by CHAD ORZEL
“So, when you make a measurement, the ion absorbs a photon, and that collapses the wavefunction?”
“Actually, the ion doesn’t need to absorb a photon at all. The Wineland group repeated the experiment starting with the ion in State 2. In that case, the ion starts out in the ‘dark’ state, and doesn’t absorb any photons during the measurements. They still got the same result—the probability of making the transition from State 2 to State 1 decreased with more measurements, exactly as predicted.”
The probability of making a transition from one state to another in the quantum Zeno effect experiment done by the Wineland group (W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 41, 2295–2300 [1990], modified and reprinted with permission). Black bars are the theoretical prediction, gray bars are the experimental result, with error bars showing the experimental uncertainty. The probability of changing states decreases as the number of measurements increases, whether the ions start in State 1 or State 2.
“Wait, not absorbing a photon is the same as absorbing a photon?”
“When it comes to thinking of the photons as measurement tools, yes. It’s just like the treat in two boxes—if you open one of the boxes, and find it empty, you know the treat has to be in the other box. That determines the state of the treat just as if you opened the box and found a treat there.”
“It’s not as much fun, though, because I don’t get the treat.”
“Yes, well, your life is very difficult.”
The quantum Zeno effect does not depend on a particular interpretation of quantum mechanics. It’s easier to discuss what’s going on using the Copenhagen language of wavefunction collapse, but we can equally well describe it in terms of the many-worlds interpretation. In the many-worlds picture, new branches of the wavefunction appear at each measurement step, but we are more likely to perceive the higher probability branch. The probability of seeing a state change is the same in both interpretations.
We can use the quantum Zeno effect to dramatically reduce the chance of a system changing states, simply by measuring it many times. We can never make the probability of transition exactly zero—there’s always a small chance that it will change in spite of the measurements—but we can make it very small, demonstrating the power of quantum measurement.
“Humans are so silly. If you want to stop the transition, wouldn’t it be easier to just turn off the microwaves?”
“Well, sure, but the point is to demonstrate that the quantum Zeno effect is real. It’s not interesting because it can stop ions from changing states; it’s interesting because of what it tells us about quantum physics.”
“Yeah, but what good is it? Can it do anything useful?”
“Well, you can use it to detect objects without having them absorb any light.”
“Objects . . . like bunnies?”
“Sure, hypothetically.”
“Oooh! I like the sound of that!”
MEASURING WITHOUT LOOKING: QUANTUM INTERROGATION
The quantum Zeno effect can be exploited to do some amazing things. A collaboration between the University of Innsbruck and Los Alamos National Laboratory has demonstrated that it’s possible to use light to detect the presence of an absorbing object without having it absorb any photons, by using the quantum Zeno effect to stop a photon moving from one place to another.
We start with a photon on the left-hand side of the apparatus, bouncing back and forth between two mirrors. There is a small chance of the photon leaking through the central mirror, so over time the photon will shift into the right-hand side of the apparatus. If there is an absorbing object (a bunny, say) on the right-hand side, though, it will prevent the photon from moving, through the quantum Zeno effect.
In the future, this technique may be used to study the properties of quantum systems that are too fragile to survive absorbing even a single photon.
Here’s a simplified version of this quantum interrogation experiment: imagine that we have a single photon bouncing back and forth between two perfect mirrors. Halfway between those two, we insert a third mirror that’s not quite perfect.
The wavefunction for this system has two pieces, one corresponding to finding the photon in the left half of the apparatus, and the other corresponding to finding the photon in the right half. If we start the experiment with a single photon in the left half, we find that over time, it will slowly move into the right half. Each time the photon hits the imperfect central mirror, there’s a small chance that it goes through, so the left-side piece of the wavefunction gets a little smaller, and the right-side piece gets bigger. Eventually, the left-side piece is reduced to zero, and there is a 100% chance of finding the photon on the right side. Then the process reverses itself. The photon will slowly “slosh” back and forth between the two sides of the apparatus, just as the ions in the NIST experiment moved between State 1 and State 2.
We can trigger the quantum Zeno effect by adding a device to measure the position of the photon, such as a bunny in the right half of the apparatus. Each time the photon hits the central mirror, the bunny measures whether the photon passed through the mirror: being very skittish, the bunny will run away if it detects even a single photon on the right side.
The “sloshing” that happens in the no-bunny case is blocked by the quantum Zeno effect when the bunny is present. If the photon does pass through the mirror, the bunny absorbs it and flees. The photon no longer exists, so its wavefunction is zero, and nothing changes after that. If it doesn’t make it through, the photon is definitely on the left-hand side, and the wavefunction is put back in the initial photon-on-the-left state, and everything starts over.
The quantum Zeno effect lets us do what any dog wants to: determine whether there’s a bunny in the apparatus without scaring it off. We start with a photon on the left side, wait long enough for it to move over to the right side, and then look at the left side of the apparatus. If there’s no photon there, there’s no bunny on the right, either because the bunny absorbed the photon and ran off, or because there never was a bunny and the photon has “sloshed” over to the right. If the photon is still in the left-hand side of the apparatus, we know that not only was there a bunny, but it is still there, and has not absorbed even a single photon of light.
