Computing with Quantum Cats (13 page)

It's worth spelling this out in detail. Imagine that we have a single electron confined in a large box. According to the Copenhagen Interpretation, the wave function fills the box evenly—the chance of finding the electron at any point in the box is the same as the chance of finding it anywhere else in the box. Now, we take a measurement to detect the electron. We find it, looking just like a little particle, at a definite point in the box.
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But as soon as we stop monitoring the electron, the wave function immediately spreads out from the point where we discovered it. If we quickly take another measurement, there is a high probability of finding the electron close to the place where we last saw it. That matches common sense—but there is still some quantifiable probability of finding it anywhere in the box. The longer we wait, the more the wave function develops, and the chances of finding the electron
anywhere
in the box even out. That's weird, but not completely crazy.

But it is just the beginning. Richard Feynman was fond of presenting what he called “the central mystery” of quantum mechanics by applying the Copenhagen Interpretation to a description of what happens to an electron (or any other quantum entity) when it passes through what he called “the experiment with two holes.” It “has in it,” he said, “the heart of quantum mechanics.”
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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 strips 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.”
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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 P
H
D 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.