The Quantum Zeno Effect

In order for a system to perform a computation, the components within the computer need to be able to interact with each other. This is fairly easy with electricity, but surprisingly hard with photons. Fortunately, the Quantum Zeno Effect enables that interaction. While very good work has been done to work out the rigorous mathematics behind this effect (such as this article), the point of this page is to explain on a more conceptual level, without giving the full mathematics.

Introduction

A popular expression is that a “watched pot never boils” referring to the psychological effect of time appearing to pass more slowly when watching and strongly anticipating an event. Classically, watched pots do boil of course, and if we talk about actual physical time as measured on a stopwatch, the time to boil is completely unaffected by looking at it. Quantum mechanically, the situation is different because systems are affected by observing them, and a quantum-watched pot really doesn’t boil. In other words, a system can be stopped by observing it. Quantum pots don’t actually exist, so in practice, it is more like “a watched atom never decays” or “a watched light particle never enters a cavity”. Various experiments have been performed to show that this is a very real effect, including by members of our team. Here “observing” is a loose term, it just means strongly interacting with something else and it doesn’t need a human or other sentient being to be aware of the result.

How does it work?

To understand the Zeno effect it is worth first learning a bit about where the name comes from. The name is after a philosophical paradox known as Zeno’s arrow paradox, developed by the ancient Greek philosopher Zeno of Elea between 500 and 400 BCE. The basic idea is (very roughly) that an arrow could not really travel through space, because if observed at any instant it appears motionless, so if we observe it at every time it can never move. While this does point to a conceptual difficulty in classical continuum mechanics, it isn’t the reality classically because arrows really can travel through space (many ancient Greeks who were shot by arrows could confirm this fact), and when viewed mathematically, it takes calculus to resolve the arrow paradox. The reason is that a classical system can be measured without having any effect on it; a watched arrow moves exactly like an unwatched arrow.

Quantum mechanically, the situation is fundamentally different. No matter how good a measurement apparatus is, it fundamentally does change the system it measures, and observing an arrow as motionless makes it motionless. This is known as projective measurement; it projects a state into the result of the measurement. However, to understand how motion is stopped we need to understand a little bit about how quantum systems work. Quantum mechanics is based on the movement of probability amplitude, which is proportional to the square root of probability. If a quantum system starts to move into a different state, this is mathematically described by amplitude starting to build up in that state. If we measure before much amplitude can build up then the amplitude squared is going to be very small, since squaring a number less than one makes it even smaller. Making a lot of quantum measurements quickly therefore can stop the system from moving entirely.

(File:Quantum Zeno effect animation.Gif. In Wikipedia. https://commons.wikimedia.org/wiki/File:Quantum_Zeno_effect_animation.gif)

With a minimal amount of equations, we can see how this works, if we take a system that would normally transition from one state to another (let’s call them state $0$ $0$ and $1$ $1$ , starting in state $0$ $0$ ). Mathematically, the system can be described as a vector with two elements corresponding to probability amplitudes. $(a_0,a_1)$ $(a_{0}, a_{1})$ . The probabilities to be in each correspond to $P_0=|a_0|^2$ $P_{0} = ∣ a_{0} ∣^{2}$ and $P_1=|a_1|^2$ $P_{1} = ∣ a_{1} ∣^{2}$ , where the absolute value is needed because these can generally contain factors of $i=\sqrt{-1}$ $i = - 1$ . Since the system must be in one state or the other $P_0+P_1=1$ $P_{0} + P_{1} = 1$ . If we measure, the probability of outcome $1$ $1$ is proportional to $|a_1|^2$ $∣ a_{1} ∣^{2}$ .

If we measure it $N$ $N$ (evenly spaced) times then the amplitude $a^2$ $a^{2}$ which can build up during each time period is proportional to $1/N$ $1/ N$ , making the probability to measure in state $1$ $1$ ( $P_1=|a_1|^2$ $P_{1} = ∣ a_{1} ∣^{2}$ ) proportional to $1/N^2$ $1/ N^{2}$ . Since we measure $N$ $N$ times, the overall, probability of ever finding the system in state $1$ $1$ will be proportional to the number of measurements times the probability of finding it in state $1$ $1$ in each measurement, this will be proportional to $N/N^2=1/N$ $N / N^{2} = 1/ N$ and therefore will go to zero as the number of measurements becomes large. Note that this needed both the fact that quantum measurements change the system and the fact that the important quantity in quantum mechanics is amplitude, not probability, to work correctly.

But how is this useful?

At this point, a completely fair question is how any of this is actually useful. Naively, stopping a system doesn’t seem all that useful. We want our quantum systems to go around doing quantum things after all. The answer was hinted at in the beginning, recall that a system does not actually need to be observed by something as sophisticated as a human. In fact, using some of our nonlinear optics systems, we can engineer a situation where a single light particle can effectively watch another light particle. In this way, we can make light particles, which are famously hard to make interact, have interactions with each other. Light has a lot of fantastic properties for computing. For example, light particles naturally move, making information transfer relatively simple, and since light particles aren’t charged they don’t interact much with noise sources. However, the weakness of light has traditionally been that it is hard to make the light particles interact with each other. In other words, we only stop a particle from moving from one component on a chip to another if a different particle is already there, and this can be used to build a quantum computer by providing an entangling gate between light particles. Zeno effects are useful in many other ways too. For example, if the measurement slightly changes each time a system can instead be guided into different states, but this is a subject for a different discussion.

Next, we will build off of this knowledge and discuss the Zeno Blockade.