Retardation effect and dark state in a waveguide QED setup with rectangle cross section

In this paper, we investigate the dynamics of a two-atom system which couples to a quasi-one dimensional waveguide with rectangle cross section. The waveguide supports different TM and TE modes and the former ones play as environment by on-demand choosing the dipole moment of the atoms. Such environment induces the interaction and collective dissipation of the atoms. When both of the two atoms are located in the middle of the waveguide, we observe a retardation effect, which is broken by moving one of the atom to be off-centered. To preserve the complete dissipation of the system via dark state mechanism, we propose a scheme where the connection of the atoms are perpendicular to the axis of the waveguide. We hope our study will be useful in quantum information processing based on state-to-art waveguide structure.