  Randomly repeated measurements during the evolution of a closed quantum
system create a sequence of probabilities for the first detection of a certain
quantum state. The related discrete monitored evolution for the return of the
quantum system to its initial state is investigated. We found that the mean
number of measurements until the first detection is an integer, namely the
dimensionality of the accessible Hilbert space. Moreover, the mean first
detected return time is equal to the average time step between successive
measurements times the mean number of measurements. Thus, the mean first
detected return time scales linearly with the dimensionality of the accessible
Hilbert space. The main goal of this work is to explain the quantization of the
mean return time in terms of a quantized Berry phase.
