer moving in a viscous medium. They fly, and they form a shower. And you look at the light they emit when they cross the laser beam, which is located 7 centimeters below, and you measure the time of flight. This time of flight is determined by gravity. It's determined by the initial position distributions that you can measure by your camera. You cannot show that has been done. Look at a slice in the molasse by pushing all atoms except in the shadow of your needle, so you have a very thin slice of cold atoms falling back, so that you reduce uncertainty due to the position distribution. And of course, the time of flight depends on the velocity distribution which is the parameter you are interested in with such a temperature. And if you fit the experimental signal which is here, which is a very good signal to noise, with a theoretical fit comparison here, you find that the temperature is 2.5 micro Kelvin plus or minus 0.6, which I think is the coldest 3D kinetic temperature ever measured. So you have a very cold temperature. Remember that the minimum temperature predicted for Doppler cooling is 120.

So what is the explanation? The explanation is you have sodium atoms are not two-level atoms. They have several Zeeman sublevels. You have optical pumping between these amount of levels. At low intensity, this optical pumping takes a long time, so you have long time general problem in addition to the lifetime of excited state which is short. And long times mean large non-adiabatic effect, a large friction mechanism. That's the general idea.

And I would like now to explain to you the more specific example, which we have developed in Paris with Jean Delibard, and which we have called sisyphus cooling. And I would like to explain it to you on a simple case of the 1D molasse Suppose that you take a 1D molasse-- you take two counterpropagating laser beams-- but now because the atoms are not two-level atoms we have to worry about the polarization. And I will take a configuration leading to a polarization gradient, and that's easy. I take two optical linear polarizations for this beam and for this beam-- two optical non-linear polarizations. At a given place, these two laser electric field will be in quadrature, giving rise to a sigma plus light.

If you move lambda over 8 to the right, the two fields will be in phase, and the polarization will be linear at 45 degrees. If you still move lambda over 8, there will be, again, in quadrature, but with a minus pi over 2k shift instead of a plus pi over 2. So they will be sigma minus. And if you will-- again, you will have a linear polarization and then a sigma plus. So you see that, in general, in a molasse where the polarization changed are different, you have this strong polarization gradient over distances of the order of the wavelength.

Let me suppose now that the atoms which are cooled have two Zeeman sublevels in the ground state, g minus 1/2 and g plus 1/2. And let me suppose that the excited state as a g equal 3/2 angular momentum, so we have four Zeeman sublevels in the upper state-- e minus 3/2, e minus 1/2, e plus 1/2, e minus 1/2. And suppose that you are in a place where the polarization is sigma plus. If the polarization is sigma plus, you see, easily, that you have optical pumping taking atom from g minus 1/2 and putting them in g plus 1/2. So all atoms will be optically pumped in g plus 1/2. You'll see also that since the laser frequency is smaller than the atomic frequency, the two levels will be light shifted down, and because the square of this Clebsch-Gordan is 3 times smaller than the square of this Clebsch-Gordan, this light shift will be 3 times smaller than this one. So the two Zeeman sublevels will no longer be degenerate. They will be separated by a splitting which I call u0, and which is right shifted. And all atomic population will be in g plus 1/2.

Now if I go to another place where the linear-- the polarization is sigma minus, the conclusions are reversed. All atoms will be particularly pumped in g minus 1/2, and the g minus 1/2 sub levels will be shifted 3 times more than the g plus 1/2 sub level. So you see that the two sub levels are shifted like that, and when you move they are shifted in the opposite way and again in the opposite way. So you see that light shifts of the Zeeman sublevels oscillate in space like that which a separation over here which is lambda over 4. That's your g plus 1/2 level. That's your g minus 1/2 level. They oscillate with an amplitude u0, and you see that also optical pumping rates oscillate because at this point optical pumping tend to put atoms in g plus 1/2. At this point, it tends to put atoms in g minus 1/2.

