  We assess the suitability of quantum and semiclassical initial value
representations, exemplified by the coupled coherent states (CCS) method and
the Herman Kluk (HK) propagator, respectively, for modeling the dynamics of an
electronic wave packet in a strong laser field, if this wave packet is
initially bound. Using Wigner quasiprobability distributions and ensembles of
classical trajectories, we identify signatures of over-the-barrier and tunnel
ionization in phase space for static and time-dependent fields and the relevant
sets of phase-space trajectories in order to model such features. Overall, we
find good agreement with the full solution of the time-dependent Schr\"odinger
equation (TDSE) for Wigner distributions constructed with both initial-value
representations. Our results indicate that the HK propagator does not fully
account for tunneling and over-the-barrier reflections. However, it is able to
partly reproduce features associated with the wave packet crossing classically
forbidden regions, although the trajectories employed in its construction
always obey classical phase-space constraints. We also show that the Coupled
Coherent States (CCS) method represents a fully quantum initial value
representation and accurately reproduces the results of a standard TDSE solver.
Furthermore, we sow that both the HK propagator and the CCS approach may be
successfully employed to compute the time-dependent dipole acceleration and
high-harmonic spectra. Nevertheless, the semiclassical propagator exhibits a
worse agreement with the TDSE than the outcome of the CCS method, as it neither
fully accounts for tunneling nor for over-the-barrier reflections. This leads
to a dephasing in the time-dependent wave function which becomes more
pronounced for longer times.
