  The investigation of the phenomenon of dephasing assisted quantum transport,
which happens when the presence of dephasing benefits the efficiency of this
process, has been mainly focused on Markovian scenarios associated with
constant and positive dephasing rates in their respective Lindblad master
equations. What happens if we consider a more general framework, where
time-dependent dephasing rates are allowed, thereby permitting the possibility
of non-Markovian scenarios? Does dephasing assisted transport still manifest
for non-Markovian dephasing? Here, we address these open questions in a setup
of coupled two-level systems. Our results show that the manifestation of
non-Markovian dephasing assisted transport depends on the way in which the
incoherent energy sources are locally coupled to the chain. This is illustrated
with two different configurations, namely non-symmetric and symmetric.
Specifically, we verify that non-Markovian dephasing assisted transport
manifested only in the non-symmetric configuration. This allows us to draw a
parallel with the conditions in which time-independent Markovian dephasing
assisted transport manifests. Finally, we find similar results by considering a
controllable and experimentally implementable system, which highlights the
significance of our findings for quantum technologies.
