  We present an effective linear response approach to pump-probe femtosecond
coherence spectroscopy in the well separated pulse limit. The treatment
presented here is based on a displaced and squeezed state representation for
the non-stationary states induced by an ultrashort pump laser pulse or a
chemical reaction. The subsequent response of the system to a delayed probe
pulse is modeled using closed form non-stationary linear response functions,
valid for a multimode vibronically coupled system at arbitrary temperature.
When pump-probe signals are simulated using the linear response functions, with
the mean nuclear positions and momenta obtained from a rigorous moment analysis
of the pump induced (doorway) state, the signals are found to be in excellent
agreement with the conventional third order response approach. The key
advantages offered by the moment analysis based linear response approach
include a clear physical interpretation of the amplitude and phase of
oscillatory pump-probe signals, a dramatic improvement in computation times, a
direct connection between pump-probe signals and equilibrium absorption and
dispersion lineshapes, and the ability to incorporate coherence such as those
created by rapid non-radiative surface crossing. We demonstrate these aspects
using numerical simulations, and also apply the present approach to the
interpretation of experimental amplitude and phase measurements on reactive and
non-reactive samples of the heme protein Myoglobin. The role played by
inhomogeneous broadening in the observed amplitude and phase profiles is
discussed in detail. We also investigate overtone signals in the context of
reaction driven coherent motion.
