Quantum trajectories under frequent measurements in non-Markovian environment

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 ($\mathrm{\Lambda }$), and find perfect “scaling” property with the measurement frequency (${\tau }^{-1}$) in terms of the scaling variable $x=\mathrm{\Lambda }\tau$. Our result bridges the gap between the existing QT theory and the Zeno effect, by rendering them as two extremes corresponding to $x\to \mathrm{\infty }$ 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.