Parametric resonances and resonant delocalisation in quasi-phase matched photon-pair generation and quantum frequency conversion

The existing widely-accepted theory of photon-pair generation via spontaneous down-conversion (SPDC) in nonlinear optical crystals and waveguides is incomplete, as it fails to account for the important physical phenomenon of parametric resonances. We demonstrate that exponential gain of classical fields in the regime of parametric resonance corresponds to resonant delocalisation in the Glauber-Fock model of quantum SPDC. We propose a quantitative measure of localisation of Floquet eigen-modes as an analogue of classical gain to identify regimes of resonant delocalisation. Using this method, we are able to reconstruct the classical 'Arnold tongues' map of domains of instabilities for SPDC. We also predict novel regimes of resonant delocalisation in the two-level model describing quantum frequency conversion processes.