# Parametric Resonances and Resonant Delocalization in Quasi-Phase Matched Photon-pair Generation and Quantum Frequency Conversion

## TL;DR

This paper reveals the role of parametric resonances in photon-pair generation and quantum frequency conversion, linking classical gain phenomena to quantum delocalization, and introduces a new measure to identify these regimes.

## Contribution

It introduces a quantitative measure of Floquet eigenmode localization to identify resonant delocalization regimes in quantum SPDC and frequency conversion, extending classical instability maps.

## Key findings

- Exponential gain corresponds to resonant delocalization in the Glauber-Fock model.
- Reconstruction of classical Arnold tongues for SPDC.
- Prediction of novel resonant delocalization regimes in quantum frequency conversion.

## Abstract

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 delocalization 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 delocalization. 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 delocalization in the two-level model describing quantum frequency conversion processes.