  Generation and control of entanglement are fundamental tasks in quantum
information processing. In this paper, we propose a novel approach to generate
controllable frequency-entangled photons by using the concept of synthetic
frequency dimension in an optical system. Such a system consists of a ring
resonator made by a tailored third-order nonlinear media to induce
photon-photon interactions and a periodic modulator to manipulate coupling
between different frequency modes. We show this system provides a unique
platform for the exploration of distinct few- or many-body quantum phases
including chaos, localization, and integrability in a highly integrable
photonics platform. In particular, we develop the potential experimental method
to calculate the spectral form factor, which characterizes the degree of chaos
in the system and differentiates between these phases based on observable
measurements. Interestingly, the transition signatures of each phase can lead
to an efficient generation of frequency-entangled multi photons. This work is
the first to explore rich and controllable quantum phases beyond single
particle in a synthetic dimension.
