Continuous fuzzy measurement on two-level systems revisited

Imposing restrictions on the Feynman paths of the monitored system has in the past been proposed as a universal model-free approach to continuous quantum measurements. Here we revisit this proposition, and demonstrate that a Gaussian restriction, resulting in a sequence of many highly inaccurate (weak) von Neumann measurements, is not sufficiently strong to ensure proximity between a readout and the Feynman paths along which the monitored system evolves. Rather, in the continuous limit, the variations of a typical readout become much larger than the separation between the eigenvalues of the measured quantity. Thus, a typical readout is not represented by a nearly constant curve, correlating with one of the eigenvalues of the measured quantity $\a$, even when decoherence, or Zeno effect are achieved for the observed two-level system, and does not point directly to the system's final state. We show that the decoherence in a "free" system can be seen as induced by a Gaussian random walk with a drift, eventually directing the system towards one of the eigenstates of $\a$. A similar mechanism appears to be responsible for the Zeno effect in a driven system, when its Rabi oscillations are quenched by monitoring. Alongside the Gaussian case, which can only be studied numerically, we also consider a fully tractable model with a "hard wall" restriction, and show the results to be similar.