  In this study, we explore optimal environments for efficient EET between end
sites in a number of dimer and trimer model pigment-protein molecular
aggregates when the EET dynamics is modeled by the HEOM-method. For these
optimal environmental parameters, we further quantify the non-Markovianity by
the BLP-measure to elucidate its possible connection to efficient EET. We also
quantify coherence in the pigment systems by means of the measure l1-norm of
coherence to analyze its interplay with environmental effects when EET
efficiency is maximal. Our aim is to investigate possible environmental design
principles for achieving efficient EET in model pigment-protein molecular
aggregates and to determine whether non-Markovianity is a possible underlying
resource in such systems. We find that the structure of the system Hamiltonian
(i.e., the pigment Hamiltonian parameter space) and especially, the
relationship between the site excitation energies, determines whether one of
two specific environmental regimes is the most beneficial in promoting
efficient EET in these model systems. In the first regime, optimal
environmental conditions are such that the EET dynamics in the system is left
as coherent as possible. In the second regime, the most advantageous role of
the environment is to drive the system towards equilibrium as fast as possible.
In reality, optimal environmental conditions may involve a combination of these
two effects. We cannot establish a relation between efficient EET and
non-Markovianity, i.e., non-Markovianity cannot be regarded as a resource in
the systems investigated in this study.
