Quantum transfer through a non-Markovian environment under frequent measurements and Zeno effect

We study transitions of a particle between two wells, separated by a reservoir, under the condition that the particle is not detected in the reservoir. Conventional quantum trajectory theory predicts that such no-result continuous measurement would not affect these transitions. We demonstrate that it holds only for Markovian reservoirs (infinite bandwidth $\Lambda$). In the case of finite $\Lambda$, the probability of the particle's interwell transition is a function of the ratio $\Lambda/\nu$, where $\nu$ is the frequency of measurements. This scaling tells us that in the limit $\nu\to\infty$, the measurement freezes the initial state (the quantum Zeno effect), whereas for $\Lambda\to\infty$ it does not affect the particle's transition across the reservoir. The scaling is proved analytically by deriving a simple formula, which displays two regimes, with the Zeno effect and without the Zeno effect. It also supports a simple explanation of the Zeno effect entirely in terms of the energy-time uncertainty relation, with no explicit use of the projection postulate. Experimental tests of our predictions are discussed.