Phonon Echo from Multi-Level Systems and Many-Body Interactions in Low-Temperature Glasses

Abstract

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.

Figures (6)

• Figure 1: Schematic illustration of (a) the two-level system and (b) the multi-level system models. Blue dots in both panels mark the potential minima. In (a), the ground-state energy of the harmonic potential is $\hbar\omega/2$, on the order of the Debye energy. The parameters $\Delta$, $d$, and $V$ denote the asymmetry between wells, their spatial separation, and the barrier height, respectively.
• Figure 2: Numerical simulations of the phonon echo within the semi-classical framework. The Newtonian dynamics are integrated using the fourth-order Runge--Kutta algorithm. A total of $10^3$ randomly generated, non-interacting, and nonlinear oscillators are simulated with parameters: oscillator mass $m=1$, mean quadratic stiffness $\bar{k}_{2,s}=1$ with standard deviation ${\rm std}(k_{2,s})=1$, cubic stiffness $k_3\in[-1,1]$ uniformly distributed, and quartic stiffness $k_4=1$. The first pulse is applied at $\tau_1=0.5=0.08T$, where $T=2\pi(m/\bar{k}_{2,s})^{1/2}$ is the oscillator period, with duration $\Delta\tau_1=0.2=0.032T$ and amplitude $I_1/\Delta\tau_1=0.5$. The second pulse is applied at $\tau_2=100T$, with duration $\Delta\tau_2=0.2=0.032T$ and amplitude $I_2/\Delta\tau_2=1$. The pulse separation is $\tau=\tau_2-\tau_1=100T$, predicting an echo signal at $\tau_1+2\tau=200T$. The numerical time step is set to $dt=0.1$. (a) Summation of the displacement--time relation for a single set of $10^3$ oscillators. (b) Ensemble-averaged dynamics obtained from $N=100$ independent realizations of randomly generated oscillator groups. (c) Same simulation as in (a), but with the second external pulse applied at $\tau_2=300T$. (d) Same ensemble averaging as in (b), with the second external pulse applied at $\tau_2=300T$.
• Figure 3: Influence of damping on the semi-classical phonon echo. (a) Echo signal using the parameters from Fig. \ref{['fig2']} with pulse separation $\tau = 100T$ and damping coefficient $\eta=5\times 10^{-4}$. (b) Echo amplitude as a function of $\eta$ ($0 \leq \eta \leq 9\times 10^{-4}$). Dots and error bars denote the mean and standard deviation from $N=100$ samples. (c) Same as (a) but with $\tau = 300T$. (d) Same as (b) but with $\tau = 300T$.
• Figure 4: Numerical results of the phonon echo within the semi-classical framework, induced solely by many-body interactions. A total of $10^3$ randomly generated, interacting nonlinear oscillators are simulated, with ensemble averaging over $N=100$ samples. The oscillators are placed on a $10\times 10\times 10$ cubic lattice, each labeled by $(n_1, n_2, n_3)$ with $1\le n_1, n_2, n_3\le 10$. Parameters are as in Fig. \ref{['fig2']}, but with $k_3=k_4=0$ to ensure each individual oscillator is purely linear. The mutual interaction coefficients satisfy $V_{nn'}^{(ss')}=0$ for $n\neq 2$ or $n'\neq 2$. In (a) $V_{22}^{(ss')}=0$ (non-interacting case); in (b), (c), and (d) $V_{22}^{(ss')}=2\xi$, with $\xi$ uniformly distributed in $[0,1]$ over each interacting pair $s,s'$. Interactions are restricted to nearest neighbors: an oscillator at $(n_1,n_2,n_3)$ couples only to the six neighbors at $(n_1\pm1,n_2,n_3)$, $(n_1,n_2\pm1,n_3)$, and $(n_1,n_2,n_3\pm1)$. This simplification does not alter the qualitative echo behavior. The pulse separation $\tau$ is $100T$, $100T$, $200T$, and $300T$ for panels (a)-(d), respectively.
• Figure 5: Numerical simulation of the phonon echo from the multi-level-system model in the quantum regime. The Liouville-von Neumann dynamics are integrated using a fourth-order Runge-Kutta algorithm with $\hbar=1$, $k_{\mathrm{B}}=1$, and temperature $T=0.1$. A total of $10^3$ randomly generated, non-interacting quantum multi-level systems are placed on a $10\times10\times10$ lattice. Each system is described by a $5\times5$ Hamiltonian (a five-level system) with eigenvalues $\epsilon = 0,0.3,1.3,3,5.1$. The orientation-averaged strain-field strengths are $\epsilon_1=0.1$ for the first pulse and $\epsilon_2=0.2$ for the second pulse. The coupling stress tensor to the external strain field is $\hat{T}_{\mathrm{ext}}^{(s)}+\delta\hat{T}_{\mathrm{ext}}^{(s)}$, where the tensorial character has been averaged over orientations. The zeroth-order part $\hat{T}_{\mathrm{ext}}^{(s)}$ is a $5\times5$ Hermitian matrix with elements $\hat{T}_{\mathrm{ext},nn'}^{(s)} = 0.2\sqrt{{n}/{2}}\,\delta_{n',n+1}$ and $\hat{T}_{\mathrm{ext},nn'}^{(s)} = \hat{T}_{\mathrm{ext},n'n}^{(s)}$. The perturbation $\delta\hat{T}_{\mathrm{ext}}^{(s)}$ is a Hermitian random $5\times5$ matrix whose elements are uniformly distributed in $[-0.5,0.5]$. (a) Pulse separation $\tau=100T$; total stress-tensor expectation value summed over $10^3$ systems. (b) Ensemble average of the signal in (a) over $N=100$ independent realizations. (c) Same as (a) but with $\tau=300T$. (d) Same as (b) but with $\tau=300T$.