On Quantum Rainbows: Density Operator in the Frequency-Bin Representation for Entangled Twin-Photons Generated With Sub-Threshold Microcombs

Kerr optical frequency combs are generated by pumping a high-
$Q$
integrated microresonator with a resonant laser. Below threshold, the pump laser field mediates the phenomenon of spontaneous four-wave mixing, where two pump photons are symmetrically up- and down-converted as twin photons that can be entangled across up to tens of eigenmodes in the spectral domain. While these room-temperature integrated photonic circuits are expected to play a central role in quantum technology, their high dimensionality and dissipative nature are a challenge for their theoretical description, therefore hindering the understanding of their properties and potential of performance. In this article, we develop a framework that permits to obtain an explicit solution for the density operator of quantum microcombs below threshold. This self-consistent theoretical description allows for their complete characterization, as well as for the analytical determination of various performance metrics such as fidelity, purity, and entropy.