Bloch-Messiah reduction for twin beams of light

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. arXiv:1903.06578v2 [quant-ph]

Multimode single-pass spatio-temporal squeezing

“…The squeezing spectra are recorded with a spectrum analyser at a sideband frequency of 10 MHz with a 100 kHz resolution bandwidth and 30 Hz video bandwidth, meaning that the measured squeezing is averaged over 15 pulses. The amount of squeezing is −0.35(3) dB in the HG 0 case, −0.25(4) dB for HG 1 , and it reduces to lower values for the higher order modes as expected from theory [28], so that we find −0.19(5) dB for HG 2 and −0.19(6) dB HG 3 . We can estimate the original amount of squeezing for HG 0 by considering the ratio between the measured squeezing and antisqueezing values and assuming pure squeezed light at the output of the source.…”

Generator of spatial evolution of the electromagnetic field

“…As it was shown in Sec. III F 2, the relation G = −P z remains valid in fourdimensional models including propagation of a free field in both directions [10,11,25,26,43]. However, for a field with sources, the form of the generator of spatial evolution is given by the quantum version of Eq.…”

“…Establishing this generator is of paramount importance though, first, because it is a fundamental problem of quantum electrodynamics important for the correct field quantization, and, second, because this generator is required for practical calculations in the Schrödinger picture. For example, finding the modes of squeezing of an optical parametric amplifier requires a diagonalization of the squeezing matrix, given by a spatial integral of this generator [10,11]. In the last years, the spatial evolution generator has acquired special importance for the description of arrays of coupled nonlinear waveguides, playing the role of the "discrete lattice Hamiltonian" [12] and helping to find the lattice supermodes [13,14], which may be topologically protected under certain conditions [15].…”

Distinguishability in quantum interference with the squeezed states

“…In general, the squeezed states have multi-mode structure, which is conveniently accounted for by the singularvalue decomposition representation of the multi-mode squeezing Hamiltonian [48] (see also Refs. [49][50][51]). There can be infinite number of singular values and the corresponding orthogonal (a.k.a.…”

“…However, the degenerate and non-degenerate squeezed states in Eqs. ( 4) and ( 6) are not entirely equivalent for quantum interference experiments, when the Schmidt modes involve the degrees of freedom not affected by a multiport (e.g., the spectral shape of photons) [39,[48][49][50][51]. In Eqs.…”