  We quantitatively study the out-of-equilibrium edge-Majorana correlation in a
linearly ramped one-dimensional Kitaev chain of finite length in a dissipative
environment. The chemical potential is dynamically ramped to drive the chain
from its topologically trivial to nontrivial phase in the presence of couplings
to nonthermal Markovian baths. We consider two distinctive situations: In the
first situation, the bath is quasilocal in the site basis (local in
quasiparticle basis) while in the other it is local. Following a Lindbladian
approach, we compute the early time dynamics as well as the asymptotic behavior
of the edge-Majorana correlation to probe the interplay between two competing
timescales - one due to the coherent ramping while the other to the dissipative
coupling. For the quasilocal bath, we establish that there is a steady
generation of Majorana correlations in asymptotic time and the presence of an
optimal ramping time which facilitates a quicker approach to the topological
steady state. In the second scenario, we analyze the action of a local
particle-loss type of bath in which we have established the existence of an
optimal ramping time which results from the competing dynamics between the
unitary ramp and the dissipative coupling. While the defect generated by the
former decays exponentially with increasing ramp duration, the later scales
linearly with the same. This linear scaling is further established through a
perturbation theory formulated using the nondimensionalized coupling to the
bath as a small parameter.
