  We investigate quantum persistence by analyzing amplitude and phase
fluctuations of the wave function governed by the time-dependent free-particle
Schr\"odinger equation. The quantum system is initialized with local random
uncorrelated Gaussian amplitude and phase fluctuations. In analogy with
classical diffusion, the persistence probability is defined as the probability
that the local (amplitude or phase) fluctuations have not changed sign up to
time $t$. Our results show that the persistence probability in quantum
diffusion exhibits exponential-like tails. More specifically, in $d=1$ the
persistence probability decays in a stretched exponential fashion, while in
$d=2$ and $d=3$ as an exponential. We also provide some insights by analyzing
the two-point spatial and temporal correlation functions in the limit of small
fluctuations. In particular, in the long-time limit, the temporal correlation
functions for both local amplitude and phase fluctuations become
time-homogeneous, i.e., the zero-crossing events correspond to those of a
stationary Gaussian process, with sufficiently fast-decaying power-law tail of
its autocorrelation function, implying an exponential-like tail of the
persistence probabilities.
