Long-time dynamics of spontaneous parametric down-conversion and quantum limitations of conversion efficiency

We analyze the long-time quantum dynamics of degenerate parametric down-con version from an initial sub-harmonic vacuum (spontaenous down-conversion). Standard linearization of the Heisenberg equations of motion fails in this case, since it is based on an expansion around an unstable classical soluti on and neglects pump depletion. Introducing a mean-field approximation we f ind a periodic exchange of energy between the pump and subharmonic mode gov erened by an anharmonic pendulum equation. From this equation the optimum i nteraction time or crystal length for maximum conversion can be determined. A numerical integration of the 2-mode Schrodinger equation using a dynamic ally optimized basis of displaced and squeezed number states verifies the c haracteristic times predicted by the mean-field approximation. In contrast to semiclassical and mean-field predictions it is found that quantum fluctu ations of the pump mode lead to a substantial limitation of the efficiency of parametric down-conversion.