Computing with Quantum Cats

by John Gribbin

  THE EXPERIMENT WITH TWO HOLES

  What Feynman called “the experiment with two holes” is more formally known as the double-slit experiment, and may be familiar from your schooldays, for it is often used to demonstrate the wave nature of light. In this version, light is shone through a small hole in a darkened room, and spreads out from the hole to fall upon a screen (maybe just a sheet of cardboard) in which there are two holes, ether pinpricks or parallel razor slits. Beyond this screen there is another sheet of cardboard, where the light spreading from the two holes makes a pattern. For the double-slit version of the experiment, the pattern is one of parallel dark and bright stripes, and is explained in terms of the interference of waves spreading out from each of the two slits. You can see exactly the same kind of wave interference if you drop two stones into a still pond simultaneously, although that pattern corresponds to the “double-pinprick” version of the experiment. All this is compelling evidence that light travels as a wave. But there is also compelling evidence that in some circumstances light behaves like a stream of particles. This is the work for which Albert Einstein received his Nobel Prize. One way of observing this is to replace the final sheet of cardboard in the experiment with a screen like that in a TV, and turn the brightness of the beam of light down really low. Now, individual “particles of light” (photons) can be seen arriving at the screen, where they make little flashes, each at a definite (within the limits of quantum uncertainty) point. It looks as if single particles are arriving at the detector, one at a time. But if you record the process over a long period of time, it becomes clear that the flashes of light are occurring more frequently in some parts of the screen than others; indeed, they build up into the usual interference pattern of light and dark stripes. Somehow, the “particles” are conspiring to produce the pattern we expect for waves.

  It gets weirder. If the photons “really are” particles, they ought to produce a quite different pattern. Imagine firing a stream of bullets through the two slits (in an armor-plated screen!) into a sandbank. There would be one pile of spent bullets behind each slit, and nothing anywhere else. So which pattern would you expect if you fired a beam of electrons through an equivalent experiment? A team from the Hitachi research laboratories and Gakushin University in Tokyo did just that in 1987. The results were exactly the same as the results for photons. The beam of electrons interfered with itself to produce the familiar pattern corresponding to waves. And when the power of the beam was turned down so low that electrons were leaving the emitter one at a time, they produced individual flashes on the screen, which built up to make the interference pattern. Essentially, the same quantum weirdness applies to electrons and to light.

  It doesn't end there. It is also possible to set up the double-slit experiment for electrons in such a way that we can tell which of the two slits an individual electron goes through. When we do this, we do not get an interference pattern on the final screen; we get two blobs of light, one behind each slit, equivalent to the heaps of spent bullets. The electrons seem to be aware that they are being monitored, and adjust their behavior accordingly. In terms of the collapse of the wave function, you can say that what happens is that by looking at the hole we make the wave function collapse into (or onto) a particle, affecting its behavior. This almost makes sense. Curiously, though, we only have to look at one of the two slits for the outcome of the whole experiment to be affected, as if the electrons passing through the other slit also knew what we were doing. This is an example of quantum “non-locality,” which means that what happens in one location seems to affect events in another location instantly. Non-locality is a key feature of the central mystery of quantum mechanics, and a vital ingredient in quantum computers.

  Well, you may say, nobody has ever seen an electron, and we can't really be sure that we have interpreted what is going on correctly. But in 2012 a large team of researchers working at the University of Vienna and the Vienna Center for Quantum Science and Technology reported that they had observed the same kind of matter-wave interference using molecules of a dye, phthalocyanine, that are so large (0.1 mm across) that they can be seen with a video camera. Just as in the case of light, electrons, and also individual atoms used in other studies, the interference pattern characteristic of waves builds up even when the molecules are sent one at a time through the experiment with two holes. And it disappears if you look to see which hole the particles go through. The “central mystery” of quantum mechanics writ large, literally.

  Feynman came up with a way to explain what is going on, and to extend it into a broader understanding of quantum reality, in what became his PhD thesis.

  INTEGRATING HISTORY

  One way of interpreting what is going on, valid “FAPP,” is to calculate the behavior of a wave of probability passing through the experiment with two holes and interfering with itself to determine where particles are allowed to arrive on the final screen. This is a straightforward thing to do using wave mechanics, and leads to the standard pattern of bright and dark stripes. In some places, the probabilities reinforce each other, and these correspond to places where the electron might be found; in other places, the probabilities cancel out, and there is no chance of finding the electron there. If you imagine cutting four parallel, equally spaced slits in the middle screen instead of two, you can do the equivalent, slightly more complicated calculation and work out the corresponding pattern. With eight slits, we would have to add up eight lots of probabilities to determine the pattern, and so on.

