  Using a perturbative approach, we investigate the parametric down-conversion
process without the semi-classical approximation. A Wigner functional
formalism, which incorporates both the spatiotemproal degrees of freedom and
the particle-number degrees of freedom is used to perform the analysis. First,
we derive an evolution equation for the down-conversion process in the
nonlinear medium. Then we use the perturbative approach to solve the equation.
The leading order contribution is equivalent to the semi-classical solution.
The next-to-leading order contribution provides a solution that includes the
evolution of the pump field as a quantum field.
