Non-Markovian dynamics of a driven three-level giant atom coupled to a semi-infinite waveguide via complex couplings

The non-Markovian effects of open quantum systems subjected to external environments are deemed to be valuable resources in quantum optics and quantum information processing. In this work, we construct the model Hamiltonian of three-level giant atom with multiple coupling points in the semi-infinite photonic waveguide via complex coupling coefficients and investigate the non-Markovian dynamics of the three-level giant atom driven by a classical laser field. In the non-Markovian regime, the traveling time of a photon between different coupling points is sufficiently large compared to the inverse of the bare relaxation rate of the giant atom. We derive the analytical expressions for the probability amplitudes of the driven three-level giant atom by solving a set of delay differential equations and obtain two independent conditions of bound states' formation that can inhibit the dissipation. We find two different types of bound states (including the static bound states and the periodic equal-amplitude oscillating bound states) and discuss the physical origins of the bound states' formation. Moreover, we discuss the case of the driven three-level giant atom interacting with the infinite photonic waveguide, where there is only one purely imaginary solution (i.e., only one bound-state condition exists) for its complex frequency (coming from the absence of mirror at one end of the waveguide) compared to that of a driven three-level giant atom coupling with a semi-infinite photonic waveguide. With this, we also find two different types of bound states, including the static bound state and the periodic equal-amplitude oscillating bound states. We study the influences of the phase in the coupling strength on the bound states in semi-infinite and infinite photonic waveguides. The presented protocol might open a way to better understand exactly the non-Markovian dynamics of driven multilevel giant atoms interacting with semi-infinite (or infinite) photonic waveguides.