  Perhaps you can see where we are going. Even with a million razor slits, you could, in principle, calculate the pattern of bright and dark places on the final screen. That would correspond, FAPP, to each single electron going through a million holes at once. Why stop there, asked Feynman? Why not take the screen away altogether, leaving an infinite number of paths for the electron to follow from one side of the experiment to the other? It is actually easier to calculate the result for such a situation than for one with a million slits, because the mathematical rules make it straightforward to work out the implications in the limit; that is, where the numbers approach infinity. Without actually doing an infinite number of calculations, it is possible to work out which kinds of paths combine together and which ones cancel each other out. The probabilities for more complicated paths turn out to be very small, and also to cancel each other out. Only a small number of possible paths, very close to one another, reinforce each other; they combine to produce a single spot on the final detector screen. The interference pattern disappears, and we are left with what looks like a classical particle traveling from one side of the experiment to the other along a single path, or trajectory. The process of adding up the probabilities for each path is known as the “path integral” approach, or sometimes as the “sum over histories” approach.

  This sounds like a mere mathematical trick. But it is actually possible to see light traveling by some of these “nonclassical” paths. All you need is a compact disc. One of the other things we learned in school is that light travels in straight lines, so that when it encounters a mirror it bounces off at the same angle that it arrives—the angle of reflection is equal to the angle of incidence. But that isn't the whole story. According to Feynman's path integral approach, when light bounces off a mirror it does so at all possible angles, including crazy paths where it arrives perpendicular to the mirror and reflects at a shallow angle to meet your eye, and paths where it arrives at a grazing angle and bounces off at a right angle to meet your eye. All the “crazy” paths cancel each other out; only paths near to the shortest distance from the light to the mirror to your eye reinforce each other, leaving the appearance of light traveling in a single straight line. But the “crazy” paths really are there. They cancel out because, except near the classical path, light waves (or probability waves) in neighboring strips of the mirror are out of step with one another (out of phase). In one strip, the probabilities go one way, but in the next strip they go the other way. If we carefully lay str
ips of black cloth over regions where the probabilities all point one way, we are left with parallel strips of mirror, separated by the strips of cloth, where the probabilities all point the other way, so there is no canceling. The spacing of the strips needed to make this work depends on the wavelength of the light involved, so it is related to the color of the light (red light has a longer wavelength than blue light).

  It really is possible to set up a simple experiment with a light source, a mirror and an observer (your eye!), so that when you choose a part of the mirror where there is no visible reflection, then cover up strips of this part of the mirror in the right way, you will see a reflection. With part of the mirror covered up, it really does seem as if less is more when it comes to seeing reflections.

  But you don't need to go to the trouble of doing this experiment to see crazy reflections. The grooves in a CD are like little strips of mirror separated by regions where there is no reflecting material, and it happens that the spacing of these grooves is just right for the effect to work. If you hold a CD under a light, you don't just see a simple image of the light as you would from an ordinary mirror; you also see a colored, rainbow pattern of light from right across the disc. The rainbow pattern is because different wavelengths are affected slightly differently, but you can see reflected light coming from “impossible” regions of the disc, just as the path integral approach predicts. But even with a conventional mirror, “light doesn't really travel only in a straight line,” says Feynman, “it ‘smells’ the neighboring paths around it, and uses a small core of nearby space.”7

  What is special about that “small core” of space? Why does light “move in straight lines”? It's all to do with something called the Principle of Least Action, which also intrigued Feynman. Indeed, it was the Principle of Least Action that started him on the path which led to his Nobel Prize.

  A PHD WITH A PRINCIPLE

  Feynman had actually learned about this principle when he was still in high school, from a teacher, Abram Bader, who appreciated his unusual ability and encouraged him to go beyond the regular syllabus. It can best be understood in terms of the flight of a ball thrown from ground level through an open window on the upper floor of a house. At any point along its trajectory, the ball possesses both kinetic energy, due to its motion, and gravitational potential energy, related to its height above the ground. The sum of these two energies is always the same, so the higher the ball goes the more slowly it moves, trading speed for height. But the difference between the two energies changes as the ball moves along its path. “Action,” in the scientific sense, relates these changing energies to the time it takes for the ball to complete its journey. The difference between the kinetic and potential energies can be calculated for any point of the trajectory, and the action is the sum of all these differences, integrated along the whole trajectory. Equivalent actions can be calculated for other cases, such as a charged particle moving under the influence of an electric force.

  The fascinating fact which the intrigued Feynman learned from Mr. Bader is that the trajectory followed by the ball (which, you may recall, is part of a parabola), is the path for which the action is least. And the same is true in general for other cases, including for electrons moving under the influence of magnetic or electric forces. The trajectory corresponding to least action is also the one corresponding to least time—for any starting speed of the thrown ball, the appropriate parabola describes the path for which the ball takes least time to get to the window. Anyone with experience of throwing balls knows that the faster you throw the flatter the trajectory has to be to hit such a target, and this is all included in the Principle of Least Action. In the guise of the Principle of Least Time, it can also be applied to light. Light always travels in straight lines, we are taught, so we don't think of it as following a parabola from the ground through an upper-story window. But it changes direction when it encounters a different material, such as when it moves from air into a glass block. The light travels in a straight line through the air up to the edge of the glass, then changes direction and travels in another straight line through the glass. The path it follows, getting from point A outside the glass to point B inside the glass, is always the one that takes least time. Which is not the shortest distance, because light travels more quickly in air than it does in glass. Just as with the ball thrown through the window, it is the whole path that is involved in determining the trajectory.

