# Bloch-Messiah reduction for twin beams of light

## TL;DR

This paper analyzes the Bloch-Messiah reduction of pulsed parametric downconversion producing twin beams, revealing eigenvalue multiplicities, addressing mode ambiguity, and proposing methods for unique mode determination with practical examples.

## Contribution

It introduces two approaches for uniquely determining squeezing eigenmodes in nondegenerate twin beams, addressing eigenvalue multiplicity and mode ambiguity issues.

## Key findings

- Eigenvalues have at least double multiplicity in this regime
- Modal functions can be derived from Schmidt modes or Hermitian eigenvalue problems
- Eigenmode structure varies near phase-matching degeneracy

## Abstract

We study the Bloch-Messiah reduction of parametric downconversion of light in the pulsed regime with a nondegenerate phase matching providing generation of twin beams. We find that in this case every squeezing eigenvalue has multiplicity at least two. We discuss the problem of ambiguity in the definition of the squeezing eigenmodes in this case and develop two approaches to unique determination of the latter. First, we show that the modal functions of the squeezing eigenmodes can be tailored from the Schmidt modes of the signal and idler beams. Alternatively, they can be found as a solution of an eigenvalue problem for an associated Hermitian squeezing matrix. We illustrate the developed theory by an example of frequency non-degenerate collinear twin beams generated in beta barium borate crystal. On this example we demonstrate how the squeezing eigenmodes can be approximated analytically on the basis of the Mehler's formula, extended to complex kernels. We show how the multiplicity of the eigenvalues and the structure of the eigenmodes are changed when the phase matching approaches the degeneracy in frequency.