  We propose to use the effect of measurements instead of their number to study
the time evolution of quantum systems under monitoring. This time redefinition
acts like a microscope which blows up the inner details of seemingly
instantaneous transitions like quantum jumps. In the simple example of a
continuously monitored qubit coupled to a heat bath, we show that this
procedure provides well defined and simple evolution equations in an otherwise
singular strong monitoring limit. We show that there exists anomalous
observable localised on sharp transitions which can only be resolved with our
new effective time. We apply our simplified description to study the
competition between information extraction and dissipation in the evolution of
the linear entropy. Finally, we show that the evolution of the new time as a
function of the real time is closely related to a stable Levy process of index
1/2.
