Bound states in the continuum in a double whispering-gallery resonator

In this work we investigate the single-photon transport through two whispering-gallery resonators (WGRs) coupled to a one-dimensional waveguide. Using the Green's-function formalism, we compute the transmission spectra and the photonic density of states for the stationary states. We also obtain the formation of two types of bound states in the continuum (BICs). The first kind is localized in the WGR and is symmetry-protected BICs. In contrast, the second depends on the distance between resonators through the waveguide and is of the Fabry-P\'erot kind. These BICs are represented as Dirac $\ensuremath{\delta}$ functions in the local density of states. Additionally, we show that quasi-BICs manifest as sharp resonances in photonic transmission due to small symmetry-breaking perturbations. Furthermore, we investigate the dynamics of a single-photon wave packet interacting with the WGRs and analyze the mechanism for storing the wave packet in the structure formed by the WGRs and the finite waveguide between them.