Analysis and numerics for some photonic and quantum interacting systems
Registration Link: https://cityu.zoom.us/meeting/register/tJIpc--rpz4qHtG4E1Y9d17ogm_FvYJtuvTs
ABSTRACT
In this talk, we discuss some recent results for photonic and quantum interacting systems. We first consider the scattering resonances in the wave interaction with nanoparticles of high refractive indices and their applications in imaging and material science. We will establish the existence and asymptotic behaviors of dielectric resonances and show that near a resonant frequency, a singular particle exhibits resonant magnetic dipole characteristics. We then discuss the topological properties of scattering resonances for the nanophotonic crystals. In particular, we prove the genericity of the Dirac dispersion cone for honeycomb lattice and the existence of embedded eigenvalue with Fano-type asymmetric peaks in transmission spectra. Further, we focus on two prototypical quantum interacting systems that are closely related to quantum optimal transport: the ground-state energy problem and open quantum dynamics. We will present an SDP relaxation hierarchy with optimized clusters for the ground-state energy of the quantum many-body Hamiltonian that admits a quantum embedding interpretation. For open quantum dynamics, we introduce quantum Beckner's inequalities to quantify its long-time behavior with respect to convex entropies. We will discuss the properties of the Beckner constant in detail and provide a uniform lower bound in terms of the spectral gap. Moreover, we define a new class of quantum Wasserstein distances such that the quantum dynamic is a gradient flow and its convergence can be characterized by entropic Ricci curvature lower bound associated with the convex entropy. Some ongoing works and open questions will also be discussed.