Analytic expression for the charge carried by a locally excited Bloch state

We consider the time evolution of Bloch electrons after a local excitation, like an interaction with a focused laser pulse that irradiates only a part of the sample. The disturbance caused by the excitation propagates along the sample towards detectors. We focus on the measurable time integral of the usually rapidly oscillating current. In the long time limit this integral is the total charge that is displaced by the excitation. We develop an analytic way for calculating this charge. The results are verified using an analytic example with quadratic dispersion in one dimension. Additionally, numerical calculations are also performed in order to visualize the relevant physical processes in a wide band gap material that is excited by a laser pulse.