  We address the effects of quantum coherences on the optical line shape of an
exciton in the presence of dynamic disorder. We consider a one-dimensional
excitonic system that consists of two levels placed at regular intervals.
Detailed analytical calculations of line shape have been carried out by using
Kubo's stochastic Liouville equation (K-QSLE). We make use of the observation
that in the site representation, the Hamiltonian of our system with constant
off-diagonal coupling J is a tridiagonal Toeplitz matrix (TDTM) whose
eigenvalues and eigen functions are known analytically. This identification is
particularly useful for long chains where the eigen values of TDTM help to
understand crossover between static and fast modulation limits. We summarize
the new results as follows. (i) In the slow modulation limit when the bath
correlation time is large, the effects of spatial correlation are not
negligible. Here the line shape is broadened and the number of peak increases
beyond the ones obtained from TDTM (constant off-diagonal coupling element J
and no fluctuation). (ii) However, in the fast modulation limit when the bath
correlation time is small, the spatial correlation is less important (iii)
Importantly, we find that the line shape can capture that quantum coherence
affects in the two limits differently.
