We investigate open quantum dynamics for a one-dimensional incommensurate
Aubry-Andr\'{e}-Harper lattice chain, a part of which is initially filled with
electrons and is further connected to dephasing probes at the filled lattice
sites. This setup is akin to a step-initial configuration where the non-zero
part of the step is subjected to dephasing. We investigate the quantum dynamics
of local electron density, the scaling of the density front as a function of
time both inside and outside of the initial step, and the growth of the total
number of electrons outside the step. We analyze these quantities in all three
regimes, namely, the de-localized, critical, and localized phases of the
underlying lattice. Outside the initial step, we observe that the density front
spreads according to the underlying nature of single-particle states of the
lattice, for both the de-localized and critical phases. For the localized
phase, the spread of the density front hints at a logarithmic behaviour in time
that has no parallel in the isolated case (\emph{i.e.}, in the absence of
probes). Inside the initial step, due to the presence of the probes, the
density front spreads in a diffusive manner for all the phases. This
combination of rich and different dynamical behaviour, outside and inside the
initial step, results in the emergence of mixed dynamical phases. While the
total occupation of electrons remains conserved, the value outside or inside
the initial step turns out to have a rich dynamical behaviour. Our work is
widely adaptable and has interesting consequences when
disordered/quasi-disordered systems are subjected to a thermodynamically large
number of probes.Comment: 18 pages, 8 figure