Zeno effect and ergodicity in finite-time quantum measurements

TL;DR

This paper explores how finite-time quantum measurements influence system dynamics, revealing a Zeno effect that traps the system or drives it to a universal steady state, depending on measurement approach.

Contribution

It introduces a novel analysis of the Zeno effect and ergodicity in finite-time quantum measurements, connecting measurement strategies to system behavior and steady states.

Findings

Measurement of time-averaged quantities can trap the system in eigensubspaces.

Long measurements lead to a universal steady state where ensemble averages match time averages.

Zeno-like behavior arises from conservation of probability and the structure of amplitude distributions.

Abstract

We demonstrate that an attempt to measure a non-local in time quantity, such as the time average $\la A\ra_T$ of a dynamical variable $A$, by separating Feynman paths into ever narrower exclusive classes traps the system in eigensubspaces of the corresponding operator $\a$. Conversely, in a long measurement of $\la A\ra_T$ to a finite accuracy, the system explores its Hilbert space and is driven to a universal steady state in which von Neumann ensemble average of $\a$ coincides with $\la A\ra_T$. Both effects are conveniently analysed in terms of singularities and critical points of the corresponding amplitude distribution and the Zeno-like behaviour is shown to be a consequence of conservation of probability.