How non-degenerate photon pair can be entangled?

Question

papers like below describe https://arxiv.org/abs/1305.0986 "A Flexible Source of Non-Degenerate Entangled Photons Based on a Two-Crystal Sagnac Interferometer" I, somehow, thought that entanglement requires photons (or generally particles) to be identical in any other degree of freedom aside one where they are entangled. Was I wrong? if particles don't have to identical why to go through all troubles like Sagnac interferometers and spatial filtering and not to use SPDC processes directly as is?

Answer 1

You absolutely don't need the particles to be identical in all other respects — in fact you can entangle extremely different systems, like a photon and an atom (for instance the polarization of a photon can be entangled with the hyperfine state of an atom's valence electron).

SPDC processes can be used "as is" in the sense that you can simply pump a nonlinear crystal and get out pairs of photons, but those photons have constrained momenta and other properties that may not be desirable for certain applications. In particular the paper you link contains this in the introduction:

Sources of entangled photon pairs based on SPDC in non-linear crystals [5] are now widely used, and several high performance entanglement sources have been based on a nonlinear crystal in a Sagnac interferometer thanks to this type of interferometer’s intrinsic phase stability [6]. However, due to problems arising from chromatic dispersion in polarization optics, such sources are challenging to build if the members of the entangled pairs are generated at widely different wavelengths. One way to overcome this problem is to use periscopes instead [7]. Here we resort to another approach, which is based on a Sagnac interferometer that includes two SPDC crystals.

I suggest you follow the citations here and, if you run into problems understanding, please ask a new and specific question.