Maxwell’s electromagnetic theory (now 150 years old) seemed in its comprehensive way to be capable of answering all of the questions one might ever pose about the theory of light. But that spell was broken in 1900 by Planck’s discovery that light beams are showers of discrete energy quanta that behave like particles. Although the theory of light was where the quantum theory began, its further course was presently filled with dilemmas and paradoxes that were only resolved by the eventual development of the quantum theory of material particles in the 1920’s. It is interesting to describe some of those apparently strange behaviors, and to discuss some additional and more novel ones that were introduced by the concept of entanglement, which is intrinsic to the full quantum theory.

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Well, I want to talk today about a subject which is not particularly historic, nor do I want to deal very much with history. But I think I should begin that way, for at least a few moments. What we knew about light, what we know about light, these are things rather different. And our understanding has taken a while to evolve. The story with which I'll begin, of course, is the great controversy of particles versus waves. In the 18th century it was quite an exciting story on the basis of remarkably little evidence. There were, of course, formidable figures maintaining both that light is a species of particle. No one less than Isaac Newton - thank you. No one less than Isaac Newton was convinced that light consists of particles. And in Holland Christiaan Huygens had arguments which were at least as telling to the effect that light consists of waves. I'm not going to follow this conflict really much further than that, at least not much further than saying that the situation seemed to be resolved at the beginning of the 19th century. Before that let me make a little mention of my title, which will deal with light quanta, and the rather strange ways they behave, these idiosyncrasies as it were. We didn't have a great deal of experience with light quanta, we had rather more with waves to begin with. Indeed, that began at the beginning of the 19th century with the remarkable experiment, a fantastic insight, by Thomas Young. Thomas Young, as many of you may know, was an all-around genius. He happened also to be a medical man and it was partly pursuing that, that he began the famous double pinhole experiment, which seemed to say in a definitive way precisely what light is. In this experiment light enters a pinhole, the purpose of that is simply to give you a unique source, in effect a point source of light, and now you have two pinholes. These days we do the experiment, all the undergraduates do this experiment, with a bit more light. They use slits instead of pinholes and the chances are that they even do without this first screen entirely and use a laser. That's a different story and I'm not going to tell you about the story of the laser very much today. The point is that one saw, in a sufficiently darkened room, a series of fringes, alternating bright and dark fringes on this furthest screen. There is a picture of that here made by Thomas Young himself, who did the experiment and developed really the convincing evidence that light consists of waves. These are the waves that he drew and his understanding was that these consecutive, this pair of circles, the successions of circles which represent ripples as it were, spreading from the individual pinholes in the second screen. That there will be certain curves along which these wave crests, for example, reinforce one another. You'll see there a bright fringe in the centre, and then there will be certain low size, certain curves, along which the maximum of one family of circles coincides with the minima in the other family. And there you'll you have a sort of cancellation and you'll have a dark fringe on the screen. That's illustrated a good deal better these days with literal ripples. Here you have it. These balls, bouncing up and down on the surface of a liquid, represent the sources of the two pinholes in the second screen. There you see there is a reinforcement to those waves in the centre. But there are these directions in which the waves cancel one another and that's where you saw darkness on the screen. That was Young's explanation of what was going on and it was, for its age, a virtually perfect one. Young was rather puzzled about what kinds of waves these really are and no one could say. It was eventually decided, because of the way light polarization interacts with this particular sort of picture, that the displacements in the waves were perpendicular to the direction of propagation. That one was, in fact, dealing with transverse waves. Well, this picture, with further embellishments, lasted really until the beginning of the 20th century. During that time there was remarkable progress. The greatest single piece of progress surely came at the hands of James Clerk Maxwell. And we are celebrating the 150th anniversary of that discovery of the electromagnetic equations, the electromagnetic theory, as developed by Maxwell. Maxwell availed himself of everything that was known of electricity and magnetism at that time, at about 1865. He had to add a new law, something which was not within the quantitative range of perception of any experiments done at that time. He thought he was doing no harm by adding this less observable term to his set of electromagnetic equations. But that term did, tiny and imperceptible as it was in those days, change the nature of the equations and he was able to show that there is indeed a propagation phenomenon implicit in electromagnetic theory. The measurements, the contemporary measurements, even told him what the speed of those waves was and it corresponded quite accurately to the existing measurement to the speed of light. Maxwell was persuaded, he had explained the nature of light and the world was persuaded equally. It was rather some time before it was possible to develop low-frequency versions of those waves that could be investigated in the laboratory. But that did happen presently and made it very clear that Maxwell had what amounted to the complete explanation, as well as anyone could tell, of the nature of light. Now, the great surprise was what was still to come. Let me