Quantum Arrival Time For Open Systems

TL;DR

This paper extends the quantum arrival time problem to open systems using decoherent histories, showing how environment-induced decoherence leads to classical-like arrival time probabilities consistent with the probability current.

Contribution

It introduces a POVM-based approach for arrival time in open quantum systems and analyzes decoherence effects on arrival time probabilities.

Findings

Arrival time probabilities become positive after the localisation time.

Decoherence ensures the consistency of arrival time probabilities with the probability current.

Fundamental limits on timing accuracy are established, obeying $E\,\delta t >> \hbar$.

Abstract

We extend previous work on the arrival time problem in quantum mechanics, in the framework of decoherent histories, to the case of a particle coupled to an environment. The usual arrival time probabilities are related to the probability current, so we explore the properties of the current for general open systems that can be written in terms of a master equation of Lindblad form. We specialise to the case of quantum Brownian motion, and show that after a time of order the localisation time the current becomes positive. We show that the arrival time probabilities can then be written in terms of a POVM, which we compute. We perform a decoherent histories analysis including the effects of the environment and show that time of arrival probabilities are decoherent for a generic state after a time much greater than the localisation time, but that there is a fundamental limitation on the accuracy, $\delta t$, with which they can be specified which obeys $E\delta t>>\hbar$. We confirm that the arrival time probabilities computed in this way agree with those computed via the current, provided there is decoherence. We thus find that the decoherent histories formulation of quantum mechanics provides a consistent explanation for the emergence of the probability current as the classical arrival time distribution, and a systematic rule for deciding when probabilities may be assigned.