Zeno: Quantum Zeno Effect / The Observer
Quantum Foundations arc, page 1 of 193. Back to The Pixelated Physics Universe
Who they are
Stare at a pot and it never boils. Quantum mechanics has its own version. Zeno embodies the quantum Zeno effect: measure a quantum system often enough and its evolution is suppressed. Zeno wears it in pixel art on a unisex tee.
Zeno is a character of the Quantum Foundations arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
Each time you measure an unstable quantum system you reset it, and right after a reset its chance of having changed grows only with the square of the time. So if you look often enough, it barely gets the chance to change at all: a watched quantum pot never boils.
$$ \begin{gathered} P_{\text{stay}}(t) \approx 1 - \left(\frac{t}{\tau_Z}\right)^{2}, \qquad \tau_Z = \frac{\hbar}{\Delta H} \\ P_{1\to 2}(n) = \tfrac{1}{2}\left[1 - \cos^{n}\!\left(\tfrac{\pi}{n}\right)\right] \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $P_{\text{stay}}$ | probability the system is still in its initial state (dimensionless) | 1 |
| $t$ | time since preparation | s |
| $\tau_Z$ | Zeno time (short-time scale of decay) | s |
| $\hbar$ | reduced Planck constant | J·s |
| $\Delta H$ | energy spread of the initial state, √(⟨H²⟩ − ⟨H⟩²) | J |
| $P_{1\to 2}(n)$ | probability of completing a driven π-pulse transition when interrupted by n ideal measurements | 1 |
| $n$ | number of equally spaced measurements during the pulse | 1 |
Sources
- Misra & Sudarshan (1977) The Zeno’s paradox in quantum theory, J. Math. Phys. 18, 756
- Itano, Heinzen, Bollinger, Wineland (1990) Quantum Zeno effect, Phys. Rev. A 41, 2295
- Itano et al. (1990) Quantum Zeno effect — APS abstract page
- Ballentine (1991) Comment on “Quantum Zeno effect”, Phys. Rev. A 43, 5165
Crossovers
Zeno meets Nutariel (Rabi Oscillation). Nutariel starts drifting from one state to the other, sin²(Ωt/2) at a time. Zeno checks on her again and again, and each look sends her back to the start before she can get far. Watched often enough, Nutariel hardly flops at all.
Zeno meets Decoherex (Interference / Quantum Decoherence). Zeno's measurements and Decoherex's environment do the same job: each one records which state the system is in. Zeno makes the records on purpose; Decoherex lets the surroundings make them for free. Either way, the interference fades.
Wear the lore
Concept keywords: Quantum Zeno Effect / The Observer, Zeno, Quantum Foundations, physics lore, Pixelated Physics