  We examine the coherent propagation of the one-dimensional Frenkel exciton
(correlated electron-hole pair system) based on a model of a quantum walker in
multi-dimensional Hilbert space. The walk is governed by a non-Hermitian coin
flip operation coupled to a generalized shift process. The dissipative coin
flip operation is associated with amplitude leakages at occupied sites, typical
of processes which occur when an exciton is transferred along dimer sites in
photosynthetic protein complexes. We analyze the characteristics probability
distribution of the one-dimensional quantum walk for various system parameters,
and examine the complex interplay between non-Markovian signatures and
amplitude leakages within the Hilbert position subspace. The visibility of
topological defects such as exceptional points, and non-Markovian signatures
via quantum tomography based spectroscopic measurements are discussed.
