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Optomechanically induced transparency in multi-cavity optomechanical system with and without one two-level atom

We analytically study the optomechanically induced transparency (OMIT) in the $N$-cavity system with the \textit{N}th cavity driven by pump, probing laser fields and the \textit{1}st cavity coupled to mechanical oscillator. We also consider that one atom could be trapped in the \textit{i}th cavity. Instead of only illustrating the OMIT in such a system, we are interested in how the number of OMIT windows is influenced by the cavities and the atom and what roles the atom could play in different cavities. In the resolved sideband regime, we find that, the number of cavities precisely determines the maximal number of OMIT windows. It is interesting that, when the two-level atom is trapped in the even-labeled cavity, the central absorptive peak (odd $N$) or dip (even $N$) is split and forms an extra OMIT window, but if the atom is trapped in the odd-labeled cavity, the central absorptive peak (odd $N$) or dip (even $N$) is only broadened and thus changes the width of the OMIT windows rather than induces an extra window.