  How long does it take a quantum particle to return to its origin? As shown
previously under repeated projective measurements aimed to detect the return,
the closed cycle yields a geometrical phase which shows that the average first
detected return time is quantized. For critical sampling times or when
parameters of the Hamiltonian are tuned this winding number is modified. These
discontinuous transitions exhibit gigantic fluctuations of the return time.
While the general formalism of this problem was studied at length, the
magnitude of the fluctuations, which is quantitatively essential, remains
poorly characterized. Here, we derive explicit expressions for the variance of
the return time, for quantum walks in finite Hilbert space. A classification
scheme of the diverging variance is presented, for four different physical
effects: the Zeno regime, when the overlap of an energy eigenstate and the
detected state is small and when two or three phases of the problem merge.
These scenarios present distinct physical effects which can be analyzed with
the fluctuations of return times investigated here, leading to a
topology-dependent time-energy uncertainty principle.
