# Large-time limit of the quantum Zeno effect

## TL;DR

This paper studies how the quantum Zeno effect transitions to diffusive behavior as measurement frequency and total observation time scale, revealing a critical threshold at  and dependence on spectral and arithmetic properties.

## Contribution

It characterizes the large-time limit of the survival probability under different scaling regimes, identifying a sharp transition at  and exploring spectral and arithmetic sensitivities.

## Key findings

- Survival probability jumps from 1 to 0 at  scaling.
- Limit probability depends on spectral properties for  scaling.
- Behavior transitions from quantum Zeno to diffusive at .

## Abstract

If very frequent periodic measurements ascertain whether a quantum system is still in its initial state, its evolution is hindered. This peculiar phenomenon is called quantum Zeno effect. We investigate the large-time limit of the survival probability as the total observation time scales as a power of the measurement frequency, $t\propto N^\alpha$. The limit survival probability exhibits a sudden jump from 1 to 0 at $\alpha=1/2$, the threshold between the quantum Zeno effect and a diffusive behavior. Moreover, we show that for $\alpha\geq 1$ the limit probability becomes sensitive to the spectral properties of the initial state and to arithmetic properties of the measurement periods.