Quantum trajectories under frequent measurements in a non-Markovian environment

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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.