Abstract: The properties of the macroscopic quantum superposition state prepared by modulating the opto-mechanical coupling in an atom-cavity ultrastrong coupled opto-mechanical system are studied, considering the effects of the initial state of atom-cavity and the atom-cavity coupling strength. Firstly, the evolution operator of the system is calculated by the method of Wei-Norm, and the analytic form of the evolution wave function of the whole system is given for any atom-cavity initial state. The results show that, assuming that the atom-cavity system is measured, the oscillator will be in a macroscopic quantum superposition state under certain conditions. Then the analytical expression of Wigner function of the macroscopic quantum superposition state of the mechanical oscillator is given, and the possible factors affecting the quantum properties of the macroscopic quantum state are calculated and analyzed. Finally, the influence of different initial states of atom cavity on the strength of macroscopic quantum coherence is discussed, and the initial state parameters with the strongest macroscopic quantum coherence are given. The influence of atom cavity coupling strength on the quantum coherence of macroscopic quantum superposition state is also discussed. It is found that the stronger the coupling strength is, the stronger the quantum coherence of macroscopic quantum superposition state will be.

Key words: quantum optics, ultra-strong coupled opto-mechanical system, cavity opto-mechanical system, preparation of macroscopic quantum superposition states, macroscopic quantum coherence