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.

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