Spatial correlations in four-wave mixing with structured light

TL;DR

This paper develops a comprehensive theoretical framework for four-wave mixing with structured light, analyzing spatial correlations and entanglement in biphoton states, and drawing parallels with parametric down-conversion.

Contribution

It introduces a detailed quantized paraxial model for FWM, linking position and momentum correlations, and compares FWM with PDC to highlight shared features and spatial entanglement properties.

Findings

FWM biphoton states exhibit position and momentum correlations similar to PDC.

The transfer of pump structure influences the spatial coincidence profile.

The framework captures the transition from near-field to far-field correlations.

Abstract

We present a detailed theoretical treatment of four-wave mixing (FWM) in a quantized paraxial framework, capturing the multi-spatial-mode nature of the biphoton state generated in the process. By analyzing the biphoton state both in position and momentum representations, we identify the conditions under which these descriptions become equivalent. We also highlight formal and physical similarities between FWM and spontaneous parametric down-conversion (PDC), showing that the transfer of pump structure to the spatial coincidence profile, an important and well-known characteristic of the biphoton state, carries over naturally to FWM. In addition, our treatment captures the transition from position correlations in the near field to momentum correlations in the far field, reflecting the underlying spatial entanglement. The measures of entanglement, including the spiral bandwidth and the Schmidt rank, are discussed. Our work consolidates known and new results on spatial correlations in FWM and provides a theoretical framework that may support future studies in nonlinear and quantum optics with structured light.