Cavity Modified Oscillating Bound States with a $\Lambda$-type giant emitter in a linear waveguide

We study a system composed by a three-level giant atom (3GA), a waveguide initially in the vacuum state, and a single-mode cavity. The 3GA-cavity system is in a strong-coupling regime, and the distance between the coupling points is comparable to the coherent length of a spontaneously emitted photon. The dynamics of the 3GA and its radiative field in the waveguide for long time are analyzed. Besides the steady value, we also found an oscillatory character of the excited state population, a signature of oscillating bound states which is generated by the superposition of two bound states in the continuum. The radiative field propagates in the cavity-like geometry formed by the coupling points. When one bound state is emergent, a sine-like interference pattern is visible for the emitted field intensity in spacetime. An oscillatory character in time and a beat in space for the emitted field intensity are observed when two bound states are emergent in a subspace. The wavelengths and the periods are controlled by the number of the photons in the cavity.