1003.6079 (J. M. Yearsley)

Quantum Arrival Time For Open Systems    [PDF]

J. M. Yearsley

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.

View original: http://arxiv.org/abs/1003.6079