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The first two successful predictions for amorphous solid experiments bytunneling-two-level-system (TTLS) was phonon echo and saturation phenomena. Inthis paper by generalizing TTLS to infinite-level-system model with certainrandomness and nonlinearity, we prove that phonon echo is a stimulated emissionprocess. The classical infinite-level-system (taking $\hbar\to 0$ limit) cannotfind saturation effect, while quantum infinite-level-system can find it withcertain nonlinearity assumed. We also prove that without randomness ornonlinearity neither phonon echo nor saturation exists in arbitraryinfinite-level-system.

Random Nonlinear Infinite-Level-System Model for Amorphous Solid Phonon Echo and Saturation Phenomena