In this paper, we discuss the multi-photon systems by employing the method of adiabatic elimination following the procedure of Lugiato. We come to the conclusion that, in both the "good cavity case" and the "bad cavity case", the result we get from the discussion contains the result drawn by the adiabatic elimination following the procedure of Gordon. More over, it can compute directly the correlation of the variable that has been adiabatically eliminated. Particularly in the "bad cavity case", the steady state value of the field variable correlation obtained by the method of Lugiato is different from that obtained by the method of Gordon. The method of Lugiato is self-consistent so it is much better.In the appendix, the formulae of the atomic squeezing and the multi-photon spectrum of transmitted light are presented.
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