We analyze the long-time quantum dynamics of degenerate parametric
down-conversion from an initial sub-harmonic vacuum (spontaenous
down-conversion). Standard linearization of the Heisenberg equations of motions
fails in this case, since it is based on an expansion around an unstable
classical solution and neglects pump depletion. Introducing a mean-field
approximation we find a periodic exchange of energy between the pump and
subharmonic mode goverened by an anharmonic pendulum equation. From this
equation the optimum interaction time or crystal length for maximum conversion
can be determined. A numerical integration of the 2-mode Schr"odinger equation
using a dynamically optimized basis of displaced and squeezed number states
verifies the characteristic times predicted by the mean-field approximation. In
contrast to semiclassical and mean-field predictions it is found that quantum
fluctuations of the pump mode lead to a substantial limitation of the
efficiency of parametric down-conversion.Comment: 5 pages, 4 figure
