On the relevance of weak measurements in dissipative quantum systems

We investigate the impact of dissipation, including energy relaxation and decoherence, on weak measurements. While weak measurements have been successful in signal amplification, dissipation can compromise their usefulness. More precisely, we show that in systems with a unique steady state, weak values always converge to an expectation value of the measured observable as dissipation time tends to infinity, in contrast to systems with multiple steady states, where the weak values can remain anomalous, i.e. outside the range of eigenvalues of the observable, even in the limit of an infinite dissipation time. In addition, we propose a method for extracting information about the dissipative dynamics of a system using weak values at short dissipation times. Specifically, we explore the amplification of the dissipation rate in a two-level system and the use of weak values to differentiate between Markovian and non-Markovian dissipative dynamics. We also find that weak measurements operating around a weak atom-cavity coupling can probe the atom dissipation through the weak value of non-Hermitian operators within the rotating-wave approximation of the weak interaction.