Cryogenic spectroscopy of ultra-low density colloidal lead chalcogenide quantum dots on chip-scale optical cavities towards single quantum dot near-infrared cavity QED

We present evidence of cavity quantum electrodynamics from a sparse density of strongly quantum-confined Pb-chalcogenide nanocrystals (between 1 and 10) approaching single-dot levels on moderately high-Q mesoscopic silicon optical cavities. Operating at important near-infrared (1500-nm) wavelengths, large enhancements are observed from devices and strong modifications of the QD emission are achieved. Saturation spectroscopy of coupled QDs is observed at 77K, highlighting the modified nanocrystal dynamics for quantum information processing.Comment: * new additional figures and text * 10 pages, 5 figure

Patching Neural Barrier Functions Using Hamilton-Jacobi Reachability

Learning-based control algorithms have led to major advances in robotics at the cost of decreased safety guarantees. Recently, neural networks have also been used to characterize safety through the use of barrier functions for complex nonlinear systems. Learned barrier functions approximately encode and enforce a desired safety constraint through a value function, but do not provide any formal guarantees. In this paper, we propose a local dynamic programming (DP) based approach to "patch" an almost-safe learned barrier at potentially unsafe points in the state space. This algorithm, HJ-Patch, obtains a novel barrier that provides formal safety guarantees, yet retains the global structure of the learned barrier. Our local DP based reachability algorithm, HJ-Patch, updates the barrier function "minimally" at points that both (a) neighbor the barrier safety boundary and (b) do not satisfy the safety condition. We view this as a key step to bridging the gap between learning-based barrier functions and Hamilton-Jacobi reachability analysis, providing a framework for further integration of these approaches. We demonstrate that for well-trained barriers we reduce the computational load by 2 orders of magnitude with respect to standard DP-based reachability, and demonstrate scalability to a 6-dimensional system, which is at the limit of standard DP-based reachability.Comment: 8 pages, submitted to IEEE Conference on Decision and Control (CDC), 202