Quantum evolution in the stroboscopic limit of repeated measurements

TL;DR

This paper derives analytical equations describing quantum system dynamics under rapid repeated measurements, revealing non-linear and decoherent effects that emerge over longer timescales in the stroboscopic limit.

Contribution

It introduces a new analytical framework for quantum evolution under repeated measurements in the stroboscopic limit, capturing non-linear and decoherent dynamics.

Findings

Derived evolution equations in the stroboscopic limit.

Identified timescale where non-linear and decoherent effects become significant.

Demonstrated effects through multiple examples.

Abstract

We consider a quantum system dynamics caused by successive selective and non-selective measurements of the probe coupled to the system. For the finite measurement rate $\tau^{-1}$ and the system-probe interaction strength $\gamma$ we derive analytical evolution equations in the stroboscopic limit $\tau \rightarrow 0$ and $\gamma^2 \tau = {\rm const}$, which can be considered as a deviation from the Zeno subspace dynamics on a longer timescale $T \sim (\gamma^2 \tau)^{-1} \gg \gamma^{-1}$. Non-linear quantum dynamics is analyzed for selective stroboscopic projective measurements of an arbitrary rank. Non-selective measurements are shown to induce the semigroup dynamics of the system-probe aggregate. Both non-linear and decoherent effects become significant at the timescale $T \sim (\gamma^2 \tau)^{-1}$, which is illustrated by a number of examples.