  The question of how quantum coherence facilitates energy transfer has been
intensively debated in the scientific community. Since natural and artificial
light-harvesting units operate under the stationary condition, we address this
question via a non-equilibrium steady-state analysis of a molecular dimer
irradiated by incoherent sunlight and then generalize the key predictions to
arbitrarily-complex exciton networks. The central result of the steady-state
analysis is the coherence-flux-efficiency relation:$\eta=c\sum_{i\neq
j}F_{ij}\kappa_j=2c\sum_{i\neq j}J_{ij}{\rm Im}[{\rho}_{ij}]\kappa_j$ with $c$
the normalization constant. In this relation, the first equality indicates that
energy transfer efficiency $\eta$ is uniquely determined by the trapping flux,
which is the product of flux $F$ and branching ratio $\kappa$ for trapping at
the reaction centers, and the second equality indicates that the energy
transfer flux $F$ is equivalent to quantum coherence measured by the imaginary
part of the off-diagonal density matrix, i.e., $F_{ij}=2J_{ij}{\rm
Im}[{\rho}_{ij}]$. Consequently, maximal steady-state coherence gives rise to
optimal efficiency. The coherence-flux-efficiency relation holds rigorously and
generally for any exciton networks of arbitrary connectivity under the
stationary condition and is not limited to incoherent radiation or incoherent
pumping. For light-harvesting systems under incoherent light, non-equilibrium
energy transfer flux (i.e. steady-state coherence) is driven by the breakdown
of detailed balance and by the quantum interference of light-excitations and
leads to the optimization of energy transfer efficiency. It should be noted
that the steady-state coherence or, equivalently, efficiency is the combined
result of light-induced transient coherence, inhomogeneous depletion, and
system-bath correlation, and is thus not necessarily correlated with quantum
beatings.
