  In this paper, we investigate the non-Markovian quantum transport dynamics of
a two-terminal Majorana device that is made of an asymmetric topological
superconducting chain coupled to two leads. This asymmetric superconducting
chain is analytically solvable and can be realized by a hybrid system of
semiconductor nanowire coupled to superconductors or by 1D transverse-field
Ising chains. In such asymmetric superconducting chains, by the change of
chemical potential, its ground state undergoes a topological quantum phase
transition from the topological Majorana bound state to the trivial Andreev
bound state while the ground state energy remains zero. We solve the exact
transient transport current and the corresponding differential conductance. The
results show that the presence or absence of the interference between the left
and right Majorana zero modes plays an important role on the topological phase
transition of conductance. It cause the edge-localized topologically trivial
states to be insulated with zero conductance, while the nonlocally distributed
topologically nontrivial states always have a quantized conductance 2e^2/h.
This dramatic change associated with topological phase transition for zero-mode
differential conductance at zero bias is independent of the structure of leads
and the coupling strength. We also examine the finite size effect of the
superconducting chain and the coherence effect between zero mode and non-zero
energy modes on the differential conductance in this two-terminal Majorana
device.
