We study the instability of a ring Bose-Einstein condensate under a periodic modulation of interatomic interactions. The condensate exhibits temporal and spatial patterns induced by the parametric resonance, which can be characterized by Bogoliubov quasi-particle excitations in the Floquet basis. As the ring geometry significantly limits the number of excitable Bogoliubov modes, we are able to capture the non-linear dynamics of the condensate using a three-mode model. We further demonstrate the robustness of the temporal and spatial patterns against disorder, which are attributed to the mode-locking mechanism under the ring geometry. Our results can be observed in cold atomic systems and are also relevant to physical systems described by the non-linear Schrödinger's equation.

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We study a system of coupled degenerate cavities with a switchable beam rotator embedded in the optical path of the main cavity. By exploiting the phase shift of the beam rotator dependent on the orbital angular momentum of the optical modes, and modulating the phase imbalance in the auxiliary cavity, it is shown that the system dynamics is equivalent to that of a charged particle in a 1D lattice subject to both static and time-dependent electrical fields. We investigate interesting physics and phenomena such as Bloch oscillations that arise due to the simulated electrical fields, and discuss how they can be used for practical purposes such as storing optical signals in a quantum memory. We also present a powerful measurement scheme to detect the system dynamics that is non-intrusive and technically easy to perform.