# Bloch-Messiah reduction for twin beams of light. (arXiv:1903.06578v1 [quant-ph])

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. 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.