first ask, what is it that happens in Young's experiment? If light is going to be a particle, or consist of particles, as for example Newton had maintained and a great many others, those quanta are not going to split and go partially through each of the two holes in Young's experiment. They have to go through the one hole or the other. Then the question is, how do you get fringes? In some statistical sense you get these fringes, we'll see a bit more of that as individual registrations, individual spots, tiny spots for example, on a photographic film. While the image is a little less clear for light, it is certainly pretty clear for particles as we will see. We'll go into that in a moment. Here we can ask, how do particles manage to generate the fringes that you saw? Do the interference effects that appear, these fringes that appear on the furthest screen, depend on the abundance of quanta present? Do they, in some way, represent interactions between those quanta? And what, in fact, would you see if you worked with so faint a beam that there was only one quantum present at a time? That, in fact, was a remarkable experiment, not very well celebrated these days, that was performed by an undergraduate at Cambridge, G. I. Taylor, whom I eventually knew as Sir Geoffrey Taylor. He was the man who invented explosive lenses, high explosive lenses, and they were very useful during the war. I met him at the Manhattan Project at that time. What Taylor did was this: he simply did a diffraction experiment. In this case he looked at the diffraction pattern of a needle, because the two-hole screen wastes a great deal of light. He didn't have much light to waste, Taylor. He used so faint a light beam that it took him three months, with such photographic plates as he had at the time, three months to accumulate enough quanta registered by his plates to see an image. What he saw was the very same diffraction pattern that he would see with a very bright source. He had chosen his numbers so that he could be sure there was never more than a single quantum, a single light quantum at a time, in transit through his experiment. That didn't change the pattern at all. Somehow or other there is something uncanny going on. The light quanta must sense the presence of the two holes, but yet they can only go through one hole at a time. If you'll permit me to skip quite a few years, the same sort of diffraction experiment was performed with particles and the result was vastly more surprising than it had been even to the earliest investigators. This is an electron diffraction picture where electrons, a very weak beam of electrons, is passed through two slits. This was done just a few years ago by Tonomura, but these experiments were first done in 1925 and 1926 at the Bell Labs. Here you have the accumulation of such an interference pattern in successively longer exposures to quite a faint electron beam. Particles are capable of forming these interference patterns and it‘s certainly not a matter of electrons dividing and going through each of two holes that's taking place. Here is the dilemma: You have the accumulation of a diffraction pattern, with time this is the sort of experiment, which, in fact, corresponds to what Taylor did and which you just saw it with electrons with Tonomura. This is the image with which we began the passage of waves through two tiny apertures, a certain distance apart. What is going on there? Well, this was extremely puzzling until the very beginning of quantum mechanics. We had to wait until about 1924, 26 for the arrival of quantum mechanics and a certain rationalization of all of these phenomena. Here I'm going to some illustrations used by Niels Bohr in a famous paper he wrote, in the 1930s actually, explaining diffraction. This is no more than a repetition of what I showed you earlier on the first slide. But Bohr was asking the question: Can we get such a pattern of fringes as we find on the screen here, and know what the particle is doing in passing through this second screen? For example, you could, by adding a little equipment to this experiment, determine which of the two holes a particle goes through. A light quantum or an electron? It's the same diffraction phenomenon in both cases. How would you do that? If you're to get any certainty that the particle has gone through let's say the upper of these two holes here, one way of doing that is determining whether its momentum has been deflected a little upward. How do you do that? You cannot make the measurement on the particle itself, you'll never see the particle again. What you could do, in principle, is measure quite precisely the behaviour of this screen. We allow this screen to be something that can move and determine whether, as the particle is deflected upward and can go through this top hole, whether in that time the screen itself, as a movable object, has been deflected downward a little bit. A downward deflection of the screen means the particle was deflected upward and is eligible to go through the upper of those two holes. There was some wonderful illustrations due, I think, to Bohr's son Aage, who lives on and who also was a Nobel Prize winner for altogether different sorts of work in nuclear physics. Here is their picture of that first slit. The first slit is movable and you are, by this scale I assume, able to discover whether that first slit has been deflected downward, meaning that the quantum went up. Or upward, meaning that the quantum was reflected downward. It's a very simple exercise with ... I tried last night putting it on one slide but unfortunately the slide didn't come out. Without the couple of equations of mathematics, let me just say this. If we go back a stage. The momentum transfer, which sends the particle or the quantum through the upper of those two slits, is very tiny. If you're going to discover by measuring the recoil of this first screen, which of these two choices is the right one, you'll have to have a very accurate measurement of the change in momentum of this screen. That can, in principle, be done but it's so tiny a change in momentum that there is associated with it a quantum uncertainty in where the first pinhole happens to be. You'll find that in order to make a measurement of this tiny a deflection, you'll have to have an extremely precise idea of what the momentum of that screen is. You'll