Controlling atom-photon bound states in a coupled resonator array with a two-level quantum emitter

We consider a one-dimensional (1D) coupled-resonator array (CRA), where a two-level quantum emitter (2LE) is electric-dipole coupled to the modes of two adjacent resonators. We investigate the energy spectrum, the photon probability distribution of the bound states, and the emission process of the 2LE into the CRA vacuum. A quantum phase transition is found which is characterized by the change of the number of the out-of-band discrete levels. The condition for this change is also presented. The photon wave functions of bound states are found to be asymmetry around the position of the 2LE when the coupling strengths between the 2LE and the resonator are not equal, and they have the same preferred directions which are primarily determined by the larger one among the coupling strengths. The presence of the atom-photon bound states is manifested in the form of a stationary oscillation or a non-vanishing constant in the long enough time.