There is always a chance that the photon will make it through and scare the bunny away, but we can make this chance as low as we like, by decreasing the probability that the photon will leak through the mirror. We’ll have to wait longer to complete the measurement, as the time required for the photon to “slosh” into the right side will increase, but the chances of successfully detecting the bunny improve dramatically. If the photon needs to bounce back and forth on the left-hand side 100 times before it “sloshes” to the right, the probability of detecting a bunny without scaring it off is 98.8%. If you repeated the experiment 1,000 times, only 12 bunnies would be scared off.
“Oooh! So, all I need to do is get some big mirrors . . .”
“No. You are not setting this experiment up in the backyard.”
“But I can use the quantum Zeno effect to sneak up on the bunnies . . .”
“No. Just . . . No. You are not putting great big mirrors across the yard, and that’s final.”
“Awww . . .”
Quantum interrogation hasn’t been used to catch bunnies, but it has been demonstrated experimentally using polarized photons, by physicists in Innsbruck, Los Alamos, and Illinois. Quantum interrogation allows you to do some incredible things—taking pictures of objects without ever bouncing light off them, for example. This probably isn’t useful for spy purposes (unless you can somehow get your enemies to obligingly store their secrets between two mirrors), but it might be essential for probing fragile quantum systems like large collections of atoms in superposition states that can’t survive the absorption of a photon.
Whether you think of it in terms of collapsing wavefunctions, or a single expanding wavefunction undergoing decoherence, the quantum Zeno effe
ct is a dramatic demonstration of the strange nature of quantum measurement. Unlike classical measurement, the act of measuring a quantum system changes the state of that system, leaving it in only one of the allowed states, which is very different than what we expect classically. With a clever arrangement of the experimental situation, this can be exploited to prevent a system from changing states, or even to extract information from a system without interacting with it directly.
“That’s really interesting. Weird, but interesting.”
“Thanks.”
“Now, if you’ll excuse me, I need to go look in my bowl.”
“Why is that?”
“Well, I’m going to use the Zeno effect to get more food. I figure, if I keep measuring my bowl to be full of kibble, I’ll always have kibble, no matter how much I eat. That will be fun.”
“Of course, if you keep measuring your bowl to be empty, it’ll always be empty, and you’ll never have kibble.”
“Oh. That would be bad. I didn’t think of that.”
“Anyway, you’d need to have some natural quantum process that caused kibble to appear in the bowl for that to work. Things aren’t going to appear for no reason, just because you want to measure them.”
“Well, you sometimes put kibble in my bowl, right? And you’re a natural process.”
“In a manner of speaking.”
“So, how about putting some kibble in my bowl?”
“Oh, all right. It’s almost dinnertime. Come on.”
“Oooh! Kibble!”
* While the summing of infinite series is accepted as the resolution of Zeno’s paradox by physicists and engineers and most mathematicians, some philosophers do not accept this as a sufficient resolution of Zeno’s paradox (Stanford Encyclopedia of Philosophy). This just proves that philosophers are crazier than mathematicians, or even cats.
* A better Greek literary allusion might be the myth of Sisyphus, who was condemned to spend eternity pushing a boulder up a hill, only to have it slip free and roll back to the bottom again. The name “Sisyphus effect” was used for something else, though, so this is called the quantum Zeno effect.
† A quarter of a second seems pretty fast to humans or dogs, but it’s really slow for an atom. Atoms usually change states in a few billionths of a second.
CHAPTER 6
No Digging Required: Quantum Tunneling
We’re sitting in the backyard, enjoying a beautiful sunny afternoon. I’m lying on a lounge chair reading a book, and Emmy is sprawled out on the grass, basking in the sun and keeping an eye out for squirrel incursions.
“Can I ask a question?” she asks.
“Hmm? Sure, go ahead.”
“What do you know about tunneling?”
“Tunneling, eh?” I put my book down. “Well, it’s a process by which a particle can get to the other side of a barrier despite not having enough energy to pass over the barrier.”
“Barrier? Like a fence?”
“Well, metaphorically, at least.”
“Like the fence between this yard and the next?” She looks really hopeful.
“Oh. Is that what this is about?”
“There are bunnies over there!” She wags her tail for a minute, then looks crestfallen. “But I can’t get to them.”
“True, but I don’t think tunneling is the answer. It works for small particles, but wouldn’t work for a dog.”
“Why not?”
“Well, you can think of a barrier in terms of potential and kinetic energy. For example, right now, all your energy is potential energy, because you’re not moving. But you could start moving, say, if you took off after a squirrel, and turned that potential into kinetic energy.”
“I’m very fast. I have lots of energy.”
“Yes, I know. You’re a great trial to us. Anyway, whether you’re sitting still, or moving, you have the same total amount of energy. It’s just a question of what form it’s in.”
“Okay, but what does this have to do with the fence?”
