Decoherence of a weak value influenced by a non-Markovian environment

Abstract

A weak value of an observable which is influenced by a surrounding environment is studied by means of an exactly solvable model, where a qubit is weakly measured and an environment consists of independent harmonic oscillators. When the environment has a finite correlation time, the correlation between environmental states before and after the weak measurement cannot be ignored. The dependence of the weak value on weak measurement and post-selection times is investigated in detail. The condition is obtained under which the anomaly of the weak value can be observed under the influence of the environment. It is found that the anomaly of the weak value can survive longer in a non-Markovian environment than in a Markovian environment. Furthermore, the effect of imperfection of pre- and post-selection on the weak value is investigated. It is shown that the weak value anomaly can be observed even if the pre-selected state of the qubit is maximally mixed.