Introduction to the Transverse Spatial Correlations in Spontaneous Parametric Down-Conversion through the Biphoton Birth Zone

TL;DR

This paper provides a detailed theoretical tutorial on the transverse spatial correlations in spontaneous parametric down-conversion, focusing on the biphoton birth zone and its implications for quantum optics experiments.

Contribution

It introduces a first-principles derivation of the biphoton correlation width and models it with a Double-Gaussian wavefunction, enhancing understanding of spatial correlations in SPDC.

Findings

Derived the transverse correlation width from first principles.

Validated the Double-Gaussian approximation against experimental data.

Discussed the tradeoff between spatial coherence and biphoton correlations.

Abstract

As a tutorial to the spatial aspects of Spontaneous Parametric Downconversion (SPDC), we present a detailed first-principles derivation of the transverse correlation width of photon pairs in degenerate collinear SPDC. This width defines the size of a biphoton birth zone, the region where the signal and idler photons are likely to be found when conditioning on the position of the destroyed pump photon. Along the way, we discuss the quantum-optical calculation of the amplitude for the SPDC process, as well as its simplified form for nearly collinear degenerate phase matching. Following this, we show how this biphoton amplitude can be approximated with a Double-Gaussian wavefunction, and give a brief discussion of the measurement statistics (and subsequent convenience) of such Double-Gaussian wavefunctions. Next, we use this approximation to get a simplified estimation of the transverse correlation width, and compare it to more accurate calculations as well as experimental results. We then conclude with a discussion of the concept of a biphoton birth zone, using it to develop intuition for the tradeoff between the first-order spatial coherence and bipohoton correlations in SPDC.