Thanks to all the organisers, this has been a really nice experience for me and my wife who is here with me. So it’s been nice to meet some of the students so far and I hope to meet more through the week. So what I’m going to talk about here is some ideas about how to make clocks out of atoms. And I’ll just summarise, I’ll first talk about why we need precise clocks. I’ll give some basic ideas of how we make the clocks. And in this limited time I’ll only use examples from the kind of things we do in our lab. But there is atomic ions, but neutral atoms are very good, too. And I’ll try to give some idea of the state of the art and where we might go in the future. So I think certainly one of the applications for precise clocks over many centuries has been in navigation. And that’s still true today. I think certainly one system we take for granted is the GPS. And these days GPSes can be very precise as you know. But then the idea is very simple. And that is that protocols are more complicated than what I say here but the basic idea is, I assume you agree, that on satellite and on the ground you have two synchronised clocks, and let’s say you agree that the satellite is going to send a pulse every second, and by measuring the time delay, just through the speed of light, then that gives the distance from this expression here. And, of course, there can be errors in the clock. So for example if the clocks are synchronised to about 10 nanosecond, 10 ^-9 seconds, then that gives an uncertainty of about 30 centimetres. And to say what that means in terms of the frequency precision is, that one nanosecond over one day is about a part in 10^14 and that says also that our clock, the frequency must, to maintain the synchronisation, the frequency must be maintained to that same level. And, of course, it’s a bit more complicated than that. One satellite gives us the distance from that satellite, but as you know with a network of satellites the system becomes over-determined. And in fact the clocks can synchronise their position and as well time and so we get three-dimensional navigation on the surface of the earth. So we all have simple notions about how clocks work and that is that basically we rely on having some steady periodic event generator and then we use a counter to measure elapsed time. And, of course, that can give us the time we’re used to. And historically the two traditional periodic event generators, for example, has been the rotation of the earth and then later pendulum clocks which were assumed to be accurate enough to do fairly precise navigation. But we’re going to use the periodic events of oscillations and atoms and so a simple case might be an electric dipole interaction where over on the right side you see the electron density versus time. And, of course, one thing to think about is that we set up an oscillation, a superposition of two states, say the ground and first excited state in an atom. And the wave function is described by the simple expression I’ve shown there where there is a phase evolution between the two states that goes proportional to the oscillation frequency. And, of course, this frequency is given by the energy difference divided by Planck’s constant. So one mode of operation for a clock then is to surround these radiating atoms with an electromagnetic cavity and have some probe, say for the microwave region, that samples the radiation. Or in the optical case we just let the radiation escape out of two mirrors in the case of a laser, and then we have some counter that measures the oscillation of the radiating atoms. Just a little bit of personal history. Actually, I was a student of Norman Ramsey who you see here. Norman was a very famous atomic physicist. And when I started graduate school in his group, I see now 50 years ago, that’s kind of scary, but anyway he and his colleague Dan Kleppner had invented and demonstrated the first hydrogen maser. So “maser” like a “laser”, just “m” stands for “microwave”. And Norman wanted to have a precise measurement of all three of the hydrogen isotopes. So my project was to make a maser out of deuterium and measure its frequency. You can see that’s me there trying to get close to the boss, this is Norman here. Anyway so the result of this project was to measure this frequency. And you can see that the precision is fairly high but still not as high as we’d like to, say, have on board of a satellite. But nevertheless this is one idea for making an atomic clock. And I should say, you know for the students out there, certainly this was a time of personal uncertainty for me and where I would fit in, into the physics role. But I knew by that time that I really liked this kind of stuff, these precisions measurements. I think one thing that still remains interesting for me is kind of the detective work you go through to figure out all the things that can cause errors and be able to improve on that. So let me come back to this, coming back to the basic idea of the clock. There’s a second mode of operation which tends to be the more common way we do the measurements of the atomic frequencies. One of the problems in the maser or the laser is that the radiating atoms are coupled to this cavity which also has a resonance frequency, and those two objects, the cavity and the atom, can pull the frequency of one or another. This problem tends to shift the