So you see that both light shifts and optical pumping rates are modulated especially, but the important point is that these two modulations are correlated. There is a strong correlation between the spatial modulation of light shifts and the special modulation of optical pumping rate, and the correlation results in the fact that atoms tend to be optically pumped from the highest of levels to the lowest one. So if you now take a moving atom, and if you suppose that the moving atom starts from the valley-- has been just pumped here. It starts from the valley of a potential curve here, g plus 1/2. It moves to the right. It climbs a potential here. It reaches the top of the potential hill, where it has the maximum probability to be optically pumped into other sublevels. That is to say, the bottom of the valley, and so on.

And you see that because of this correlation between light shift and optical pumping rates, you achieve a situation where the atom is always climbing, as Sisyphus in the Greek mythology. This is why we call these cooling mechanisms the Sisyphus cooling mechanism, because it's exactly the atomic realization of the Sisyphus myth. And you see that during the climbing, the kinetic energy is reduced to the benefit of the potential energy, and the gain of potential energy is then dissipated by the optical photon which is emitted. Because the emitted photon has an energy which is larger than the absorbed photon, so the potential energy is dissipated by spontaneous emission, and so on. And you see that at each step, you decrease the total energy by an amount u0, and the total energy of the atom decreases until the energy of the atom reaches a value corresponding to u0. And then the atom is trapped.

So you expect that the temperature obtained in this way should be kB T of the order of u0. That is to say, you remember the light shift is omega 1 squared divided by delta, proportional to the laser intensity I L and inversely proportional to the tuning, so that's nice. When you decrease the intensity of the light, you decrease the temperature. Could you go to zero and get a zero temperature? Of course not, because the gain-- the loss of potential energy after each cycle, which is of the order of u0. If you want to have a cooling, the loss of potential energy must be larger than the gain of kinetic energy to the recoil, because when the atoms spontaneously emit a photon it recoils. So it has a kinetic energy at least h bar squared k squared over 2m. And you see that if u0 is too small-- if u0 is smaller than the recoil-- you can no longer use this mechanism for cooling. But you see that now that this mechanism linked to temperature which scale as the recoil energy and not as h bar gamma.

And of course it had been possible to make a complete theory of this cooling scheme with Yvan Castin and Jean Dalibard, and when predicted that there is a universal curve giving p squared average divided by a unit of h bar squared k squared versus u0-- the light shift divided by ER, the recoil energy. And we predict that you must have a straight line, and that a very good agreement with the experimental results-- you get here a plot of temperature measured versus the light shift in unit of gamma, and you'll see that the temperature is really a linear function of the light shift. And the different points correspond to different values of the detuning, which can vary by a factor of 20. So we have really checked that depends only on this parameter omega 1 squared over the time. And when the intensity is too small, then the curve starts again to increase.

One interesting thing about this scheme is that the lowest temperature which can achieve correspond to values of the parameter, where the oscillation frequency of the atom in the bottom of the potential wells-- the oscillation frequency is larger than the damping rate due to optical pumping. The oscillation period is smaller than the optical pumping time. You are in a new situation in this kind of problem, where the characteristic time of the external motions are shorter than the characteristic time of the internal motion. And quantum mechanically, this mean that in the potential well which is here at the bottom is nearly parabolic. You have this height energy levels, and omega oscillation large compared to 1 over tau p, meaning that the splitting between these energy levels is larger than the width of this level due to the rate of absorption. This means that we have a quantisation of atomic motion in the light potential associated with the laser beam.

And just a few weeks ago, we have observed such effects. We have observed by an experiment, which I will describe to you briefly. We have submitted a letter where we have measured in a 1D molar-- which is in blue here-- we have measured the absorption spectrum of a probe beam in red here, and we have measured the absorption of the weak probe beam versus its frequency omega probe. And we have observed a curve like that again in absorption-- two curves like that. And these two sidebands, we interpret them as Raman processes. This one, for example, we interpret-- we have these height levels here. This level is more populated than this one. So we have processes where the atom in these vibrational levels can absorb a probe photon, and make a simulated emission of a molasse photon, and go to the v equal 1 level.