  We study the dynamics of the pump mode in the down-conversion Hamiltonian
using the cumulant expansion method, perturbation theory, and the full
numerical simulation of systems with a pump mean photon number of up to one
hundred thousand. We particularly focus on the properties of the pump-mode such
as depletion, entanglement, and squeezing for an experimentally relevant
initial state in which the pump mode is initialized in a coherent state.
Through this analysis, we obtain the short-time behaviour of various quantities
and derive timescales at which the above-mentioned features, which cannot be
understood through the parametric approximation, originate in the system. We
also provide an entanglement witness involving moments of bosonic operators
that can capture the entanglement of the pump mode. Finally, we study the
photon-number statistics of the pump and the signal/idler modes to understand
the general behaviour of these modes for experimentally relevant time scales.