observed frequency in such a way that we can’t always control it well enough to be just the atomic frequency. So another way to make the clock is to, say, have the atoms in a container. I’ll come back to this in a minute. And then what we do is we think about starting the atom in the ground state, as you see on the lower left. And then we apply radiation that will be near the atomic resonance frequency for a short time. And then basically all we do is we measure the probability of the atom being excited. And when the maximum transition probability is maximum then we know that the frequency of the radiation we’re applying is equal to the resonance frequency of the atom. So one basic recipe for making an atomic clock is then to have atoms contained in some sort of container. And in fact some of the early clocks based on microwave radiation were exactly this. It was a glass cell with some typically interior, say a rubidium clock would be, the interior of the cell would be coated with some low polarizability in material like paraffin. The atoms would bounce around in there and we’d apply microwave radiation here and as the frequency of that radiation was too near the resonance of the atoms and the transmitted radiation would be decreased. So in this picture up here the idea is that there would be some absorption feature. Of course, it’s not infinitely narrow, maybe limited typically by the lifetime of the atoms in the excited state. But nevertheless we can make a simple servo to basically make the frequency of the radiation be that of the atomic resonators. In fact it’s a little, slightly more complicated than that, if we sit right on the top of this absorption feature, of course, there’s the slope, the discriminator has less sensitivity, so typically the only slight complication then is we typically measure on one side of the line and then on the other side. And basically we just make the signals out, be exactly the same. And then we know that the mean frequency of those two frequencies is equal to the resonance frequency. And it’s literally no more complicated than that, the way we typically make clocks based on this second mode of operation where we look for absorption. Ok, so what’s good about atomic clocks, and I’m here comparing to a pendulum clock. Things like quartz crystals have the same issues that I’ll describe here. So, of course, the pendulum clock is given, the frequency is given by this expression here. First of all one thing we have to worry about is various environmental effect. For example look at temperature. Even if we have a fairly low expansion material, say, that has a temperature coefficient of about 10^-8 per degree C and that’s typically about more than 100 times better than most metals, for example. Even with that fairly low expansion material the frequency shift due to temperature changes is given by the expression on the lower left there, a little less than a part in 10^8th per degree C. We also have to worry about temperature of our atoms in our container. And this is certainly one of the more interesting kind of effects. We learned from Einstein about relativistic time dilation, if the atoms are moving in time relative to us in the lab, the time moves slower for the atoms and that gives the so-called second order Doppler time dilation shift that Einstein told us about. And, for example, for caesium which is the current definition of a second based on a microwave transition in caesium that then gives rise to temperature coefficient of about a part in 10^15 per degree C, so substantially better than we can do with a pendulum clock. Ok, the last point is reproducibility of the clocks. And for a pendulum clock, for example, it depends on the length, depends on the manufacturing tolerances and certainly the local value of the acceleration of gravity. And also on wear, if the bearing that holds the pendulum wears, if the length changes a little bit. So there’s some changes due to that over time. The nice thing about atoms is that as far as we know any atom of a particular isotope, they’re exactly identical. Of course, atoms don’t wear out, we can continue to use the same atoms and they won’t change. Well, actually atomic clocks are not a new idea. This is one, I’m not sure how far this goes back but it goes back to at least this work by Lord Kelvin and his colleague Tait. They attributed this idea to Maxwell but basically at that time they were starting to realise the properties of atoms having vibrations, and in this case they were thinking about sodium. And that actually meant the optical oscillations that they were thinking of when the idea was that they realised that sodium atoms would be exactly reproducible. And they can be excused a little bit on this last part, being independent of the position. They didn’t know about relativity yet, but they certainly had the basic idea for atomic clocks, so were playing off on these early ideas. So this was actually after graduate school and then a postdoc. I went to what was then the National Bureau of Standards, now called NIST, the National Institute of Standards and Technology. This was our group at that time. We were starting to do experiments on atomic ions with the idea of making clocks. Unfortunately I see again, this is after quite a while and my two colleagues, Wayne Itano and Jim Bergquist on the left there, they look pretty much