have to detect a very small change. That small change that you are able to detect in order to be assured that the particle has gone through the upper of the two slits is related to an uncertainty in position of the slit; that's the quantum uncertainty principle. Of course, it would not have been known of in this context before about 1925 or thereabouts. In fact, due to Heisenberg, the uncertainty principle, you'll find that by the time you're able to measure that small a momentum transfer, you in consequence of the uncertainty principle, become uncertain of where this first pinhole or slit is. You'll be come so uncertain of it that that uncertainty is at least equal to the distance between those two slits. Now, what that will mean is that you see no interference pattern. That by the time you have discovered which of the two slits the particle has gone through, you will see no possibility, with such equipment you'll see no possibility of any fringes. You'll get more or less uniform illumination on the screen. The quantum theory, the new quantum theory, keeps its own secrets as it were. We have learned that indeed particles do share many of the properties of waves but that doesn't mean that a particle goes through both slits at all and here is Bohr's observation. The interference fringes can only be seen as such when it is literally impossible to determine which of the two slits the particle went through. The quantum theory, in effect, protects its own secrets. Some physicist friend of ours thought he would illustrate this with a road sign and I have to tell you that that road sign is all together incorrect. The particles do not in any sense go through both paths, through both slits. They choose, in each case, one or the other. But our science keeps the secret of which one they did go through and we'll never find out. That's what the sign should be saying, but it would take about a paragraph of prose I'm afraid. To go on with the discoveries that came, beginning in 1925, I'll talk about several of these. Here is one of the most familiar ones. Suppose you have an atom which is just sitting still and it radiates a quantum because it happens to be excited. A very reasonable thing to expect would be that this atom, about which we have no directional information whatever, would tend to radiate its quantum in a spherically symmetrical way. This corresponds, of course, to the circular fringes we were talking about earlier. If, on the other hand, this quantum goes off not as a wave but as a particle you know it's going to carry momentum in a unique direction, wherever the particle goes. That's going to mean that the atom, in order to conserve momentum, is going to have to go in the opposite direction. How in the world do you reconcile these two images? The answer is, you do that in virtue of something we call 'entanglement', which is another way of saying that the light quantum and the atom are coupled in a way, by conservation of momentum in this case. And what actually happens is that the atom chooses a random direction in which to send its quantum. The atom suffers a recoil in the opposite direction. This is a spherically symmetrical picture. This is certainly not. The actual situation of an atom radiating allows it to radiate in arbitrary directions. Here I've indicated five of them but, of course, there is an infinite number covering a whole sphere around this black atom, initially at rest, which emitted the quantum. There should be many, many more arrows filling out that entire sphere. These are all the possibilities and that is how the spherical symmetry is preserved. Now what is it that permits us to diagnose what's going on? We put a counter, a photon counter - it’s something which registers a single light quantum - somewhere around the radiating atom and what happens? It detects a quantum. When it detects the quantum, the atom, of course, recoils in the opposite direction. In this way the spherical symmetry is broken and indeed our detecting the quantum is really what has broken the symmetry. What it does it to say all those other possible waves radiating from the central point are absent. That measurement wipes out that part of the wave packet, it's called sometimes the reduction of the wave packet. But in a way more remarkably it casts the atom into a particular momentum state. The atom is suddenly going in a particular direction which has, in fact, been determined by where we put our counter. We have interacted with the atom in a sense, but we've never touched it. This is the phenomenon of entanglement. There is a quantum correlation implied by the momentum conservation. The photon goes in one direction, the atom must recoil in the opposite direction. And this remarkable phenomenon, in effect, amounts to accelerating the atom in a particular direction without ever touching it. This is one of the many mysteries associated with entanglement and we'll talk a bit about a few others. These are the various things that came on the scene with quantum mechanics in the middle 20s. Now let's talk about a few of them. Here is one that really eventually elicited a great deal of criticism from no less than Albert Einstein working with two post docs, Podolsky and Rosen. Suppose we have two atoms that are more or less stuck together but, for example by photo dissociation process, are somehow made to fly apart with equal and opposite momenta. What can we measure? We can, for example, wait a certain length of time and measure the position of one of them. Say the one that started out as a blue atom. We've discovered it here, but that immediately tells us something about the other atom which is entangled by the conservation of momentum. That other atom must be in a mirror image position. We have a choice of making a different sort of measurement if you like. We might measure not the position of this first atom after a certain time, but we might measure its momentum. Its momentum says that it corresponds to a certain wave with a specific wave length going in that direction. We're making the same statement then about the red atom going here. Here we have a choice of experiments that we can perform. And it leads to two altogether different states of this red particle, which we have never touched. This is again the effect of entanglement and it's a uniquely quantum mechanical effect, which really