Accurate relative-phase and time-delay maps all over the emission cone of hyperentangled photon source

TL;DR

This paper introduces a numerical method to accurately map the relative phase and time delays of hyperentangled photon pairs emitted over the entire SPDC emission cone, considering complex spatial and spectral entanglement.

Contribution

It presents a vectorial numerical approach to determine phase and delay maps across the emission cone, including self-compensation effects, in a low-gain SPDC regime.

Findings

Relative-phase map approximates a quadratic function in polar angle.

Maps are valid over the entire emission cone, including self-compensation scenarios.

Two-photon emission directions are coplanar with the pump beam, spreading around phase-matching directions.

Abstract

High flux of hyperentangled photons entails collecting the two-photon emission over relatively wide extent in frequency and transverse space within which the photon pairs are simultaneously entangled in multiple degrees of freedom. In this paper, we present a numerical approach to determining the spatial-spectral relative-phase and time-delay maps of hyperentangled photons all over the spontaneous parametric down conversion (SPDC) emission cone. We consider the hyperentangled-photons produced by superimposing noncollinear SPDC emissions of two crossed and coherently-pumped nonlinear crystals. We adopt a vectorial representation for all parameters of concern. This enables us to study special settings such as the self-compensation via oblique pump incidence. While rigorous quantum treatment of SPDC emission requires Gaussian state representation, in low-gain regime (like the case of the study), it is well approximated to the first order to superposition of vacuum and two-photon states. The relative phase and time-delay maps are then calculated between the two-photon wavepackets created along symmetrical locations of the crystals. Assuming monochromatic plane-wave pump field, the mutual signal-idler relations like energy conservation and transverse-momentum conservation define well one of the two-photon with reference to its conjugate. The weaker conservation of longitudinal momentum (due to relatively thin crystals) allows two-photon emission directions coplanar with the pump beam while spreading around the perfect phase-matching direction. While prior works often adopt first-order approximation, it is shown that the relative-phase map is a very well approximated to a quadratic function in the polar angle of the two-photon emission while negligibly varying with the azimuthal angle.