the same, but those other guys didn’t fare so well over time. So anyway, one thing I would say is it’s nice, although that was the size of our group when we started at that time, one nice thing for me and I think for all of us, we basically have spent our whole careers together. And I’ll say a little bit about where we are now. But anyway, around 1981 we were thinking, we were playing with mercury ions. And mercury was interesting because it has a fairly high hyperfine frequency which would make a microwave clock around 40 gigahertz. But it also had this optical transition which would be interesting. And the basic idea here or at least one of the features of using optical transitions is that the oscillation rate, the tick rate of the clock is much faster. So you can define any unit of time into much finer increments if you use higher frequency. So anyway, we were interested in this transition, this electric quadrupole transition. The upper state has a lifetime of about 1/10 of a second so the line width is around a hertz at about 10^15 hertz. And the basic idea, we’d been playing with single ions and this, I won’t describe this trap structure, but we make a simple electrode structure that uses oscillating and static electric fields to hold the atoms. And then we can radiate the atoms with, in this case ultraviolet light and try to excite this transition here. And you obviously would want more atoms to have a higher signal rate and I’ll comment a little later – this is one of the advantages of the current neutral atom clocks, they typically deal with more atoms than we do. But in any case, the reason we are stuck with one ion is just because of the systematic shifts. And in this case with our mercury ions, if we have two ions in the trap the upper state in this transition of mercury has a quadruple shape like an American football. And in that the electric field gradients from one ion acting on this quadrupole causes a shift. It’s about a kilohertz for typical conditions where the ions are, in these traps where two ions would be separated by a few microns. And we’d like to get down to about a millihertz precision. And so that’s why we’ve stuck with 1 ion so far. Another feature of this is that, and to lead into how we detect transitions, is that we also have a transition in mercury where the lifetime of the upper state is very short. So we can scatter a lot of photons on that transition. It does two things for us, it allows us to cool the ions, to laser-cool the ions to about a millikelvin. And we can also see our ions here. Since they fluoresce in ultraviolet we can’t see with our eyes. But we can make use of simple ultraviolet video camera to see the ions. And so the basic ideas, we can use this other transition to detect transitions on our favourite clock transition. And the basic idea is that if we start the atom in the ground state, the s state, and then we apply the radiation on this clock transition near that transition frequency, if the radiation was mistuned and the atom remains in the ground state, then we turn on this cooling and detection light. And if it remains in the ground state then we’d see fluorescence. On the other hand, if it’s been promoted to the excited state then when we turn on this cooling detection laser, we don’t see fluorescence. And in fact we can get fairly good discrimination. You can see in the lower signal there that when the atom fluoresces it’s quite a bit different and we see a little bit of background light. But we can essentially tell with 100% efficiency whether the atom has made the transition. That doesn’t mean the signal-to-noise is perfect because we’re always left with the quantum fluctuations. We make a superposition state after we apply the radiation, but then when we project or measure the ion there’s always the quantum fluctuations of which state it’s in even though we detect the state with 100% efficiency. Well, anyway, as I mentioned one reason for going after optical transitions is if the tick rate is much higher. Here’s on the upper part of the figure there, we show we measure the frequency of the ion to a precision of about a part in 10^15. And actually this was, we were kind of proud of this, this was around 2005 and we were proud of this, because it was the first time over many decades that another type of atomic clock would actually have higher accuracy than that of the caesium clock. The other thing which I’m leaving out of this, to count these very high frequencies, both Ted Hänsch and John Hall and their colleagues really made this wonderful frequency comb. And it was an astounding development, because the way to count frequencies, I don’t have time to go into, was extremely complicated, these very high optical frequencies. But the device they made, these frequency combs, allowed us to actually have a counter of these very high frequencies which then could be used to make our clock. And as I say it was an amazing development, because after they made their developments, within a year or two many labs could build these counters and have them in their lab. Well, I’m leaving out a lot of details. One of our favourite projects is to use aluminum ions which has some advantages over the mercury ion. And we got down a few years ago to about 8 parts in 10^18 uncertainty. So, as I said before, there’s many things we have to worry about. A lot of categories