never received any attention prior to about the late 20s. This entanglement - entanglement is not a particularly good choice, it's a word for it, but it is a phenomenon which is explicitly quantum mechanical. It has no analogue in classical theory at all. It represents the origin of a good many paradoxes that we have to deal with these days. This is perhaps the first one which was clearly stated by Einstein, Podolsky and Rosen. Here we can find other examples of the same sort of thing. Angular momentum conservation also leads to entanglement. Suppose we have two atoms joined together, well let's say one atom in this case, one atom does it. One atom in a state of zero angular momentum, positive parody. It emits a red quantum which has a certain direction in space perhaps, but it has certainly a particular energy. Here is the second radiation by that same atom. The two spins of the spin angular momentum since we are going from a zero state to a zero state will have to add up to zero. That, again, is such a quantum correlation and once again, of course, it leads to entanglement. Let me show you how that works. I'm going to assume here that the photons are given off in opposite directions and that they have the same helicity. Helicity is the scalar product between the angular momentum and the momentum direction. These two directions K1 and K2 are equal and opposite - just because we're choosing to talk about such a simple example oppositely emitted photons. The state that these two quanta are in, are therefore either positive helicity that gives us zero angular momentum, or negative helicity. But we have to expect that the two can occur with equal likelihood. And so what one gets is a familiar sort of state which expresses entanglement of the two particles. Now, let me go on and show you what happens if we use not helicity states. Well, I'll try ... Here I've put in the states of ordinary linear polarization. Here are the familiar states of polarization in the X and Y direction. The algebra connecting them to these circular states is familiar and obvious. If you do that, you will find that these photon pairs can be either. For example, polarized in the X direction or both polarized in the Y direction - the algebra is all there. These are entangled states in the usual sense. Now I want to show you how different this sort of entanglement is from the usual situation. Both photon beams are unpolarised but we put in polaroid sheets to make measurements on these entangled photons. There are different images you might use. One is, you might imagine, that the passage of any polarized, transversed polarised photon unpolarised through a polaroid sheet has a probability one-half. Then the probability would be one-quarter that both photons get through. Alternatively, you might say, let us assume that these are independent events getting the two photons through. They are random events but they must be the same event in the two polaroids. If you did that you would be averaging - in order to find out the probability of getting both through, you would be averaging the fourth power of a cosine. If you do that you get three-eighths for seeing both. In actual fact what happens is that if one photon goes through using that entangled state, the other one must also go through. The probability that you get both photons through is one-half, it's neither of these two fairly obvious mistakes, which have been made by many people, I can show you. Here is the illustration. There is our light bulb. One fails to get through, the other fails to get through, but now if one gets through the other must get through. That is a remarkable effect of entanglement. It's a purely quantum mechanical effect. It has no classical analogue at all. There is an analogous effect that was in fact mistaken in the decay of positronium, an electron positron atom. Years earlier, and let me just say John Wheeler did this calculation, let me show you what it's like for scalar emission. If we have an atom here at the centre it sends out its two gamma rays. They must be scattered but in general those scattering planes will be different. The question is what is the distribution of the angle between those? The correct calculation was only done as late as 1948. The original proposal of this experiment with the wrong calculation was done in the 1930s. Here is an illustration of what happens in ... Okay, this is an example of what happens in non-linear optics where one photon into the device turns into two photons which are related, of course, by the conservation laws and they are inevitably entangled. There are pairs of entangled photons that make up this picture for a non-linear medium. If you add to the non-linearity, double refraction, then you get two sets of cones and what happens is that you get here: one cone comes, let's say, from vertical polarization, the other comes from horizontal polarization. They can indeed have the same colour and what you get out are pairs of photons. One here and one here and with entangled polarizations. I can show this at some later point. There are those photons, for example, in the green. There's one photon here and another one here but they are described by precisely such entangled wave functions. From here I would just go to other things. I'll run through the remaining slides right now. Let me assume we have a 50/50 beam splitter, and send the photon down. Here are two things that can happen. Either you get reflection or transmission. This is the same thing from the other side. Suppose you have both of those photons appearing on the beam slitter at the same time, you would imagine that you could get this, but in fact it never happens. What happens is that you get either two photons out on one side, or two photons out on the other side. You can show that these particular amplitudes add up to zero and what we are left with is this and this. I will stop here because there are a couple of other slides, but they show things that are still more arcane (Applause). Thank you very much and I think that the discussions will continue in the afternoon when you can show the rest of the slides also for this very interesting phenomenon.

# Roy J. Glauber (2015)

## Light Quanta, and Their Idiosyncrasies

# Roy J. Glauber (2015)

## Light Quanta, and Their Idiosyncrasies

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