  The simulation of spectroscopic observables for molecular aggregates with
strong and structured coupling of electronic excitation to vibrational degrees
of freedom is an important but challenging task. The hierarchy of pure states
(HOPS) provides a formally exact solution based on local, stochastic
trajectories. Exploiting the localization of HOPS for the simulation of
absorption spectra in large aggregares requires a formulation in terms of
normalized trajectories. Here we provide a normalized dyadic equation where the
ket- and bra-states are propagated in different electronic Hilbert spaces. This
work opens the door to apply adaptive HOPS methods for the simulation of
absorption spectra and also to a formulation for non-linear spectroscopy that
is perturbative with respect to interactions with the electric field.
