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CHAPTER 5
Are We There Yet?
The Quantum Zeno Effect
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I’ve had a long, annoying day at work, running from one committee meeting to another, and I come home with a pounding headache. I have an hour or so until Kate gets home, and all I want is a nap. Emmy is ecstatic to see me, and does the Happy Dance all over the living room.
“Hooray! You’re home! Yippeee!” She’s wagging her tail so hard she almost loses her balance. This happens every afternoon when I come home.
“It’s good to see you, too.”
“Let’s do something fun! Let’s play fetch! Let’s go for a walk! Let’s play fetch on a walk!”
“Let’s let me take a nap.” She stops bounding immediately, and looks crestfallen. Her ears and tail droop.
“No walk?”
“Not right now,” I say, lying down on the sofa. “Let me sleep for half an hour, and then we’ll do something fun.”
“Promise?”
“I promise. Now be quiet. The sooner I get to sleep, the sooner we’ll do something fun.”
“Oh. Okay.”
I lie down and get comfortable on the sofa. I’m just starting to settle in for my nap, when a cold, wet nose pokes me in the face.
“Are you asleep?”
“No, I’m not asleep.”
“Oh.” A minute passes.
Poke. “Are you asleep?”
“No.” Another minute passes.
Poke. “Are you asleep?”
“No!” I sit back up. “And I’m never going to get to sleep if you don’t stop poking me with your nose and asking that question.”
“Why not?”
“Every time you poke me, you wake me back up, and I have to start over again. If you keep waking me up, I don’t get all the way asleep, and you don’t get to do anything fun.” “Oh.” She brightens up. “It’s just like the Zero Effect!” “The what?”
“You know. The paradox with the bloke who can’t catch the turtle because he has to go half the way there, and then another half, and so on, so he never gets anywhere.”
“You mean Zeno’s paradox. Zeno, with an n as in ‘nap.’ The Zero Effect is a film with Bill Pullman and Ben Stiller.”
“Whatever. I don’t spell so well.”
“Anyway, what you’re thinking of is the quantum Zeno effect, and yes, this is sort of like that. If you have a system that’s moving from one state to another, with the probability of being in the second state increasing over time, you can prevent the state change by repeated measurements. Every time you measure it to be in the first state, you restart the process.”
“Right. So when I ask if you’re asleep, I collapse your wavefunction back to the ‘awake’ state, and you need to start napping again.”
“Or you find yourself perceiving the branch of the wavefunction in which I’m awake again, in the many-worlds picture. But yes, that’s the basic idea, and a good analogy.”
“I’m a philosophical dog!”
“Yes, you’re very clever. Now shut up and let me sleep.”
“Okay. I won’t ask if you’re asleep any more.”
“Thank you.”
I settle back down onto the sofa, and start to feel warm and cozy, and feel myself drifting off…
Poke. “Are you awake?”
Whether you prefer the Copenhagen interpretation, many-worlds, or one of the many others, something happens when you make a measurement. Whether you think this involves the physical collapse of a wavefunction, or just limiting your perception to a single branch of an expanding and evolving wavefunction, measurement is an active process. Before you measure an object’s state, it exists in a quantum superposition of all possible states, while immediately after the measurement, you observe one and only one state.
In this chapter, we’ll look at the most dramatic consequence of active measurement, the “quantum Zeno effect.”
We’ll see that making repeated measurements of a quantum particle can prevent it from changing its state. We can also use the quantum Zeno effect to detect the presence of objects without hitting them with even a single photon of light.
YOU CAN’T GET ANYWHERE FROM HERE: ZENO’S PARADOX
The name of the effect is a reference to the famous paradoxes of the Greek philosopher Zeno of Elea, who lived in the fifth century B.C.E. There are several different versions of the paradoxes, but all of them purport to show that motion is impossible.
Here’s a modern canine version of the argument: in order to reach a treat on the far side of the room, a dog first needs to cross half the width of the room, which takes a finite time. Then, she needs to cross half of the remaining distance, which takes a finite time, and then half of the remaining distance, and so on. The distance across the room is divided into an infinite number of half steps, each requiring a finite time to cross. If you add together an infinite number of steps, each taking a finite time, it should take an infinite amount of time to cross the room. Thus, it’s impossible for the poor dog to ever get all the way to the tasty treat.
Happily for hungry dogs everywhere, there’s a mathematical solution to the apparent paradox: as the distance gets smaller, the time required to cross it also gets smaller. If it takes one second to cross half the width of the room, it takes half a second to cross the next quarter, and a quarter of a second to cross the next eighth, and so on. Adding together all those times, we find that:
The total time is the sum of an infinite number of terms, but the terms get smaller as you go. Mathematicians learned how to add this sort of series when calculus was invented in the seventeenth and eighteenth centuries. The infinite sum gives a finite result: the dog crosses the room in two seconds. Motion is possible after all, and a good dog can always reach her treats.*
WATCHED POTS AND MEASURED ATOMS:THE QUANTUM ZENO EFFECT
The quantum Zeno effect uses the active nature of quantum measurement to prevent a quantum object (like an atom) from moving from one state to another, by making repeated measurements. If we measure the atom a very short time after the transition starts, it will most likely be found in the initial state. The act of measuring the atom projects it back into the initial state, as we saw in chapter 3, and the transition starts over.
