Phonon Echo from Multi-Level 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 multi-level 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 multi-level systems, leading to nonlinear eigen-energies that enhance the echo signal. Meanwhile, finite-temperature thermal fluctuations cause dephasing, resulting in a decay of echo amplitude over time. The analytical and numerical results are consistent across semi-classical and quantum regimes. Our work validates the multi-level-system model and underscores the role of many-body interactions in low-temperature glassy dynamics.