Hilbert-Space Ergodicity in Driven Quantum Systems: Obstructions and Designs

Abstract

Despite its long history, a canonical formulation of quantum ergodicity that applies to general classes of quantum dynamics, including driven systems, has not been fully established. Here we introduce and study a notion of quantum ergodicity for closed systems with time-dependent Hamiltonians, defined as statistical randomness exhibited in their longtime dynamics. Concretely, we consider the temporal ensemble of quantum states (time-evolution operators) generated by the evolution, and investigate the conditions necessary for them to be statistically indistinguishable from uniformly random states (operators) in the Hilbert space (space of unitaries). We find that the number of driving frequencies underlying the Hamiltonian needs to be sufficiently large for this to occur. Conversely, we show that statistical pseudo-randomness -- indistinguishability up to some large but finite moment, can already be achieved by a quantum system driven with a single frequency, i.e., a Floquet system, as long as the driving period is sufficiently long. Our work relates the complexity of a time-dependent Hamiltonian and that of the resulting quantum dynamics, and offers a fresh perspective to the established topics of quantum ergodicity and chaos from the lens of quantum information.

24 pages, 5 figures. (As published)

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Cited by in corpus (8)

• Periodically and aperiodically Thue-Morse driven long-range systems: from dynamical localization to slow dynamics
• Non-Clifford Cost of Random Unitaries
• Critically Slow Hilbert-Space Ergodicity in Quantum Morphic Drives
• Late-time ensembles of quantum states in quantum chaotic systems
• On Computational Complexity of Unitary and State Design Properties
• Quantum birthmarks: Ergodicity breaking beyond scarring
• Equilibration of Non-interacting Photons and Quantum Signatures of Chaos
• Observation of hierarchy of Hilbert space ergodicities in the quantum dynamics of a single spin