High-order corrections on the laser cooling limit in the Lamb-Dicke regime

We investigate corrections on the cooling limit of high-order Lamb-Dicke (LD) parameters in the double electromagnetically induced transparency (EIT) cooling scheme. Via utilizing quantum interferences, the single-phonon heating mechanism vanishes and the system evolves to a double dark state, from which we will obtain the mechanical occupation on the single-phonon excitation state. In addition, the further correction induced by two-phonon heating transitions is included to achieve a more accurate cooling limit. There exist two pathways of two-phonon heating transitions: direct two-phonon excitation from the dark state and further excitation from the single-phonon excited state. By adding up these two parts of correction, the obtained analytical predictions show a well consistence with numerical results. Moreover, we find that the two pathways can destructively interfere with each other, leading to the elimination of two-phonon heating transitions and achieving a lower cooling limit.