  Is it possible to infer the time evolving quantum state of a
multichromophoric system from a sequence of two-dimensional electronic spectra
(2D-ES) as a function of waiting time? Here we provide a positive answer for a
tractable model system: a coupled dimer. After exhaustively enumerating the
Liouville pathways associated to each peak in the 2D-ES, we argue that by
judiciously combining the information from a series of experiments varying the
polarization and frequency components of the pulses, detailed information at
the amplitude level about the input and output quantum states at the waiting
time can be obtained. This possibility yields a quantum process tomography
(QPT) of the single-exciton manifold, which completely characterizes the open
quantum system dynamics through the reconstruction of the process matrix. This
is the first of a series of two articles. In this manuscript, we specialize our
results to the case of a homodimer, where we prove that signals stemming from
coherence to population transfer and viceversa vanish upon isotropic averaging,
and therefore, only a partial QPT is possible in this case. However, this fact
simplifies the spectra, and it follows that only two polarization controlled
experiments (and no pulse-shaping requirements) suffice to yield the elements
of the process matrix which survive under isotropic averaging. The angle
between the two site transition dipole moments is self-consistently obtained
from the 2D-ES. Model calculations are presented, as well as an error analysis
in terms of the angle between the dipoles and peak overlap. In the second
article accompanying this study, we numerically exemplify the theory for
heterodimers and carry out a detailed error analysis for such case. This
investigation provides an important benchmark for more complex proposals of
quantum process tomography (QPT) via multidimensional spectroscopic
experiments.