  All this applies to “classical” (that is, non-quantum) physics. Feynman's contribution was to incorporate the idea of least action into quantum physics, coming up with a new formulation of quantum physics different from, and in many ways superior to, those of the pioneers, Heisenberg and Schrödinger. This was something which might have been the crowning achievement of a lesser scientist, but in Feynman's case was “merely” his contribution as a PhD student, completed in a rush before going off to war work at Los Alamos.

  In his Nobel Lecture,8 Feynman said that the seed of the idea which became his PhD thesis was planted when he was an undergraduate at MIT. At that time, the problem of an electron's “self-interaction” was puzzling physicists. The strength of an electric force is proportional to 1 divided by the distance from an electric charge, but the distance of an electron from itself is 0, and since 1 divided by 0 is infinity, the force of its self-interaction ought to be infinite. “Well,” Feynman told his audience in Stockholm,

  it seemed to me quite evident that the idea that a particle acts on itself, that the electrical force acts on the same particle that generates it, is not a necessary one—it is a sort of a silly one, as a matter of fact. And so, I suggested to myself that electrons cannot act on themselves; they can only act on other electrons…. It was just that when you shook one charge another would shake later. There was a direct interaction between charges, albeit with a delay…. Shake this one, that one shakes later. The sun atom shakes; my eye electron shakes eight minutes later, because of a direct interaction.

  The snag with this idea was that it was too good. It meant that when an electron (or other charged particle) interacted with another charged particle by ejecting a photon (which is the way charged particles interact with one another) there would be no back-reaction to produce a recoil of the first electron. This would violate the law of conservation of energy, so there had to be some way to provide just the right amount of interaction to produce a kick of the first electron (equivalent to the kick of a rifle when it is fired9) without being plagued by infinities. Stuck, but convinced that there must be a way around the problem, Feynman carried the idea with him to Princeton, where he discussed it with Wheeler; together they came up with an ingenious solution.

  The starting point was the set of equations, discovered by James Clerk Maxwell in the nineteenth century,10 which describe the behavior of light and other forms of electromagnetic radiation. It is a curious feature of these equations that they have two sets of solutions, one corresponding to an influence moving forward in time (the “retarded solution”) and another corresponding to an influence moving backward in time (the “advanced solution”). If you like, you can think of these as waves moving either forward or backward in time, but it is better to avoid such images if you can. Since Maxwell's time, most people had usually ignored the advanced solution, although mathematically inclined physicists were aware that a combination of advanced and retarded solutions could also be used in solving problems involving electricity, magnetism and light. Wheeler suggested that Feynman might try to find such a combination that would produce the precise feedback he needed to balance the energy budget of an emitting electron.

  Feynman found that there is indeed such a solution, provided that the Universe absorbs all the radiation that escapes out into it, and that the solution is disarmingly simple. It is just a mixture of one-half retarded and one-half advanced interaction. For a pair of interacting electrons (or other charged particles), half the interaction travels forward in time from electron A to electron B, and half travels backward in time f
rom electron B to electron A. As Feynman put it, “one is to use the solution of Maxwell's equation which is symmetrical in time.” The overall effect is to produce an interaction including exactly the amount of back-reaction (the “kick”) needed to conserve energy. But the crucial point is that the whole interaction has to be considered as—well, as a whole. The entire process, from start to finish, is a seamless and in some sense timeless entity. This is like the way the whole path of the ball thrown through the window has to be considered to determine the action, and (somehow!) for nature to select the path with least action. Indeed, Feynman was able to reformulate the whole story of interacting electrons in terms of the Principle of Least Action.

  The discovery led to Feynman's thesis project, in which he used the Principle of Least Action to develop a new understanding of quantum physics. He started from the basis that “fundamental (microscopic) phenomena in nature are symmetrical with respect to the interchange of past and future” and pointed out that, according to the ideas I have just outlined, “an atom alone in empty space would, in fact, not radiate…all of the apparent quantum properties of light and the existence of photons may be nothing more than the result of matter interacting with matter directly, and according to quantum mechanical laws,” before presenting those laws in all their glory. The thesis, he emphasized, “is concerned with the problem of finding a quantum mechanical description applicable to systems in which their classical analogues are expressible by a principle of least action.” He was helped in this project when a colleague, Herbert Jehle, showed him a paper written by Paul Dirac11 that used a mathematical function known as the Lagrangian, which is related mathematically to action, in which Dirac said that a key equation “contains the quantum analogue of the action principle.” Feynman, being Feynman, worried about the meaning of the term “analogue” and tried making the function equal to the action. With a minor adjustment (he had to put in a constant of proportionality), he found that this led to the Schrödinger equation. The key feature of this formulation of quantum mechanics, which many people (including myself) regard as the most profound version, is, as Feynman put it in his thesis, that “a probability amplitude is associated with an entire motion of a particle as a function of time, rather than simply with a position of a particle at a particular time.” But everything, so far, ignored the complications caused by including the effects of the special theory of relativity.