A quantum version of Zeno's paradox

The quantum Zeno effect describes the situation where an unstable particle, say a radioactive atom, won’t decay if it is observed continuously. More generally, it says that if you repeatedly interact with a quantum system through measurement then you can effectively freeze its quantum state.

The phenomenon is named after a Greek philosopher of ancient times, Zeno of Elea. Zeno is known best for a set of paradoxes (we know of 9 of them) that he posed as arguments  against Aristotle’s concept of motion. Here we are interested in the arrow or fletcher’s paradox.

If you observe an arrow flying through the air at some particular instant in time, then it would have a definite position, meaning it isn’t moving at that specific moment. However, you can think of the arrow’s motion as happening one moment at a time. This says that motion must be impossible since it is made up of this long sequence of motionless moments. 

Of course, as far as we can tell, the world is not static and objects in it are not forever motionless. What’s lacking with Zeno’s assertion is the mathematical notion of continuity. Motion is possible because time doesn’t flow like a series of separate frames in a film but more like the seamless current of a steady stream.

(You might be more familiar with the paradox involving a race between a tortoise and Achilles. The idea is if Achilles gives the tortoise, he will never catch up because when you consider each time he halves the distance between him and the tortoise, it’s always at a location behind where the tortoise. Of course, the paradox is solved by the concept of a convergent series.) 
 

Roughly speaking, the quantum Zeno effect says that the state of a quantum system does not change if you keep checking whether the state has changed or not. 

They say it’s like when you’re heating some water and waiting for it to boil. If you keep staring at the water, it never seems to boil but if you look away long enough then it finally does boil. Of course, in this case it’s merely the perception: when you want something to happen immediately, it feels like it takes forever to happen. Your gaze does not prevent the water from boiling, and you will see heated water boil if you wait long enough.

However, if you repeatedly measure a quantum system often enough, you can keep it from evolving into a different state.

To describe the effect, we employ the qubit represented by the spin of the valence electron of a silver atom in the Stern-Gerlach apparatus described in the previous post. There we learned that if you applied a uniform magnetic field to a silver atom, it exerts a torque to the spin of the atom’s valence electron, causing it to rotate about an axis that’s parallel to the direction of the magnetic field. We used this to construct a flipper, a device that flips the spin from pointing up to pointing down as shown below.

An up-down flipper, one of the Stern-Gerlach devices discussed in the previous blog post. It flips the spin state of an atom from |u) to |d) and vice-versa.

We can change the duration of the magnetic field if we want to control how much the spin rotates. For example, if we apply the uniform field half as long as we would for a flipper, then the spin is rotated only halfway towards the opposite direction. If our electron was initially at state |u) and the uniform field is pointed away from your monitor, then it would end up at state |+) = 1/sqrt(2) |u) + 1/sqrt(2) |d).

Schematic diagram for the up x-selector, a device that rotates a spin pointing up an angle x about an axis parallel to the uniform magnetic field.

In general, we want to imagine a rotator that rotates a spin pointing up x° along an axis pointing to the right of this page. We shall call this an up x-rotator for short. The Stern-Gerlach apparatus for an up x-rotator is depicted above.  For example, the device which takes a spin from state |u) to state |+) is an up 90°-rotator.

Suppose we have a beam of silver atoms prepared in the state |u). What we want to do is flip the spin from up to down using two 90°-rotators (since a flip requires an angle of 180°) but in between the rotators,  let’s check if the spin is still pointing up. The setup for that experiment is shown in the figure below: the beam is sent through a sequence of three devices, an up-selector sandwiched by two 90°-selectors.

In this Stern-Gerlach experiment, we have a beam of atoms with spin pointing up, written in shorthand form as (1,0) on the left of the figure. We have two 90°-rotators that will attempt to flip the spin in two steps but in between we want to "look" at the atoms and check if they are still pointing up. Because about half the beam gets through the up-selector and the final state of the atoms is |+), you could say that 1/4 of the original atoms "survived" in the initial state.

The scenario is simple enough that we can just follow step by step how the state of the atoms changes with each device:

1.  The first rotator essentially turns the spin from state |u) into state |+).
2.  Next, the up-selector checks every atom if it is “|u) or |d)” and transmits it only if the spin is pointing up.
3. About half the beam will pass the selector and another rotator converts the state from |u) to |+).

Of the remaining atoms, roughly half will be measured in the |u) state.  All in all, ¼ of the atoms in the original beam gets through. That's what happens if we check once.

Now what we really want is to repeatedly check if the atoms are still in the |u) state. It means we want to have as many selectors as possible. It also means that we can only apply a uniform magnetic field for a brief period between adjacent selectors, so the angle x gets smaller as we add more selectors.

In this case, it is helpful to know how our rotator works for any angle. Since we always start with a beam of atoms in the state |u), it is simple to describe what happens to the state: it goes from |u) to cos(x/2) |u) + sin(x/2)|d). Using the shorthand notation for up-down qubits, we would write these states as (1,0) and (cos(x/2), sin(x/2)), respectively.

A Stern-Gerlach experiment for demonstrating the quantum Zeno effect. We have n x-rotators that attempt to flip the spin of the initial beam of atoms in the state |u) (that is, nx = 180°). In between every pair of adjacent x-rotators, an up-selector checks whether the atoms are still in their initial state. Because a fraction |cos(x/2)|^2 of the atoms get transmitted each time an up-selector is used and there are n selectors if we also measure the final beam, the probability any particular atom survives the experiment is [|cos(x/2))^2]^n, which is very nearly 100% when n is large enough. Roughly speaking, the act of repeatedly observing if the state of the atoms changed causes it not to change.

Let's see what happens in an experiment where we check the state of the atoms n times. Again, since we are trying to flip the spins while also repeatedly measuring if they are still pointing up, we have sequence of n rotators with x = 180°/n with an up-selector between every pair of rotators, as depicted in the diagram above. 

We are interested in large values of n so suppose we have n = 100. Then x = 1.8° and so each rotator changes the state from |u) := (1,0) to 

 |x) := (cos(x/2), sin(x/2)) = (0.9999, 0.0157).

That is, once the beam hits the first selector, we expect 99.97% of the atoms to be measured in the state |u) and transmitted. This process is repeated n = 100 times, which tells us that the probability any single atom makes it to the end is

prob("atom is intact") = [|cos(x/2)|^2]^100 = 0.9756.

Therefore, we expect most of the atoms in the original beam to get through.

Despite our attempts to flip the spin state from |u) to |d), almost all of the atoms we get at the end of the experiment will be found in state |u) just because we kept observing them. If you take a larger value of n (which means x gets smaller at the same time), you get even closer to 100%. Because we have effectively frozen the state of the atoms. in some sense we have achieved something similar to Zeno’s arrow.

We generally expect the state to be altered by measurement, since a measurement "forces" an atom to choose one of the outcome states. What happens here, however, is that as we include more and more selectors, there is less time for each rotator to try and rotate the spins away from pointing up. By trying to do more (more rotators and selectors) we end up doing less (the state is more likely to remain the same).

The quantum Zeno effect might seem surprising at first, especially since it suggests that we can prevent a radioactive element such as uranium from decaying if we keep “looking” at it (with the right measurement) often enough. But all you need is a simple experiment with atoms and magnets to illustrate how it reasonably follows from the rules of quantum theory. By examining details more carefully, we find it’s not so surprising after all.