Single-Photon Scattering in a Waveguide Coupled to a Lossy or Gain Giant Atom

This work investigates single-photon scattering in a one-dimensional coupled-resonator waveguide coupled to a giant atom with a complex on-site energy. Within the generalized projection operator formalism, we derive analytical expressions for the scattering coefficients. We find that a lossy giant atom absorbs the incident wave, whereas a gain giant atom not only amplifies the incident wave but also leads to scattering divergence at certain energies, corresponding to spectral singularities. We explore the critical scattering dynamics associated with these singularities, and attribute the persistent wave emission to the existence of a stationary bound state in the continuum. Due to the presence of this bound state, the conventional time-independent scattering theory proves inadequate for such a non-Hermitian system. Furthermore, we show that the system with gain always features at least one time-growing bound state, which dominates the long-time dynamics, and we verify our time-dependent theoretical predictions via numerical simulations of Gaussian wave packet scattering.