Smale flows on S2×S1\mathbb{S}^2\times\mathbb{S}^1

In this paper, we use abstract Lyapunov graphs as a combinatorial tool to obtain a complete classification of Smale flows on $\mathbb{S}^2\times\mathbb{S}^1$. This classification gives necessary and sufficient conditions that must be satisfied by an abstract Lyapunov graph in order for it to be associated to a Smale flow on $\mathbb{S}^2\times\mathbb{S}^1$

Modeling quasi-dark states with Temporal Coupled-Mode Theory

Coupled resonators are commonly used to achieve tailored spectral responses and allow novel functionalities in a broad range of applications, from optical modulation and filtering in integrated photonic circuits to the study of nonlinear dynamics in arrays of resonators. The Temporal Coupled-Mode Theory (TCMT) provides a simple and general tool that is widely used to model these devices and has proved to yield very good results in many different systems of low-loss, weakly coupled resonators. Relying on TCMT to model coupled resonators might however be misleading in some circumstances due to the lumped-element nature of the model. In this article, we report an important limitation of TCMT related to the prediction of dark states. Studying a coupled system composed of three microring resonators, we demonstrate that TCMT predicts the existence of a dark state that is in disagreement with experimental observations and with the more general results obtained with the Transfer Matrix Method (TMM) and the Finite-Difference Time-Domain (FDTD) simulations. We identify the limitation in the TCMT model to be related to the mechanism of excitation/decay of the supermodes and we propose a correction that effectively reconciles the model with expected results. A comparison with TMM and FDTD allows to verify both steady-state and transient solutions of the modified-TCMT model. The proposed correction is derived from general considerations, energy conservation and the non-resonant power circulating in the system, therefore it provides good insight on how the TCMT model should be modified to eventually account for the same limitation in a different coupled-resonator design. Moreover, our discussion based on coupled microring resonators can be useful for other electromagnetic resonant systems due to the generality and far-reach of the TCMT formalism.Comment: 7 pages, 4 figure

Generative Temporal Models with Spatial Memory for Partially Observed Environments

In model-based reinforcement learning, generative and temporal models of environments can be leveraged to boost agent performance, either by tuning the agent's representations during training or via use as part of an explicit planning mechanism. However, their application in practice has been limited to simplistic environments, due to the difficulty of training such models in larger, potentially partially-observed and 3D environments. In this work we introduce a novel action-conditioned generative model of such challenging environments. The model features a non-parametric spatial memory system in which we store learned, disentangled representations of the environment. Low-dimensional spatial updates are computed using a state-space model that makes use of knowledge on the prior dynamics of the moving agent, and high-dimensional visual observations are modelled with a Variational Auto-Encoder. The result is a scalable architecture capable of performing coherent predictions over hundreds of time steps across a range of partially observed 2D and 3D environments.Comment: ICML 201

Spectral Engineering with Coupled Microcavities: Active Control of Resonant Mode-Splitting

Optical mode-splitting is an efficient tool to shape and fine-tune the spectral response of resonant nanophotonic devices. The active control of mode-splitting, however, is either small or accompanied by undesired resonance shifts, often much larger than the resonance-splitting. We report a control mechanism that enables reconfigurable and widely tunable mode-splitting while efficiently mitigating undesired resonance shifts. This is achieved by actively controlling the excitation of counter-traveling modes in coupled resonators. The transition from a large splitting (80 GHz) to a single-notch resonance is demonstrated using low power microheaters (35 mW). We show that the spurious resonance-shift in our device is only limited by thermal crosstalk and resonance-shift-free splitting control may be achieved.Comment: 4 pages, 3 figure