Phonon echo from multilevel systems and many-body interactions in low-temperature glasses.

At low temperatures, glasses exhibit distinctive properties compared to crystalline solids. A notable example is the phonon echo, a phenomenon that motivated the two-level-system (TLS) model. This model has successfully explained many universal anomalies in glasses. Here, we extend the TLS framework to a multilevel system and show that phonon echoes persist when nonlinear energy structures and disorder are included. By incorporating virtual phonon exchange, we introduce many-body interactions between these multilevel systems, leading to nonequidistant energy spacings that enhance the echo signal. Interestingly, we find that phonon echo coherence within the TLS model is suppressed under many-body interactions, originating from the fixed natural frequency inherent to each TLS unit. In stark contrast, the multilevel-system model exhibits remarkable robustness against such interactions, arising from anharmonicity-induced frequency adaptability and multiple parallel transition channels. Even if one transition falls out of resonance, the remaining channels can still preserve quantum coherence collectively. Meanwhile, finite-temperature thermal fluctuations cause dephasing, resulting in a decay of echo amplitude over time. Analytical and numerical results, examined perturbatively in both semiclassical and quantum regimes, show consistent agreement. Our work validates this multilevel extension of the TLS framework and underscores the role of many-body interactions in low-temperature glassy dynamics.