Nonexponential dephasing in a local random matrix model

Deviations from exponential decay dynamics have been proposed for a wide variety of systems in high-energy, atomic, and molecular physics. This work examines the quantum dynamics of a simple hierarchical local random matrix model. The hierarchical structure is imposed by two physically motivated constraints: an exponential size scaling of matrix elements, and a quantum number "triangle rule," which introduces correlations in the quantum-state space by mimicking the nodal structure of wave functions in a coordinate Hamiltonian. These correlations lead to a systematic slowing of dephasing dynamics compared to exponential decays. A generalized Lorentzian line shape is introduced as the Fourier transform of a polynomial survival amplitude to describe the average behavior of these decays. The model is brought into a representation that can be compared directly with the golden rule. In this representation, the deviations from exponentiality arise from energy-dependent correlations among the coupling matrix elements that persist even for large systems. Finally, the effects of relaxing the size scaling and "triangle rule" constraints are studied. Sparsity of the random matrix alone is not sufficient to produce slow asymptotic dynamics; both types of constraints are required.