Parametric down-conversion beyond the semi-classical approximation

TL;DR

This paper develops a quantum formalism for parametric down-conversion that extends beyond the semi-classical approximation, incorporating quantum effects of the pump field and spatiotemporal degrees of freedom.

Contribution

It introduces a Wigner functional formalism and derives an evolution equation for the process, including quantum pump field effects beyond semi-classical models.

Findings

Leading order matches semi-classical results

Next-to-leading order includes quantum pump evolution

Provides a more complete quantum description of down-conversion

Abstract

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.