Randomization Timescales under Quantum Chaotic Hamiltonian Evolution

Initially unentangled states evolving under quantum-chaotic Hamiltonian dynamics are typically expected to become featureless at late times, specifically with respect to the statistical properties of Haar random states. While extensive numerical studies over the past two decades have confirmed this expectation at the level of averaged, local observables, in a recent work [1] we showed that this expectation also holds at the level of higher statistical moments and non-local observables. This results holds as long as the (midspectrum) initial state spreads uniformly across the eigenbasis of the symmetry operator. In this talk, I will discuss the timescales over which randomization occurs, showing that the randomization of quantum information—at the level of finite statistical moments of states sampled from evolution at finite times—takes place on polynomial (in system size N) timescales, much shorter than the exponentially long times required for states to ergodically explore the full accessible Hilbert space. Interestingly, for special classes of initial unentangled states, we find that randomization occurs on O(N) timescales. We also find that the prefactor of this timescale is sensitive to microscopic details and attains its minimum in models that were previously identified as “maximally chaotic”. Our new results, together with those of [1], suggest that local Hamiltonian systems described by sparse, non-random matrices can still generate a large amount of randomness within polynomial times.

[1] Ghosh, Langlett, Hunter-Jones, Rodriguez-Nieva PRB 112, 094302 (2025).