Giant atoms, where the dipole approximation ceases to be valid, allow us to
observe unconventional quantum optical phenomena arising from interference and
time-delay effects. Most previous studies consider giant atoms coupling to
conventional materials with right-handed dispersion. In this study, we first
investigate the quantum dynamics of a giant atom interacting with left-handed
superlattice metamaterials. Different from those right-handed counterparts, the
left-handed superlattices exhibit an asymmetric band gap generated by anomalous
dispersive bands and Bragg scattering bands. First, by assuming that the giant
atom is in resonance with the continuous dispersive energy band, spontaneous
emission will undergo periodic enhancement or suppression due to the
interference effect. At the resonant position, there is a significant
discrepancy in the spontaneous decay rates between the upper and lower bands,
which arises from the differences in group velocity. Second, we explore the
non-Markovian dynamics of the giant atom by considering the frequency of the
emitter outside the energy band, where bound states will be induced by the
interference between two coupling points. By employing both analytical and
numerical methods, we demonstrate that the steady atomic population will be
periodically modulated, driven by variations in the size of the giant atom. The
presence of asymmetric band edges leads to diverse interference dynamics.
Finally, we consider the case of two identical emitters coupling to the
waveguide and find that the energy within the two emitters undergoes exchange
through the mechanism Rabi oscillations.Comment: 11 pages, 7 figure