“Well, you can think of the fence as being a place where you can only go if you have enough energy. For you to be at the spot where the fence is, you would have to jump very high or else occupy the same space as the fence, and either would take an awful lot of energy.”
“I can’t jump that high. That’s why I can’t get the bunnies.”
“Right. You don’t have enough energy to get over the fence. And because you don’t have enough energy, you can’t end up in the neighbors’ yard, and everybody is much happier that way, believe me.”
“Except me.” She pouts.
“Yes, well, except you.” I scratch behind her ears by way of apology. “Anyway, quantum mechanics predicts that even though you don’t have enough energy to go over the fence, there’s still a chance that you could end up on the other side. You could just sort of . . . pass through the fence, as if it weren’t there.”
“Like the bunnies do!”
“Pardon?”
“The bunnies. They go back and forth through the fence all the time.”
“Yes, well, that’s because they fit between the bars of the fence. It has nothing to do with quantum tunneling.” I stop for a moment. “Of course, it’s not a bad analogy. The bunnies don’t have enough energy to go over the fence, either, but they can go through it, and end up on the other side. Which is sort of like tunneling.”
“So how do I tunnel through the fence?”
“Well, you could eat fewer treats, and get skinny enough to pass between the bars like the bunnies do.”
“I don’t like that plan. I’m a good dog. I deserve the treats I get.”
“And you get the treats you deserve. The other option would be quantum tunneling through the fence, but quantum tunneling isn’t something you do, it’s something that just happens. If you send a whole bunch of particles at the barrier, a small number of them will show up on the other side. But which ones go through is completely random. It’s all about probability.”
“So, I just need to run at the fence enough times, and I’ll end up on the other side?”
“I wouldn’t try it. The probability of a particle tunneling through a barrier depends on the thickness of the barrier and the quantum wavelength of the particle. The probability of a fifty-pound dog passing through a half-inch aluminum barrier would be something like one over e to the power of ten to the thirty-six. Do you know what that is?”
“What?”
“Zero. Or near enough to make no difference. So don’t go throwing yourself at the fence.”
She’s quiet for a minute.
“Anyway, I hope that answers your question.” I pick my book back up.
“Sort of.”
“Sort of?”
“Well, the quantum stuff was interesting, and all, but I was thinking of classical tunneling.”
“Classical tunneling?”
“I was going to dig a hole under the fence.”
“Oh.”
“It’s a good plan!” She wags her tail enthusiastically, and looks very pleased with herself.
“No, it’s not. Only bad dogs dig holes.”
“Oh.” Her tail stops, and her head droops. “But I’m a good dog, right?”
“Yes, you’re a very good dog. You’re the best.”
“Rub my belly?” She flips over on her back, and looks hopeful.
“Oh, okay . . .” I put my book back down, and lean over to rub her belly.
“Tunneling” is one of the most unexpected quantum phenomena, where a particle headed at some sort of obstacle—say, a dog running toward a fence—will pass right through it as if it weren’t there. This odd behavior is a direct consequence of the underlying wave nature of quantum particles seen in chapter 2.
In this chapter, we’ll talk about the essential physics concept of energy, and how energy determines where particles can be found. We’ll see that the wave nature of matter allows quantum particles to turn up in places that classical physics says they can’t reach, pas
sing into or even through solid objects. We’ll also see how tunneling lets scientists build microscopes that can study the structure of matter, making possible revolutionary developments in biochemistry and nanotechnology.
THE ABILITY TO GET THINGS DONE: ENERGY
In order to explain quantum tunneling, we need to first talk about the classical physics of energy. While the term “energy” has passed from physics into more general use, its physics meaning is slightly different from its everyday, conversational use.
A one-sentence definition of the term “energy” in physics might be: “The energy content of an object is a measure of its ability to change its own motion or the motion of another object.” An object can have energy because it is moving, or because it is held stationary in a place where it might start moving. Every object has some energy simply because it has mass (Einstein’s E = mc2) and because its temperature is above absolute zero.* All of these forms of energy can be used to set a stationary object into motion, or to stop or deflect an object that is moving.
The most obvious form of energy is kinetic energy, the energy associated with a moving object. The kinetic energy of an object moving at an everyday sort of speed is equal to half its mass times the velocity squared, or as it’s usually written:
KE = ½ mv2
Kinetic energy is always a positive number, and increases as you increase either the mass or the speed. A Great Dane has more kinetic energy than a little Chihuahua moving at the same speed, while a hyperactive Siberian husky has more kinetic energy than a sleepy old bloodhound of the same mass. Kinetic energy is similar to momentum, but it increases faster as you increase the velocity, and unlike momentum, it doesn’t depend on the direction of motion.
Objects that are not already moving have the potential to start moving due to interactions with other objects. We describe this as potential energy. A heavy object on a table has potential energy: it’s not moving, but it can acquire kinetic energy if a hyperactive dog bumps into the table and it falls on her. Two magnets held close to each other have potential energy: when released, they’ll either rush together or fly apart. A dog always has potential energy, even when sleeping: at the slightest sound, she can leap up and start barking at nothing.