  In this article, we use quantum Langevin equations to provide a theoretical
understanding of the non-classical behavior of Kerr optical frequency combs
when pumped below and above threshold. In the configuration where the system is
under threshold, the pump field is the unique oscillating mode inside the
resonator, and triggers the phenomenon of spontaneous four-wave mixing, where
two photons from the pump are symmetrically up- and down-converted in the
Fourier domain. This phenomenon can only be understood and analyzed from a
fully quantum perspective as a consequence of the coupling between the field of
the central (pumped) mode and the vacuum fluctuations of the various sidemodes.
We analytically calculate the power spectra of the spontaneous emission noise,
and we show that these spectra can be either single- or double peaked depending
on the parameters of the system. We also calculate as well the overall
spontaneous noise power per sidemode, and propose simplified analytical
expressions for some particular cases. In the configuration where the system is
pumped above threshold, we investigate the phenomena of quantum correlations
and multimode squeezed states of light that can occur in the Kerr frequency
combs originating from stimulated four-wave mixing. We show that for all
stationary spatio-temporal patterns, the side-modes that are symmetrical
relatively to the pumped mode in the frequency domain display quantum
correlations that can lead to squeezed states of light. We also explicitly
determine the phase quadratures leading to photon entanglement, and
analytically calculate their quantum noise spectra. We finally discuss the
relevance of Kerr combs for quantum information systems at optical
telecommunication wavelengths, below and above threshold.
