Quantum trajectories under frequent measurements in non-Markovian environment

TL;DR

This paper extends quantum trajectory theory to non-Markovian environments modeled by Lorentzian spectral density, revealing a scaling property that unifies QT theory and the Zeno effect and clarifies the theory's validity conditions.

Contribution

It introduces a scaling framework for quantum trajectories in non-Markovian environments, connecting QT theory with the Zeno effect through a new $x$-dependent criterion.

Findings

Identifies a perfect scaling property with measurement frequency and environment bandwidth.

Bridges QT theory and the Zeno effect as two limits of the scaling variable.

Provides a quantitative criterion for the validity of conventional QT theory.

Abstract

In this work we generalize the quantum trajectory (QT) theory from Markovian to non-Markovian environments. We model the non-Markovian environment by using a Lorentzian spectral density function with bandwidth ($\Lambda$), and find perfect "scaling" property with the measurement frequency ($\tau^{-1}$) in terms of the scaling variable $x=\Lambda\tau$. Our result bridges the gap between the existing QT theory and the Zeno effect, by rendering them as two extremes corresponding to $x\to\infty$ and $x\to 0$, respectively. This $x$-dependent criterion improves the idea of using $\tau$ alone, and quantitatively identifies the validity condition of the conventional QT theory.