If we keep measuring the state of the atom, we keep putting it back where it started. The atom is in a predicament reminiscent of Zeno’s paradox—taking an infinite number of steps toward some goal, but never getting there.* As the old saying has it, a watched pot never boils, at least as long as it’s a quantum pot.
This is dramatically different from classical physics. Measuring the state of a classical object does not change the state—if a pot of water is 50% of the way to boiling when the measurement is made, it’s still 50% of the way to boiling after the measurement. The quantum Zeno effect works only because of the active nature of quantum measurement—the water in a quantum pot is either boiling or not boiling. If you find that it isn’t boiling, you need to start over again, as if you had never heated it.
The definitive quantum Zeno effect experiment was done in 1990 by Wayne Itano in Dave Wineland’s group at the US National Institute of Standards and Technology (NIST) in Colorado, using beryllium ions. Ions are just atoms with one electron removed, and like all atoms, they have a collection of allowed energy states, which they move between by absorbing or emitting light. Itano’s experiment collected a few thousand beryllium ions, and made them move slowly from one state to another by exposing them to microwaves.
Left unmeasured, the ions took 256 milliseconds to complete the transition from State 1 to State 2.* Their state during this process was described by a wavefunction with two parts, corresponding to the probability of finding the atom in State 1 and State 2. At the start of the experiment, the atoms were 100% in State 1, and at the end, they were 100% in State 2. In between, the probability of State 2 steadily increased, while the probability of State 1 steadily decreased.
The experimenters measured the state of the ions using an ultraviolet laser with its frequency chosen so that an ion in State 1 would happily absorb light, while ions in State 2 would not absorb any light. Ions in State 1 absorbed photons from the laser and re-emitted them a few nanoseconds later, making a bright spot on a camera pointed at the ions. Ions in State 2, on the other hand, produced no light when illuminated by the laser. The total amount of light reaching the camera, then, was a direct measurement of the number of ions in State 1.
To demonstrate the quantum Zeno effect, the NIST group trapped a large number of ions, all in State 1. Then they turned on the microwaves, waited 256 milliseconds, and pulsed on the laser. None of the ions produced any light, indicating that 100% of the sample had moved to State 2, as expected. Then they repeated the experiment, with two laser pulses: one after 128 ms (halfway through the move to State 2), and one after 256 ms. In this case, they saw half as much light after 256 ms, indicating that only 50% of the sample had made the transition to State 2.
The decreased probability is explained by the quantum Zeno effect. The laser pulse halfway through measured the state of the ions. Many of them were found in State 1, and the measurement destroyed the State 2 part of the wavefunction. These atoms were now 100% in State 1, so the transition had to start over again, with the probability of State 2 increasing slowly. After another 128 ms, the probability of finding the ions in State 2 was only 50%.
The probability of moving from State 1 to State 2 decreased further with more measurements. With four pulses (at 64, 128, 192, and 256 ms), only 35% of the atoms made the transition. With eight pulses, only 19% made the transition. With a total of 64 laser pulses over the full experimental interval (one every 4 ms), fewer than 1% of the atoms made the transition. All of these probabilities were in excellent agreement with the theoretical predictions of the quantum Zeno effect, as shown in the figure below.
“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.”
“Wait, not absorbing a photon is the same as absorbing a photon?”
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.
“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, yes, 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.”
“Yes, 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?”
“Yes, hypothetically.”
“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
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 rabbit, say) on the right-hand side, though, it will prevent the photon from moving, through the quantum Zeno effect.
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. 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 rabbit in the right half of the apparatus. Each time the photon hits the central mirror, the rabbit measures whether the photon passed through the mirror: being very skittish, the rabbit will run away if it detects even a single photon on the right side.
The “sloshing” that happens in the no-rabbit case is blocked by the quantum Zeno effect when the rabbit is present. If the photon does pass through the mirror, the rabbit 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 again.
The quantum Zeno effect lets us do what any dog wants to: determine whether there’s a rabbit 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 rabbit on the right, either because the rabbit absorbed the photon and ran off, or because there never was a rabbit 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 rabbit, 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 rabbit 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 rabbit 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 rabbit without scaring it off is 98.8%. If you repeated the experiment 1,000 times, only 12 rabbits would be scared off.
“Great! So, all I need to do is get some big mirrors…”
“No. You are not setting this experiment up in the back garden.”
“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 garden, and that’s final.”
“Awww…”
Quantum interrogation hasn’t been used to catch rabbits, 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 effect 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 and look in my bowl.”
“Why is that?”
“Well, I’m going to use the Zeno effect to get more food. I reckon, if I keep measuring my bowl to be full of dog food, I’ll always have food, 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 food.”
“Oh. That would be bad. I didn’t think of that.”
“Anyway, you’d need to have some natural quantum process that caused food 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 food in my bowl, right? And you’re a natural process.”
“In a manner of speaking.”
“So, how about putting some food in my bowl?”
“Oh, all right. It’s almost dinnertime. Come on.”
* 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 madder 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.
