  In classical probability theory, the term "cutoff" describes the property of
some Markov chains to jump from (close to) their initial configuration to
(close to) completely mixed in a very narrow window of time. We investigate how
coherent quantum evolution affects the mixing properties in two fermionic
quantum models (the "gain/loss" and "topological" models), whose time evolution
is governed by a Lindblad equation quadratic in fermionic operators, allowing
for a straightforward exact solution. We check that the phenomenon of cutoff
extends to the quantum case and examine with some care how the mixing
properties depend on the initial state, drawing different regimes of our models
with qualitatively different behaviour. In the topological case, we further
show how the mixing properties are affected by the presence of a long-lived
edge zero mode when taking open boundary conditions.