of electric and magnetic field shifts. Some of the more interesting ones, I’ve talked about the time dilation shift, the second entry on the view graph there. Actually, an interesting one we have to worry about also is the first order Doppler shift. And the way that manifests itself in the lab is that we will have lasers on one end of the table and the ion trap on the other. And, for example, there’s a frequency, a Doppler shift associated with the fact that when the temperature in the room changes, the table shrinks and contracts. And we’re sensitive to about less than a nanometre per second and we have to worry about this. So actually what we do is to compensate for this, we use a technique that was used in satellite ranging and we got the idea from this experiment here to compensate for this Doppler shift. So there’s another very interesting one, also from Einstein, in addition to the time dilation shift from movement. In his theory of general relativity, of course, he explained that clocks also run in different rates and different gravitational potentials. And one way to say this is – I mean it’s not a very significant effect on our ordinary day lives – but a good example is, I suppose, for you students out there, if you had a twin sibling and you were separated at birth. Then suppose your twin lived in Boulder Colorado, about a mile above sea level. In fact, after 80 years your twin would only be about a millisecond older than you. So it’s nothing to get too worried about. But nevertheless, for the precisions we have in our atomic clocks, we have to take account of this. So anyway, one fun way to demonstrate that in our experiments on aluminum clocks – this shows the table that holds the optics and ion trap for one of our clocks. We had another clock in an adjacent room. And we could measure the frequency ratio of the two transitions, optical transitions in the aluminum clocks. And we measured the ratio you see on the bottom entry there. So here’s James Chou who was a postdoc at that time. He’s going to raise one of these clocks with the jacks, and in fact when we then compare the frequency we can see that, obviously it’s not great precision here, but in fact we can see the frequency shift due to the second order Doppler shift. So for a while we held the record at this 8 parts in 10^18, but I’ll mention very briefly there’s many, first of all there’s many other groups working on ions. There’s many groups working on neutral atoms. All these experiments are very interesting in many ways and the way that neutral atom experiments work. The principle is actually not so much different than how we trap our charged atoms, but basically they can apply certain laser fields that will trap the atoms. Sort of a two-dimensional analogue of the way these optical lattices work is that it looks like the atoms are held in kind of like an egg crate. Anyway, they use that to hold their atoms. And just to give you kind of an idea, this doesn’t represent all the many groups working on it, but just to give you an idea. Professor Katori from Japan was the one to first be able to utilise the idea that if you chose the lasers that are used for this trapping, if you choose the wavelength appropriately, the energy levels are shifted by these trapping fields. But if you choose the wavelength appropriate then it turns out the ground and excited state are shifted in an equal way so you get rid of the perturbing effects of these trapping fields. And he has been developing this idea for many years. Jun Ye at JILA, down the street from us, has also been working on the system. And other groups are as well. So they have actually the world record right now on accuracies held by Jun Ye’s group at about 2 parts in 10^18. There’s a group, our counter part of NIST in Germany, the PTB, they also have an ion clock which is pretty close to the same performance. And this game will never end. We’ll keep trying to push each other to increase the precision. And in the future, I think, you know, you’d say, well, do we really need better navigation and certainly it’s good for most of our daily needs. One interesting idea is to, if we can do this, to increase this level of precision of navigation, we could measure the relative positons of two locations on the earth down to a centimetre or less precision. For example, this might be useful for, certainly we know that the strain between various locations is a precursor of earth quakes. So if we could do it at this precision, it might be used in earth quake prediction. The gravitational red shift might be used in geodesy. There’s many interesting ideas to use precise clocks to look for basic effects. And we’re always trying to prove Einstein wrong as a sport and so far he’s doing just fine. But nevertheless we always want to check if at some level there might be deviations to what Einstein predicted. With that I’ll conclude this. The group gradually grew over the years. You see there’s quite a few people in addition to the original group now. I do want to say, I think both Bill Phillips and I, we are in the same laboratory of NIST, called the precision measurement lab. And our laboratory directory Katharine Gebbie – basically a measure of her success is I’m the 4th person to win a Nobel Prize under her direction, including Bill and Eric Cornell and John Hall. So with that I’ll conclude. Thank you.