Phonon Effects on the Weak Measurement of Charge States in Quantum Dots with a Quantum Point Contact

We study theoretically the process of a weak measurement of charge states in a double quantum dot performed by coupling to a quantum point contact. We focus on the interplay of the measurement process with phonon-induced dephasing of delocalized superpositions of electron states and compare the time scales relevant for the dynamics of the open system and for the measurement process: the decoherence rate, the localization rate of the state conditioned on the measurement outcome and the time scale on which information on the system state is gained in the process of the measurement.