Single-photon scattering in a waveguide coupled to a giant atom with loss or gain

This work investigates single-photon scattering in a one-dimensional coupled-resonator waveguide interacting with 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 the interference effects of a giant atom can decouple the waveguide from the atom under specific conditions, resulting in perfect transmission, distinguishing it from a small atom. Otherwise, a giant atom with loss absorbs the incident wave, whereas a giant atom with gain 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 a spectral-singularity state. Furthermore, we show that the system with gain supports at least one time-growing bound state, which dominates the long-time dynamics. Due to the presence of these generalized bound states, the conventional time-independent scattering theory proves inadequate for such a non-Hermitian system in the gain regime. We verify our time-dependent theoretical predictions via numerical simulations of Gaussian wave packet scattering.