Continuous fuzzy measurement on two-level systems revisited

TL;DR

This paper critically revisits the concept of continuous quantum measurement via path restrictions, demonstrating that Gaussian restrictions do not reliably reflect the system's eigenstates, and explores alternative models and mechanisms underlying decoherence and the Zeno effect.

Contribution

The paper shows that Gaussian restrictions are insufficient for representing measurement readouts and introduces a tractable 'hard wall' model to better understand continuous quantum measurement effects.

Findings

Gaussian restrictions lead to large readout variations, not reflecting eigenstates.

Decoherence can be modeled as a Gaussian random walk with drift towards eigenstates.

Hard wall restriction model yields similar results to Gaussian case.

Abstract

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.